{"id":248,"date":"2021-06-02T01:24:26","date_gmt":"2021-06-02T01:24:26","guid":{"rendered":"https:\/\/opentextbc.ca\/businesstechnicalmath\/chapter\/introduction-to-real-numbers\/"},"modified":"2021-08-31T21:17:25","modified_gmt":"2021-08-31T21:17:25","slug":"introduction-to-real-numbers","status":"publish","type":"chapter","link":"https:\/\/opentextbc.ca\/businesstechnicalmath\/chapter\/introduction-to-real-numbers\/","title":{"raw":"1.6 Roots and Radicals","rendered":"1.6 Roots and Radicals"},"content":{"raw":"[latexpage]\n<div class=\"textbox textbox--learning-objectives\"><header class=\"textbox__header\">\n<p class=\"textbox__title\">Learning Objectives<\/p>\n\n<\/header>\n<div class=\"textbox__content\">\n\nBy the end of this section it is expected that you will be able to:\n<ul>\n \t<li>Simplify expressions with roots<\/li>\n \t<li>Estimate and approximate roots<\/li>\n \t<li>Use radicals in applications<\/li>\n<\/ul>\n<\/div>\n<\/div>\n<h1 data-type=\"title\">Simplify Expressions with Square Roots<\/h1>\n<p id=\"fs-id1170654963188\">Remember that when a number <em data-effect=\"italics\">n<\/em> is multiplied by itself, we write \\({n}^{2}\\) and read it \u201cn squared.\u201d The result is called the <span data-type=\"term\">square<\/span> of <em data-effect=\"italics\">n<\/em>. For example,<\/p>\n\n<div id=\"fs-id1170654905253\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\">\\(\\begin{array}{cccc}{8}^{2}\\hfill &amp; &amp; &amp; \\text{read}\\phantom{\\rule{0.2em}{0ex}}\\text{\u2018}8\\phantom{\\rule{0.2em}{0ex}}\\text{squared\u2019}\\hfill \\\\ 64\\hfill &amp; &amp; &amp; 64\\phantom{\\rule{0.2em}{0ex}}\\text{is called the}\\phantom{\\rule{0.2em}{0ex}}\\mathit{\\text{square}}\\phantom{\\rule{0.2em}{0ex}}\\text{of}\\phantom{\\rule{0.2em}{0ex}}8.\\hfill \\end{array}\\)<\/div>\n<p id=\"fs-id1170655353155\">Similarly, 121 is the square of 11, because \\({11}^{2}\\) is 121.<\/p>\n\n<div id=\"fs-id1170655158009\" data-type=\"note\">\n<div data-type=\"title\">\n<div class=\"textbox textbox--key-takeaways\"><header class=\"textbox__header\">\n<p class=\"textbox__title\">Square of a Number<\/p>\n\n<\/header>\n<div class=\"textbox__content\">\n\nIf \\({n}^{2}=m,\\) then <em data-effect=\"italics\">m<\/em> is the <strong data-effect=\"bold\">square<\/strong> of <em data-effect=\"italics\">n<\/em>.\n\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1170654957861\">Complete the following table to show the squares of the counting numbers 1 through 15.<\/p>\n<span id=\"fs-id1170655354103\" data-type=\"media\" data-alt=\"There is a table with two rows and 17 columns. The first row reads from left to right Number, n, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, and 15. The second row reads from left to right Square, n squared, blank, blank, blank, blank, blank, blank, blank, 64, blank, blank, 121, blank, blank, blank, and blank.\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/06\/CNX_ElemAlg_Figure_01_08_016_img_new.jpg\" alt=\"There is a table with two rows and 17 columns. The first row reads from left to right Number, n, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, and 15. The second row reads from left to right Square, n squared, blank, blank, blank, blank, blank, blank, blank, 64, blank, blank, 121, blank, blank, blank, and blank.\" data-media-type=\"image\/jpeg\"><\/span>\n<p id=\"fs-id1170655105477\">The numbers in the second row are called perfect square numbers. It will be helpful to learn to recognize the perfect square numbers.<\/p>\n<p id=\"fs-id1170655202790\">The squares of the counting numbers are positive numbers. What about the squares of negative numbers? We know that when the signs of two numbers are the same, their product is positive. So the square of any negative number is also positive.<\/p>\n\n<div id=\"fs-id1170655229994\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\">\\(\\begin{array}{cccccccccc}{\\left(-3\\right)}^{2}=9\\hfill &amp; &amp; &amp; {\\left(-8\\right)}^{2}=64\\hfill &amp; &amp; &amp; {\\left(-11\\right)}^{2}=121\\hfill &amp; &amp; &amp; {\\left(-15\\right)}^{2}=225\\hfill \\end{array}\\)<\/div>\n<p id=\"fs-id1170654940925\">Did you notice that these squares are the same as the squares of the positive numbers?<\/p>\n<p id=\"fs-id1170655007029\">Sometimes we will need to look at the relationship between numbers and their squares in reverse. Because \\({10}^{2}=100,\\) we say 100 is the square of 10. We also say that 10 is a <em data-effect=\"italics\">square root<\/em> of 100. A number whose square is \\(m\\) is called a <span data-type=\"term\">square root<\/span> of <em data-effect=\"italics\">m<\/em>.<\/p>\n\n<div id=\"fs-id1170654966160\" data-type=\"note\">\n<div data-type=\"title\">\n<div class=\"textbox textbox--key-takeaways\"><header class=\"textbox__header\">\n<p class=\"textbox__title\">Square Root of a Number<\/p>\n\n<\/header>\n<div class=\"textbox__content\">\n\nIf \\({n}^{2}=m,\\) then <em data-effect=\"italics\">n<\/em> is a <strong data-effect=\"bold\">square root<\/strong> of <em data-effect=\"italics\">m<\/em>.\n\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1170655151795\">Notice \\({\\left(-10\\right)}^{2}=100\\) also, so \\(-10\\) is also a square root of 100. Therefore, both 10 and \\(-10\\) are square roots of 100.<\/p>\n<p id=\"fs-id1170655227720\">So, every positive number has two square roots\u2014one positive and one negative. What if we only wanted the positive square root of a positive number? The <span class=\"no-emphasis\" data-type=\"term\">radical sign<\/span>, \\(\\sqrt{m},\\) denotes the positive square root. The positive square root is called the <span class=\"no-emphasis\" data-type=\"term\">principal square root<\/span>. When we use the radical sign that always means we want the principal square root.<\/p>\n<p id=\"fs-id1170655228645\">We also use the radical sign for the square root of zero. Because \\({0}^{2}=0,\\) \\(\\sqrt{0}=0.\\) Notice that zero has only one square root.<\/p>\n\n<div id=\"fs-id1170654985840\" data-type=\"note\">\n<div data-type=\"title\">\n<div class=\"textbox textbox--key-takeaways\"><header class=\"textbox__header\">\n<p class=\"textbox__title\">Square Root Notation<\/p>\n\n<\/header>\n<div class=\"textbox__content\">\n<p id=\"fs-id1170655111452\">\\(\\sqrt{m}\\) is read \u201cthe square root of <em data-effect=\"italics\">m<\/em>\u201d<\/p>\n<span data-type=\"media\" data-alt=\"A square root is given, with an arrow to the radical sign (it looks like a checkmark with a horizontal line extending from its long end) denoted radical sign and an arrow to the number under the radical sign, which is marked radicand.\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/06\/CNX_ElemAlg_Figure_01_08_001_img_new.jpg\" alt=\"A square root is given, with an arrow to the radical sign (it looks like a checkmark with a horizontal line extending from its long end) denoted radical sign and an arrow to the number under the radical sign, which is marked radicand.\" data-media-type=\"image\/jpeg\"><\/span>\n<p id=\"fs-id1170655207964\">If \\(m={n}^{2},\\) then \\(\\sqrt{m}=n,\\) for \\(n\\ge 0.\\)<\/p>\n<p id=\"fs-id1170654872607\">The square root of <em data-effect=\"italics\">m<\/em>, \\(\\sqrt{m},\\) is the positive number whose square is <em data-effect=\"italics\">m<\/em>.<\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1170655230646\">Since 10 is the principal square root of 100, we write \\(\\sqrt{100}=10.\\) You may want to complete the following table to help you recognize square roots.<\/p>\n<span id=\"fs-id1170655162081\" data-type=\"media\" data-alt=\"There is a table with two rows and 15 columns. The first row reads from left to right square root of 1, square root of 4, square root of 9, square root of 16, square root of 25, square root of 36, square root of 49, square root of 64, square root of 81, square root of 100, square root of 121, square root of 144, square root of 169, square root of 196, and square root of 225. The second row consists of all blanks except for the tenth cell under the square root of 100, which reads 10.\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/06\/CNX_ElemAlg_Figure_01_08_017_img_new.jpg\" alt=\"There is a table with two rows and 15 columns. The first row reads from left to right square root of 1, square root of 4, square root of 9, square root of 16, square root of 25, square root of 36, square root of 49, square root of 64, square root of 81, square root of 100, square root of 121, square root of 144, square root of 169, square root of 196, and square root of 225. The second row consists of all blanks except for the tenth cell under the square root of 100, which reads 10.\" data-media-type=\"image\/jpeg\"><\/span>\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\n<p class=\"textbox__title\">EXAMPLE 1<\/p>\n\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1170655105180\" data-type=\"problem\">\n\nSimplify: a) \\(\\sqrt{25}\\) b) \\(\\sqrt{121}.\\)\n\n<\/div>\n<div id=\"fs-id1170655192917\" data-type=\"solution\">\n<div data-type=\"title\"><\/div>\n<div data-type=\"title\"><strong>Solution<\/strong><\/div>\n<table id=\"eip-105\" style=\"height: 64px; width: 100%;\" summary=\".\">\n<tbody>\n<tr style=\"height: 32px;\">\n<td style=\"height: 32px; width: 145.203px;\"><span class=\"token\">a)<\/span><span data-type=\"newline\">\n<\/span>Since \\({5}^{2}=25\\)<\/td>\n<td style=\"height: 32px; width: 285.297px;\">\\(\\begin{array}{c}\\sqrt{25}\\\\ 5\\end{array}\\)<\/td>\n<\/tr>\n<tr style=\"height: 32px;\">\n<td style=\"height: 32px; width: 145.203px;\"><span class=\"token\">b)<\/span><span data-type=\"newline\">\n<\/span>Since \\({11}^{2}=121\\)<\/td>\n<td style=\"height: 32px; width: 285.297px;\">\\(\\begin{array}{c}\\sqrt{121}\\\\ 11\\end{array}\\)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 1<\/p>\n\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1170655222336\" data-type=\"problem\">\n<p id=\"fs-id1170655205656\">Simplify: a) \\(\\sqrt{36}\\) b) \\(\\sqrt{169}.\\)<\/p>\n\n<\/div>\n<div id=\"fs-id1170655113123\" data-type=\"solution\"><details><summary>Show answer<\/summary>\n<p id=\"fs-id1170654981435\"><span class=\"token\">a)<\/span> 6 b) 13<\/p>\n\n<\/details><\/div>\n<\/div>\n<\/div>\nWe know that every positive number has two square roots and the radical sign indicates the positive one. We write \\(\\sqrt{100}=10.\\) If we want to find the negative square root of a number, we place a negative in front of the radical sign. For example, \\(\\text{-}\\sqrt{100}=-10.\\) We read \\(\\text{-}\\sqrt{100}\\) as \u201cthe opposite of the square root of 100.\u201d\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\n<p class=\"textbox__title\">EXAMPLE 2<\/p>\n\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1170654988401\" data-type=\"problem\">\n<p id=\"fs-id1170655112232\">Simplify: a) \\(-\\sqrt{49}\\) b) \\(-\\sqrt{144}.\\)<\/p>\n\n<\/div>\n<div id=\"fs-id1170655155062\" data-type=\"solution\">\n<div data-type=\"title\"><\/div>\n<div data-type=\"title\"><strong>Solution<\/strong><\/div>\n<table id=\"eip-995\" style=\"width: 100%;\" summary=\".\">\n<tbody>\n<tr>\n<td><span class=\"token\">a)<\/span><span data-type=\"newline\">\n<\/span>The negative is in front of the radical sign.<\/td>\n<td>\\(\\begin{array}{c}-\\sqrt{49}\\\\ -7\\end{array}\\)<\/td>\n<\/tr>\n<tr>\n<td><span class=\"token\">b)<\/span><span data-type=\"newline\">\n<\/span>The negative is in front of the radical sign.<\/td>\n<td>\\(\\begin{array}{c}-\\sqrt{144}\\\\ -12\\end{array}\\)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 2<\/p>\n\n<\/header>\n<div class=\"textbox__content\">\n<div data-type=\"problem\">\n<p id=\"fs-id1170654901222\">Simplify: a) \\(-\\sqrt{16}\\) b) \\(-\\sqrt{225}.\\)<\/p>\n\n<\/div>\n<div id=\"fs-id1170655125652\" data-type=\"solution\"><details><summary>Show answer<\/summary><span class=\"token\">a) <\/span>\\(-4\\) b) \\(-15\\)\n\n<\/details><\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1169148889441\" class=\"bc-section section\" data-depth=\"1\">\n<p id=\"fs-id1169149123590\">Can we simplify \\(\\sqrt{-49}?\\) Is there a number whose square is \\(-49?\\)<\/p>\n\n<div id=\"fs-id1169149004881\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\">\\({\\left(\\phantom{\\rule{0.2em}{0ex}}\\right)}^{2}=-49\\)<\/div>\n<p id=\"fs-id1169148861223\">Any positive number squared is positive. Any negative number squared is positive. There is no real number equal to \\(\\sqrt{-49}.\\) The square root of a negative number is not a real number.<\/p>\n\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\n<p class=\"textbox__title\">EXAMPLE 3<\/p>\n\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1169148820580\" data-type=\"problem\">\n<p id=\"fs-id1169148859897\">Simplify: a) \\(\\sqrt{-196}\\) b) \\(-\\sqrt{64}.\\)<\/p>\n\n<\/div>\n<div id=\"fs-id1169144560004\" data-type=\"solution\">\n<p id=\"fs-id1169148836684\"><strong><span class=\"token\">Solution<\/span><\/strong><\/p>\n<span class=\"token\">a)<\/span>\n<table style=\"border-collapse: collapse; width: 100%; height: 48px;\" border=\"0\">\n<tbody>\n<tr style=\"height: 16px;\">\n<td style=\"width: 50%; height: 16px;\"><\/td>\n<td style=\"width: 50%; height: 16px;\">\\(\\sqrt{-196}\\)<\/td>\n<\/tr>\n<tr style=\"height: 32px;\">\n<td style=\"width: 50%; height: 32px;\">There is no real number whose square is \\(-196\\).<\/td>\n<td style=\"width: 50%; height: 32px;\">\\(\\sqrt{-196}\\).<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<span class=\"token\">b)<\/span>\n<div data-type=\"newline\">\n<table style=\"border-collapse: collapse; width: 100%;\" border=\"0\">\n<tbody>\n<tr>\n<td style=\"width: 50%;\"><\/td>\n<td style=\"width: 50%;\">\\(-\\sqrt{64}\\)<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 50%;\">The negative is in front of the radical.<\/td>\n<td style=\"width: 50%;\">\\(-8\\)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 3<\/p>\n\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1169146638717\" data-type=\"problem\">\n<p id=\"fs-id1169148929622\">Simplify: a) \\(\\sqrt{-169}\\) b) \\(-\\sqrt{81}.\\)<\/p>\n\n<\/div>\n<div id=\"fs-id1169144491036\" data-type=\"solution\"><details open=\"open\"><summary>Show answer<\/summary>\n<p id=\"fs-id1169146823304\"><span class=\"token\">a)<\/span> not a real number b) \\(-9\\)<\/p>\n\n<\/details><\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1169148859228\">So far we have only talked about squares and square roots. Let\u2019s now extend our work to include higher powers and higher roots.<\/p>\n<p id=\"fs-id1169149014185\">Let\u2019s review some vocabulary first.<\/p>\n<p id=\"fs-id1169148861279\">\\(\\begin{array}{cccccc}\\text{We write:}\\hfill &amp; &amp; &amp; &amp; &amp; \\text{We say:}\\hfill \\\\ \\hfill {n}^{2}\\hfill &amp; &amp; &amp; &amp; &amp; n\\phantom{\\rule{0.2em}{0ex}}\\text{squared}\\hfill \\\\ \\hfill {n}^{3}\\hfill &amp; &amp; &amp; &amp; &amp; n\\phantom{\\rule{0.2em}{0ex}}\\text{cubed}\\hfill \\\\ \\hfill {n}^{4}\\hfill &amp; &amp; &amp; &amp; &amp; n\\phantom{\\rule{0.2em}{0ex}}\\text{to the fourth power}\\hfill \\\\ \\hfill {n}^{5}\\hfill &amp; &amp; &amp; &amp; &amp; n\\phantom{\\rule{0.2em}{0ex}}\\text{to the fifth power}\\hfill \\end{array}\\)<\/p>\n<p id=\"fs-id1169148823401\">The terms \u2018squared\u2019 and \u2018cubed\u2019 come from the formulas for area of a square and volume of a cube.<\/p>\n<p id=\"fs-id1169149038030\">It will be helpful to have a table of the powers of the integers from \u22125 to 5. See <a class=\"autogenerated-content\" href=\"#CNX_IntAlg_Figure_08_01_002\">(Table 1)<\/a>.<\/p>\n\n<div id=\"CNX_IntAlg_Figure_08_01_002\" class=\"bc-figure figure\">\n\n[caption id=\"\" align=\"aligncenter\" width=\"806\"]<img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/06\/CNX_IntAlg_Figure_08_01_002_img_new.jpg\" alt=\"The figure contains two tables. The first table has 9 rows and 5 columns. The first row is a header row with the headers \u201cNumber\u201d, \u201cSquare\u201d, \u201cCube\u201d, \u201cFourth power\u201d, and \u201cFifth power\u201d. The second row contains the expressions n, n squared, n cubed, n to the fourth power, and n to the fifth power. The third row contains the number 1 in each column. The fourth row contains the numbers 2, 4, 8, 16, 32. The fifth row contains the numbers 3, 9, 27, 81, 243. The sixth row contains the numbers 4, 16, 64, 256, 1024. The seventh row contains the numbers 5, 25, 125 625, 3125. The eighth row contains the expressions x, x squared, x cubed, x to the fourth power, and x to the fifth power. The last row contains the expressions x squared, x to the fourth power, x to the sixth power, x to the eighth power, and x to the tenth power. The second table has 7 rows and 5 columns. The first row is a header row with the headers \u201cNumber\u201d, \u201cSquare\u201d, \u201cCube\u201d, \u201cFourth power\u201d, and \u201cFifth power\u201d. The second row contains the expressions n, n squared, n cubed, n to the fourth power, and n to the fifth power. The third row contains the numbers negative 1, 1 negative 1, 1, negative 1. The fourth row contains the numbers negative 2, 4, negative 8, 16, negative 32. The fifth row contains the numbers negative 3, 9, negative 27, 81, negative 243. The sixth row contains the numbers negative 4, 16, negative 64, 256, negative 1024. The last row contains the numbers negative 5, 25, negative 125, 625, negative 3125.\" width=\"806\" height=\"189\" data-media-type=\"image\/jpeg\"> Table 1[\/caption]\n\n<\/div>\n<p id=\"fs-id1169148994337\">Notice the signs in the table. All powers of positive numbers are positive, of course. But when we have a negative number, the <em data-effect=\"italics\">even<\/em> powers are positive and the <em data-effect=\"italics\">odd<\/em> powers are negative. We\u2019ll copy the row with the powers of \u22122 to help you see this.<\/p>\n<img class=\"aligncenter\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_IntAlg_Figure_08_01_003_img_new.jpg\" alt=\"The image contains a table with 2 rows and 5 columns. The first row contains the expressions n, n squared, n cubed, n to the fourth power, and n to the fifth power. The second row contains the numbers negative 2, 4, negative 8, 16, negative 32. Arrows point to the second and fourth columns with the label \u201cEven power Positive result\u201d. Arrows point to the first third and fifth columns with the label \u201cOdd power Negative result\u201d.\" width=\"375\" height=\"103\" data-media-type=\"image\/jpeg\">\n<p id=\"fs-id1169149155980\">We will now extend the square root definition to higher roots.<\/p>\n\n<div id=\"fs-id1169149108455\" data-type=\"note\">\n<div data-type=\"title\"><\/div>\n<div data-type=\"title\">\n<div class=\"textbox textbox--key-takeaways\"><header class=\"textbox__header\">\n<p class=\"textbox__title\"><em data-effect=\"italics\">n<\/em><sup>th<\/sup> Root of a Number<\/p>\n\n<\/header>\n<div class=\"textbox__content\">\n\n\\(\\begin{array}{}\\\\ \\\\ \\hfill \\text{If}\\phantom{\\rule{0.2em}{0ex}}{b}^{n}=a,\\phantom{\\rule{0.2em}{0ex}}\\text{then}\\phantom{\\rule{0.2em}{0ex}}b\\phantom{\\rule{0.2em}{0ex}}\\text{is an}\\phantom{\\rule{0.2em}{0ex}}{n}^{th}\\phantom{\\rule{0.2em}{0ex}}\\text{root of}\\phantom{\\rule{0.2em}{0ex}}a.\\hfill \\\\ \\hfill \\text{The principal}\\phantom{\\rule{0.2em}{0ex}}{n}^{th}\\phantom{\\rule{0.2em}{0ex}}\\text{root of}\\phantom{\\rule{0.2em}{0ex}}a\\phantom{\\rule{0.2em}{0ex}}\\text{is written}\\phantom{\\rule{0.2em}{0ex}}\\sqrt[n]{a}.\\hfill \\\\ \\hfill n\\phantom{\\rule{0.2em}{0ex}}\\text{is called the}\\phantom{\\rule{0.2em}{0ex}}\\mathbf{\\text{index}}\\phantom{\\rule{0.2em}{0ex}}\\text{of the radical.}\\hfill \\end{array}\\)\n\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1169149368114\">Just like we use the word \u2018cubed\u2019 for <em data-effect=\"italics\">b<\/em><sup>3<\/sup>, we use the term \u2018cube root\u2019 for \\(\\sqrt[3]{a}.\\)<\/p>\n<p id=\"fs-id1169149310383\">We can refer to <a class=\"autogenerated-content\" href=\"#CNX_IntAlg_Figure_08_01_002\">(Table 1)<\/a> to help find higher roots.<\/p>\n\n<div id=\"fs-id1169146874884\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\">\\(\\begin{array}{ccc}\\hfill {4}^{3}&amp; =\\hfill &amp; 64\\hfill \\\\ \\hfill {3}^{4}&amp; =\\hfill &amp; 81\\hfill \\\\ \\hfill {\\left(-2\\right)}^{5}&amp; =\\hfill &amp; -32\\hfill \\end{array}\\phantom{\\rule{6em}{0ex}}\\begin{array}{ccc}\\hfill \\sqrt[3]{64}&amp; =\\hfill &amp; 4\\hfill \\\\ \\hfill \\sqrt[4]{81}&amp; =\\hfill &amp; 3\\hfill \\\\ \\hfill \\sqrt[5]{-32}&amp; =\\hfill &amp; -2\\hfill \\end{array}\\)<\/div>\n<p id=\"fs-id1169148827466\">Could we have an even root of a negative number? We know that the square root of a negative number is not a real number. The same is true for any even root. <em data-effect=\"italics\">Even<\/em> roots of negative numbers are not real numbers. <em data-effect=\"italics\">Odd<\/em> roots of negative numbers are real numbers.<\/p>\n\n<div id=\"fs-id1169147027485\" data-type=\"note\">\n<div data-type=\"title\">\n<div class=\"textbox textbox--key-takeaways\"><header class=\"textbox__header\">\n<p class=\"textbox__title\">Properties of \\(\\sqrt[n]{a}\\)<\/p>\n\n<\/header>\n<div class=\"textbox__content\">\n<p id=\"fs-id1169149326612\">When <em data-effect=\"italics\">n<\/em> is an even number and<\/p>\n\n<ul id=\"fs-id1169149002308\" data-bullet-style=\"bullet\">\n \t<li>\\(a\\ge 0,\\) then \\(\\sqrt[n]{a}\\) is a real number.<\/li>\n \t<li>\\(a&lt;0,\\) then \\(\\sqrt[n]{a}\\) is not a real number.<\/li>\n<\/ul>\n<p id=\"fs-id1169148898687\">When <em data-effect=\"italics\">n<\/em> is an odd number, \\(\\sqrt[n]{a}\\) is a real number for all values of <em data-effect=\"italics\">a<\/em>.<\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1169149096279\">We will apply these properties in the next two examples.<\/p>\n\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\n<p class=\"textbox__title\">EXAMPLE 4<\/p>\n\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1169146654571\" data-type=\"problem\">\n<p id=\"fs-id1169148924022\">Simplify: a) \\(\\sqrt[3]{64}\\) b) \\(\\sqrt[4]{81}\\) c) \\(\\sqrt[5]{32}.\\)<\/p>\n\n<\/div>\n<div id=\"fs-id1169146648492\" data-type=\"solution\">\n\n<strong><span class=\"token\">Solution<\/span><\/strong>\n<p id=\"fs-id1169149329984\"><span class=\"token\">a)<\/span><\/p>\n\n<table style=\"border-collapse: collapse; width: 100%; height: 32px;\" border=\"0\">\n<tbody>\n<tr style=\"height: 16px;\">\n<td style=\"width: 50%; height: 16px;\"><\/td>\n<td style=\"width: 50%; height: 16px;\">\\(\\sqrt[3]{64}\\)<\/td>\n<\/tr>\n<tr style=\"height: 16px;\">\n<td style=\"width: 50%; height: 16px;\">Since \\({4}^{3}=64\\).<\/td>\n<td style=\"width: 50%; height: 16px;\">\u00a04<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<span class=\"token\">b)<\/span>\n<table style=\"border-collapse: collapse; width: 100%;\" border=\"0\">\n<tbody>\n<tr>\n<td style=\"width: 50%;\"><\/td>\n<td style=\"width: 50%;\">\\(\\sqrt[4]{81}\\)<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 50%;\">Since \\(\\left(3\\right)}^{4}=81\\)<\/td>\n<td style=\"width: 50%;\">\u00a03<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<span class=\"token\">c)<\/span>\n<table style=\"border-collapse: collapse; width: 100%;\" border=\"0\">\n<tbody>\n<tr>\n<td style=\"width: 50%;\"><\/td>\n<td style=\"width: 50%;\">\\(\\sqrt[5]{32}\\)<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 50%;\">Since \\(\\left(2\\right)}^{5}=32\\)<\/td>\n<td style=\"width: 50%;\">2<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 4<\/p>\n\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1169148843404\" data-type=\"problem\">\n<p id=\"fs-id1169148929680\">Simplify: a) \\(\\sqrt[3]{27}\\) b) \\(\\sqrt[4]{256}\\) c) \\(\\sqrt[5]{243}.\\)<\/p>\n\n<\/div>\n<div id=\"fs-id1169146658975\" data-type=\"solution\"><details open=\"open\"><summary>Show answer<\/summary>\n<p id=\"fs-id1169149157689\"><span class=\"token\">a)<\/span> 3 b) 4 c) 3<\/p>\n\n<\/details><\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1165927757740\">In this example be alert for the negative signs as well as even and odd powers.<\/p>\n\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\n<p class=\"textbox__title\">EXAMPLE 5<\/p>\n\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1169149204958\" data-type=\"problem\">\n<p id=\"fs-id1169144560227\">Simplify: a) \\(\\sqrt[3]{-125}\\) b) \\(\\sqrt[4]{-16}\\) c) \\(\\sqrt[5]{-243}.\\)<\/p>\n\n<\/div>\n<div id=\"fs-id1169149065347\" data-type=\"solution\">\n\n<strong><span class=\"token\">Solution<\/span><\/strong>\n<p id=\"fs-id1169146662763\"><span class=\"token\">a)<\/span><\/p>\n\n<table style=\"border-collapse: collapse; width: 100%;\" border=\"0\">\n<tbody>\n<tr>\n<td style=\"width: 50%;\"><\/td>\n<td style=\"width: 50%;\">\\(\\sqrt[3]{-125}\\)<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 50%;\">Since \\(\\left(-5\\right)}^{3}=-125\\)<\/td>\n<td style=\"width: 50%;\">\u00a0\\(-5\\)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<span class=\"token\">b)<\/span>\n<table style=\"border-collapse: collapse; width: 100%;\" border=\"0\">\n<tbody>\n<tr>\n<td style=\"width: 50%;\"><\/td>\n<td style=\"width: 50%;\">\\(\\sqrt[4]{-16}\\)<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 50%;\">Think, \\(\\left(?\\right)}^{4}=-16\\). No real number raised to the fourth power is negative.<\/td>\n<td style=\"width: 50%;\">Not a real number.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<span class=\"token\">c)<\/span>\n<table style=\"border-collapse: collapse; width: 100%;\" border=\"0\">\n<tbody>\n<tr>\n<td style=\"width: 50%;\"><\/td>\n<td style=\"width: 50%;\">\\(\\sqrt[5]{-243}\\)<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 50%;\">Since \\(left(-3\\right)}^{5}=-243\\).<\/td>\n<td style=\"width: 50%;\">\\(-3\\)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 5<\/p>\n\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1169144873604\" data-type=\"problem\">\n<p id=\"fs-id1169149349661\">Simplify: a) \\(\\sqrt[3]{-27}\\) b) \\(\\sqrt[4]{-256}\\) c) \\(\\sqrt[5]{-32}.\\)<\/p>\n\n<\/div>\n<div id=\"fs-id1169144379098\" data-type=\"solution\"><details open=\"open\"><summary>Show answer<\/summary>\n<p id=\"fs-id1169148990626\"><span class=\"token\">a) <\/span>\\(-3\\) b)\u00a0 not real c)\u00a0 \\(-2\\)<\/p>\n\n<\/details><\/div>\n<\/div>\n<\/div>\n<span style=\"font-family: Helvetica, Arial, 'GFS Neohellenic', sans-serif; font-size: 1.2em; font-weight: bold;\">Estimate and Approximate Roots<\/span>\n\n<\/div>\n<div id=\"fs-id1169148831408\" class=\"bc-section section\" data-depth=\"1\">\n<p id=\"fs-id1169149288530\">When we see a number with a radical sign, we often don\u2019t think about its numerical value. While we probably know that the \\(\\sqrt{4}=2,\\) what is the value of \\(\\sqrt[]{21}\\) or \\(\\sqrt[3]{50}?\\) In some situations a quick estimate is meaningful and in others it is convenient to have a decimal approximation.<\/p>\n<p id=\"fs-id1169149092400\">To get a numerical estimate of a square root, we look for <span class=\"no-emphasis\" data-type=\"term\">perfect square<\/span> numbers closest to the radicand. To find an estimate of \\(\\sqrt{11},\\) we see 11 is between perfect square numbers 9 and 16, <em data-effect=\"italics\">closer<\/em> to 9. Its square root then will be between 3 and 4, but closer to 3.<\/p>\n<span id=\"fs-id1169147066070\" data-type=\"media\" data-alt=\"The figure contains two tables. The first table has 5 rows and 2 columns. The first row is a header row with the headers \u201cNumber\u201d and \u201cSquare Root\u201d. The second row has the numbers 4 and 2. The third row is 9 and 3. The fourth row is 16 and 4. The last row is 25 and 5. A callout containing the number 11 is directed between the 9 and 16 in the first column. Another callout containing the number square root of 11 is directed between the 3 and 4 of the second column. Below the table are the inequalities 9 is less than 11 is less than 16 and 3 is less than square root of 11 is less than 4. The second table has 5 rows and 2 columns. The first row is a header row with the headers \u201cNumber\u201d and \u201cCube Root\u201d. The second row has the numbers 8 and 2. The third row is 27 and 3. The fourth row is 64 and 4. The last row is 125 and 5. A callout containing the number 91 is directed between the 64 and 125 in the first column. Another callout containing the number cube root of 91 is directed between the 4 and 5 of the second column. Below the table are the inequalities 64 is less than 91 is less than 125 and 4 is less than cube root of 91 is less than 5.\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_IntAlg_Figure_08_01_004_img_new.jpg\" alt=\"The figure contains two tables. The first table has 5 rows and 2 columns. The first row is a header row with the headers \u201cNumber\u201d and \u201cSquare Root\u201d. The second row has the numbers 4 and 2. The third row is 9 and 3. The fourth row is 16 and 4. The last row is 25 and 5. A callout containing the number 11 is directed between the 9 and 16 in the first column. Another callout containing the number square root of 11 is directed between the 3 and 4 of the second column. Below the table are the inequalities 9 is less than 11 is less than 16 and 3 is less than square root of 11 is less than 4. The second table has 5 rows and 2 columns. The first row is a header row with the headers \u201cNumber\u201d and \u201cCube Root\u201d. The second row has the numbers 8 and 2. The third row is 27 and 3. The fourth row is 64 and 4. The last row is 125 and 5. A callout containing the number 91 is directed between the 64 and 125 in the first column. Another callout containing the number cube root of 91 is directed between the 4 and 5 of the second column. Below the table are the inequalities 64 is less than 91 is less than 125 and 4 is less than cube root of 91 is less than 5.\" data-media-type=\"image\/jpeg\"><\/span>\n<p id=\"fs-id1169148932974\">Similarly, to estimate \\(\\sqrt[3]{91},\\) we see 91 is between perfect cube numbers 64 and 125. The cube root then will be between 4 and 5.<\/p>\n\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\n<p class=\"textbox__title\">EXAMPLE 6<\/p>\n\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1169149031465\" data-type=\"problem\">\n<p id=\"fs-id1169149037560\">Estimate each root between two consecutive whole numbers: a) \\(\\sqrt{105}\\) b) \\(\\sqrt[3]{43}.\\)<\/p>\n<strong><span class=\"token\">Solution<\/span><\/strong>\n\n<\/div>\n<div id=\"fs-id1169149109449\" data-type=\"solution\">\n<p id=\"fs-id1169149029219\"><span class=\"token\">a)<\/span> Think of the perfect square numbers closest to 105. Make a small table of these perfect squares and their squares roots.<\/p>\n\n<table id=\"fs-id1169149097063\" style=\"width: 100%;\" summary=\"The table has 5 rows and 2 columns. The first row is a header row with the headers \u201cNumber\u201d and \u201cSquare Root\u201d. The second row has the numbers 81and 9. The third row is 100 and 10. The fourth row is 121 and 11. The last row is 144 and 12. A callout containing the number 105 is directed between the 100 and 121 in the first column. Another callout containing the number square root of 105 is directed between the 10 and 11 of the second column. Below the table are the inequalities 100 is less than 105 is less than 121 and 10 is less than square root of 105 is less than 11.\" data-label=\"\">\n<tbody>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span id=\"fs-id1169149014939\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_IntAlg_Figure_08_01_005a_img.jpg\" alt=\".\" data-media-type=\"image\/jpeg\"><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span id=\"fs-id1169148870813\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_IntAlg_Figure_08_01_005b_img.jpg\" alt=\".\" data-media-type=\"image\/jpeg\"><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Locate 105 between two consecutive perfect squares.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span id=\"fs-id1169144379992\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_IntAlg_Figure_08_01_005c_img.jpg\" alt=\".\" data-media-type=\"image\/jpeg\"><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">\\(\\sqrt{105}\\) is between their square roots.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span id=\"fs-id1169149101962\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_IntAlg_Figure_08_01_005d_img.jpg\" alt=\".\" data-media-type=\"image\/jpeg\"><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1169148993521\"><span class=\"token\">b)<\/span> Similarly we locate 43 between two perfect cube numbers.<\/p>\n\n<table id=\"fs-id1169148970427\" style=\"width: 100%;\" summary=\"The table has 5 rows and 2 columns. The first row is a header row with the headers \u201cNumber\u201d and \u201cCube Root\u201d. The second row has the numbers 8 and 2. The third row is 27 and 3. The fourth row is 64 and 4. The last row is 125 and 5. A callout containing the number 43 is directed between the 27 and 64 in the first column. Another callout containing the number cube root of 43 is directed between the 3 and 4 of the second column. Below the table are the inequalities 27 is less than 43 is less than 64 and 3 is less than cube root of 43 is less than 4.\" data-label=\"\">\n<tbody>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span id=\"fs-id1169149115401\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_IntAlg_Figure_08_01_006a_img.jpg\" alt=\".\" data-media-type=\"image\/jpeg\"><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span id=\"fs-id1169149005295\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_IntAlg_Figure_08_01_006b_img.jpg\" alt=\".\" data-media-type=\"image\/jpeg\"><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Locate 43 between two consecutive perfect cubes.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span id=\"fs-id1169149108611\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_IntAlg_Figure_08_01_006c_img.jpg\" alt=\".\" data-media-type=\"image\/jpeg\"><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">\\(\\sqrt[3]{43}\\) is between their cube roots.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span id=\"fs-id1169148992327\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_IntAlg_Figure_08_01_006d_img.jpg\" alt=\".\" data-media-type=\"image\/jpeg\"><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 6<\/p>\n\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1169148915764\" data-type=\"problem\">\n<p id=\"fs-id1169149105146\">Estimate each root between two consecutive whole numbers:<\/p>\n<p id=\"fs-id1169149144160\"><span class=\"token\">a) <\/span>\\(\\sqrt{38}\\)b)\u00a0 \\(\\sqrt[3]{93}\\)<\/p>\n\n<\/div>\n<div id=\"fs-id1169148898667\" data-type=\"solution\"><details open=\"open\"><summary>Show answer<\/summary>\n<p id=\"fs-id1169149117873\"><span class=\"token\">a) <\/span>\\(6&lt;\\sqrt{38}&lt;7\\)<\/p>\n\n<\/details>\n<div data-type=\"newline\"><\/div>\n<span class=\"token\">b) <\/span>\\(4&lt;\\sqrt[3]{93}&lt;5\\)\n\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1169148871159\"><img class=\"alignnone wp-image-6857\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/06\/business-color_calculator_icon-icons.com_53466.png\" alt=\"\" width=\"18\" height=\"18\"> There are mathematical methods to approximate square roots, but nowadays most people use a calculator to find square roots. To find a square root you will use the \\(\\sqrt{x}\\) key on your calculator. To find a cube root, or any root with higher index, you will use the \\(\\sqrt[y]{x}\\) key.<\/p>\n<p id=\"fs-id1169146646664\">When you use these keys, you get an approximate value. It is an approximation, accurate to the number of digits shown on your calculator\u2019s display. The symbol for an approximation is \\(\\approx \\) and it is read \u2018approximately\u2019.<\/p>\n<p id=\"fs-id1169144382250\">Suppose your calculator has a 10 digit display. You would see that<\/p>\n\n<div id=\"fs-id1169149014395\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\">\\(\\begin{array}{c}\\hfill \\sqrt{5}\\approx 2.236067978\\phantom{\\rule{0.2em}{0ex}}\\text{rounded to two decimal places is}\\phantom{\\rule{0.2em}{0ex}}\\sqrt{5}\\approx 2.24\\hfill \\\\ \\hfill \\sqrt[4]{93}\\approx 3.105422799\\phantom{\\rule{0.2em}{0ex}}\\text{rounded to two decimal places is}\\phantom{\\rule{0.2em}{0ex}}\\sqrt[4]{93}\\approx 3.11\\hfill \\end{array}\\)<\/div>\n<p id=\"fs-id1169149303690\">How do we know these values are approximations and not the exact values? Look at what happens when we square them:<\/p>\n\n<div id=\"fs-id1169149121409\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\">\\(\\begin{array}{ccc}\\hfill {\\left(2.236067978\\right)}^{2}&amp; =\\hfill &amp; 5.000000002\\hfill \\\\ \\hfill {\\left(2.24\\right)}^{2}&amp; =\\hfill &amp; 5.0176\\hfill \\end{array}\\phantom{\\rule{4em}{0ex}}\\begin{array}{ccc}\\hfill {\\left(3.105422799\\right)}^{4}&amp; =\\hfill &amp; 92.999999991\\hfill \\\\ \\hfill {\\left(3.11\\right)}^{4}&amp; =\\hfill &amp; 93.54951841\\hfill \\end{array}\\)<\/div>\n<p id=\"fs-id1169148926161\">Their squares are close to 5, but are not exactly equal to 5. The fourth powers are close to 93, but not equal to 93.<\/p>\n\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\n<p class=\"textbox__title\">EXAMPLE 7<\/p>\n\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1169149294811\" data-type=\"problem\">\n<p id=\"fs-id1169149344377\">Round to two decimal places: a) \\(\\sqrt{17}\\) b) \\(\\sqrt[3]{49}\\) c) \\(\\sqrt[4]{51}.\\)<\/p>\n<strong><span class=\"token\">Solution<\/span><\/strong>\n\n<\/div>\n<div id=\"fs-id1169149360529\" data-type=\"solution\">\n<p id=\"fs-id1169149305330\"><span class=\"token\">a)<\/span><\/p>\n\n<table style=\"border-collapse: collapse; width: 100%;\" border=\"0\">\n<tbody>\n<tr>\n<td style=\"width: 50%;\"><\/td>\n<td style=\"width: 50%;\">\\(\\sqrt{17}\\)<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 50%;\">Use the calculator square root key.<\/td>\n<td style=\"width: 50%;\">\\(4.123105626\\)...<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 50%;\">Round to two decimal places.<\/td>\n<td style=\"width: 50%;\">\\(4.12\\)<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 50%;\"><\/td>\n<td style=\"width: 50%;\">\\(\\sqrt{17}\\approx 4.12\\)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<span class=\"token\">b)<\/span>\n<table style=\"border-collapse: collapse; width: 100%; height: 64px;\" border=\"0\">\n<tbody>\n<tr style=\"height: 16px;\">\n<td style=\"width: 50%; height: 16px;\"><\/td>\n<td style=\"width: 50%; height: 16px;\">\\(\\sqrt[3]{49}\\)<\/td>\n<\/tr>\n<tr style=\"height: 16px;\">\n<td style=\"width: 50%; height: 16px;\">Use the calculator \\(\\sqrt[y]{x}\\) key.<\/td>\n<td style=\"width: 50%; height: 16px;\">\\(3.659305710\\)...<\/td>\n<\/tr>\n<tr style=\"height: 16px;\">\n<td style=\"width: 50%; height: 16px;\">Round to two decimal places.<\/td>\n<td style=\"width: 50%; height: 16px;\">\\(3.66\\)<\/td>\n<\/tr>\n<tr style=\"height: 16px;\">\n<td style=\"width: 50%; height: 16px;\"><\/td>\n<td style=\"width: 50%; height: 16px;\">\u00a0\\(\\sqrt[3]{49}\\approx 3.66\\)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<span class=\"token\">c)<\/span>\n\n<\/div>\n<table style=\"border-collapse: collapse; width: 100%;\" border=\"0\">\n<tbody>\n<tr>\n<td style=\"width: 50%;\"><\/td>\n<td style=\"width: 50%;\">\\(\\sqrt[4]{51}\\)<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 50%;\">Use the calculator \\(\\sqrt[y]{x}\\) key.<\/td>\n<td style=\"width: 50%;\">\\(2.6723451177\\)<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 50%;\">Round to two decimal places.<\/td>\n<td style=\"width: 50%;\">\\(2.67\\)<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 50%;\"><\/td>\n<td style=\"width: 50%;\">\\(\\sqrt[4]{51}\\approx 2.67\\)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 7<\/p>\n\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1169144874512\" data-type=\"problem\">\n<p id=\"fs-id1169146731256\">Round to two decimal places:<\/p>\n<p id=\"fs-id1169144645848\"><span class=\"token\">a) <\/span>\\(\\sqrt{11}\\) b) \\(\\sqrt[3]{71}\\) c) \\(\\sqrt[4]{127}.\\)<\/p>\n\n<\/div>\n<div id=\"fs-id1169149348031\" data-type=\"solution\"><details open=\"open\"><summary>Show answer<\/summary>\n<p id=\"fs-id1169149097346\"><span class=\"token\">a) <\/span>\\(\\approx 3.32\\) b) \\(\\approx 4.14\\)<\/p>\n\n<\/details>\n<div data-type=\"newline\"><\/div>\n<span class=\"token\">c) <\/span>\\(\\approx 3.36\\)\n\n<\/div>\n<\/div>\n<\/div>\n<h1 data-type=\"title\">Use Radicals in Applications<\/h1>\n<p id=\"fs-id1169148985032\">As you progress through your college or university courses, you\u2019ll encounter formulas that include radicals in many disciplines. We will modify our Problem Solving Strategy for Geometry Applications slightly to give us a plan for solving applications with formulas from any discipline.<\/p>\n\n<div id=\"fs-id1169149040822\" class=\"howto\" data-type=\"note\">\n<div data-type=\"title\">\n<div class=\"textbox textbox--key-takeaways\"><header class=\"textbox__header\">\n<p class=\"textbox__title\">Use a problem solving strategy for applications with formulas.<\/p>\n\n<\/header>\n<div class=\"textbox__content\">\n<ol id=\"fs-id1169144543890\" class=\"stepwise\" type=\"1\">\n \t<li><strong data-effect=\"bold\">Read<\/strong> the problem and make sure all the words and ideas are understood. When appropriate, draw a figure and label it with the given information.<\/li>\n \t<li><strong data-effect=\"bold\">Identify<\/strong> what we are looking for.<\/li>\n \t<li><strong data-effect=\"bold\">Name<\/strong> what we are looking for by choosing a variable to represent it.<\/li>\n \t<li><strong data-effect=\"bold\">Translate<\/strong> into an equation by writing the appropriate formula or model for the situation. Substitute in the given information.<\/li>\n \t<li><strong data-effect=\"bold\">Solve the equation<\/strong> using good algebra techniques.<\/li>\n \t<li><strong data-effect=\"bold\">Check<\/strong> the answer in the problem and make sure it makes sense.<\/li>\n \t<li><strong data-effect=\"bold\">Answer<\/strong> the question with a complete sentence.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n&nbsp;\n\n<\/div>\n<\/div>\n<p id=\"fs-id1169149118428\">One application of radicals has to do with the effect of <span class=\"no-emphasis\" data-type=\"term\">gravity<\/span> on falling objects. The formula allows us to determine how long it will take a fallen object to hit the gound.<\/p>\n\n<div id=\"fs-id1169148957158\" data-type=\"note\">\n<div data-type=\"title\">\n<div class=\"textbox textbox--key-takeaways\"><header class=\"textbox__header\">\n<p class=\"textbox__title\">Falling Objects<\/p>\n\n<\/header>\n<div class=\"textbox__content\">\n<p id=\"fs-id1169144729653\">On Earth, if an object is dropped from a height of <em data-effect=\"italics\">h<\/em> feet, the time in seconds it will take to reach the ground is found by using the formula<\/p>\n\n<div id=\"fs-id1169149103897\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\">\\(t=\\frac{\\sqrt{h}}{4}.\\)<\/div>\n<\/div>\n<\/div>\n&nbsp;\n\n<\/div>\n<\/div>\n<p id=\"fs-id1169146628740\">For example, if an object is dropped from a height of 64 feet, we can find the time it takes to reach the ground by substituting \\(h=64\\) into the formula.<\/p>\n\n<table class=\"unnumbered unstyled aligncenter\" summary=\"Since h equals 64 we rewrite the formula, replacing h with the number 64. The formula then becomes t equals square root of 64 divided by 4. Taking the square root of 64 we get t equals 8 divided by 4. Simplifying the fraction we get t equals 2. It would take 2 seconds for an object dropped from a height of 64 feet to reach the ground.\" data-label=\"\">\n<tbody>\n<tr valign=\"top\">\n<td style=\"text-align: center;\" data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span id=\"fs-id1169149065270\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_IntAlg_Figure_08_06_010a_img.jpg\" alt=\".\" data-media-type=\"image\/jpeg\"><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span id=\"fs-id1169149032044\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_IntAlg_Figure_08_06_010b_img.jpg\" alt=\".\" data-media-type=\"image\/jpeg\"><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Take the square root of 64.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span id=\"fs-id1169149342082\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_IntAlg_Figure_08_06_010c_img.jpg\" alt=\".\" data-media-type=\"image\/jpeg\"><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Simplify the fraction.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_IntAlg_Figure_08_06_010d_img.jpg\" alt=\".\" data-media-type=\"image\/jpeg\"><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1169149190917\" style=\"text-align: center;\">It would take 2 seconds for an object dropped from a height of 64 feet to reach the ground.<\/p>\n\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\n<p class=\"textbox__title\">EXAMPLE 8<\/p>\n\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1169149120935\" data-type=\"problem\">\n<p id=\"fs-id1169148973896\">Marissa dropped her sunglasses from a bridge 400 feet above a river. Use the formula \\(t=\\frac{\\sqrt{h}}{4}\\) to find how many seconds it took for the sunglasses to reach the river.<\/p>\n<strong><span class=\"token\">Solution<\/span><\/strong>\n\n<\/div>\n<div id=\"fs-id1169148890736\" data-type=\"solution\">\n<table id=\"fs-id1169149376727\" style=\"width: 100%;\" summary=\"The first step in the process is to read the problem. Step 2 is to identify what we are looking for. We are looking for the time it takes the sunglasses to reach the river. Step 3 is to name what we are looking for. Let t equal the time. Step 4 is to translate into an equation by writing the appropriate formula and substitute in the given information. t equals the square root of h divided by 4 and h equals 400. So t equals the square root of 400 divided by 4. Step 5 is to solve the equation. So t equals 20 divided by 4. So t equals 5. Step 6 is to check the answer in the problem and make sure it makes sense. Does 5 equal the square root of 400 divided 4. Since 5 equals 20 divided by 4, the answer is a solution to the equation. Does 5 seconds seem like a reasonable length of time? Yes. Step 7 is to answer the question. It will take 5 seconds for the sunglasses to reach the river.\" data-label=\"\">\n<tbody>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Step 1. Read<\/strong> the problem.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Step 2. Identify<\/strong> what we are looking for.<\/td>\n<td data-valign=\"top\" data-align=\"left\">the time it takes for the\n<div data-type=\"newline\"><\/div>\nsunglasses to reach the river<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Step 3. Name<\/strong> what we are looking.<\/td>\n<td data-valign=\"top\" data-align=\"left\">Let \\(t=\\) time.<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Step 4. Translate<\/strong> into an equation by writing the\n<div data-type=\"newline\"><\/div>\nappropriate formula. Substitute in the given\n<div data-type=\"newline\"><\/div>\ninformation.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span id=\"fs-id1169148938836\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_IntAlg_Figure_08_06_011b_img.jpg\" alt=\".\" data-media-type=\"image\/jpeg\"><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Step 5. Solve the equation.<\/strong><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span id=\"fs-id1169146744203\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_IntAlg_Figure_08_06_011c_img.jpg\" alt=\".\" data-media-type=\"image\/jpeg\"><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span id=\"fs-id1169144561652\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_IntAlg_Figure_08_06_011d_img.jpg\" alt=\".\" data-media-type=\"image\/jpeg\"><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Step 6. Check<\/strong> the answer in the problem and make\n<div data-type=\"newline\"><\/div>\nsure it makes sense.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span id=\"fs-id1169149157046\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_IntAlg_Figure_08_06_011a_img.jpg\" alt=\".\" data-media-type=\"image\/jpeg\"><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Does 5 seconds seem like a reasonable length of\n<div data-type=\"newline\"><\/div>\ntime?<\/td>\n<td data-valign=\"top\" data-align=\"left\">Yes.<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Step 7. Answer<\/strong> the question.<\/td>\n<td data-valign=\"top\" data-align=\"left\">It will take 5 seconds for the\n<div data-type=\"newline\"><\/div>\nsunglasses to reach the river.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 8<\/p>\n\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1169149115845\" data-type=\"problem\">\n<p id=\"fs-id1169149089351\">A helicopter dropped a rescue package from a height of 1,296 feet. Use the formula \\(t=\\frac{\\sqrt{h}}{4}\\) to find how many seconds it took for the package to reach the ground.<\/p>\n\n<\/div>\n<div id=\"fs-id1169144450967\" data-type=\"solution\"><details open=\"open\"><summary>Show answer<\/summary>\n<p id=\"fs-id1169138945430\">9 seconds<\/p>\n\n<\/details><\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1169148957152\">Police officers investigating car accidents measure the length of the skid marks on the pavement. Then they use square roots to determine the <span class=\"no-emphasis\" data-type=\"term\">speed<\/span>, in miles per hour, a car was going before applying the brakes.<\/p>\n\n<div id=\"fs-id1169149012194\" data-type=\"note\">\n<div data-type=\"title\">\n<div class=\"textbox textbox--key-takeaways\"><header class=\"textbox__header\">\n<p class=\"textbox__title\">Skid Marks and Speed of a Car<\/p>\n\n<\/header>\n<div class=\"textbox__content\">\n<p id=\"fs-id1169144555978\">If the length of the skid marks is <em data-effect=\"italics\">d<\/em> feet, then the speed, <em data-effect=\"italics\">s<\/em>, of the car before the brakes were applied can be found by using the formula<\/p>\n\n<div id=\"fs-id1169149092734\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\">\\(s=\\sqrt{24d}\\)<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"equation\" data-label=\"\">\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\n<p class=\"textbox__title\">EXAMPLE 9<\/p>\n\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1169149311099\" data-type=\"problem\">\n<p id=\"fs-id1169147133499\">After a car accident, the skid marks for one car measured 190 feet. Use the formula \\(s=\\sqrt{24d}\\) to find the speed of the car before the brakes were applied. Round your answer to the nearest tenth.<\/p>\n<strong><span class=\"token\">Solution<\/span><\/strong>\n\n<\/div>\n<div id=\"fs-id1169148993220\" data-type=\"solution\">\n<table id=\"fs-id1169144381624\" class=\"unnumbered unstyled can-break\" style=\"height: 350px;\" summary=\"The first step in the process is to read the problem. Step 2 is to identify what we are looking for. We are looking for the speed of the car. Step 3 is to name what we are looking for. Let s equal the speed. Step 4 is to translate into an equation by writing the appropriate formula and substitute in the given information. s equals the square root of the quantity 24 d in parentheses, and d equals 190. So s equals the square root of the quantity 24 times 190 in parentheses. Step 5 is to solve the equation. So s equals the square root of 4560. So s is approximately equal to 67.52777. Rounding to 1 decimal place we et s equal to 67.5. Step 6 is to check the answer in the problem and make sure it makes sense. Does the square root of 4560 equal the square root of the quantity 24 times 190 in parentheses? It does. Does 67.5 mph seem like a reasonable speed? Yes. Step 7 is to answer the question. The car was traveling approximately 67.5 mph before the brakes were applied.\" data-label=\"\">\n<tbody>\n<tr style=\"height: 15px;\" valign=\"top\">\n<td style=\"height: 15px; width: 349.757px;\" data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Step 1. Read<\/strong> the problem<\/td>\n<td style=\"height: 15px; width: 396.424px;\" data-valign=\"top\" data-align=\"left\"><\/td>\n<\/tr>\n<tr style=\"height: 15px;\" valign=\"top\">\n<td style=\"height: 15px; width: 349.757px;\" data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Step 2. Identify<\/strong> what we are looking for.<\/td>\n<td style=\"height: 15px; width: 396.424px;\" data-valign=\"top\" data-align=\"left\">the speed of a car<\/td>\n<\/tr>\n<tr style=\"height: 15px;\" valign=\"top\">\n<td style=\"height: 15px; width: 349.757px;\" data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Step 3. Name<\/strong> what we are looking for,<\/td>\n<td style=\"height: 15px; width: 396.424px;\" data-valign=\"top\" data-align=\"left\">Let \\(s=\\) the speed.<\/td>\n<\/tr>\n<tr style=\"height: 92px;\" valign=\"top\">\n<td style=\"height: 92px; width: 349.757px;\" data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Step 4. Translate<\/strong> into an equation by writing\n<div data-type=\"newline\"><\/div>\nthe appropriate formula. Substitute in the\n<div data-type=\"newline\"><\/div>\ngiven information.<\/td>\n<td style=\"height: 92px; width: 396.424px;\" data-valign=\"top\" data-align=\"left\"><span id=\"fs-id1169147109998\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_IntAlg_Figure_08_06_012a_img.jpg\" alt=\".\" data-media-type=\"image\/jpeg\"><\/span><\/td>\n<\/tr>\n<tr style=\"height: 21px;\" valign=\"top\">\n<td style=\"height: 21px; width: 349.757px;\" data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Step 5. Solve the equation.<\/strong><\/td>\n<td style=\"height: 21px; width: 396.424px;\" data-valign=\"top\" data-align=\"left\"><span id=\"fs-id1169148971507\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_IntAlg_Figure_08_06_012b_img.jpg\" alt=\".\" data-media-type=\"image\/jpeg\"><\/span><\/td>\n<\/tr>\n<tr style=\"height: 17px;\" valign=\"top\">\n<td style=\"height: 17px; width: 349.757px;\" data-valign=\"top\" data-align=\"left\"><\/td>\n<td style=\"height: 17px; width: 396.424px;\" data-valign=\"top\" data-align=\"left\"><span id=\"fs-id1169149219635\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_IntAlg_Figure_08_06_012c_img.jpg\" alt=\".\" data-media-type=\"image\/jpeg\"><\/span><\/td>\n<\/tr>\n<tr style=\"height: 16px;\" valign=\"top\">\n<td style=\"height: 16px; width: 349.757px;\" data-valign=\"top\" data-align=\"left\">Round to 1 decimal place.<\/td>\n<td style=\"height: 16px; width: 396.424px;\" data-valign=\"top\" data-align=\"left\"><span id=\"fs-id1169144558132\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_IntAlg_Figure_08_06_012d_img.jpg\" alt=\".\" data-media-type=\"image\/jpeg\"><\/span><\/td>\n<\/tr>\n<tr style=\"height: 98px;\" valign=\"top\">\n<td style=\"height: 98px; width: 349.757px;\" data-valign=\"top\" data-align=\"left\"><\/td>\n<td style=\"height: 98px; width: 396.424px;\" data-valign=\"top\" data-align=\"left\"><span id=\"fs-id1169146594591\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_IntAlg_Figure_08_06_012e_img.jpg\" alt=\".\" data-media-type=\"image\/jpeg\"><\/span><\/td>\n<\/tr>\n<tr style=\"height: 61px;\" valign=\"top\">\n<td style=\"height: 61px; width: 349.757px;\" data-valign=\"top\" data-align=\"left\"><\/td>\n<td style=\"height: 61px; width: 396.424px;\" data-valign=\"top\" data-align=\"left\">The speed of the car before the brakes were applied\n<div data-type=\"newline\"><\/div>\nwas 67.5 miles per hour.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1169144382040\" class=\"try\" data-type=\"note\">\n<div id=\"fs-id1169144382958\" data-type=\"exercise\">\n<div id=\"fs-id1169149330072\" data-type=\"problem\">\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 9<\/p>\n\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1169149330072\" data-type=\"problem\">\n<p id=\"fs-id1169149354829\">An accident investigator measured the skid marks of the car. The length of the skid marks was 76 feet. Use the formula \\(s=\\sqrt{24d}\\) to find the speed of the car before the brakes were applied. Round your answer to the nearest tenth.<\/p>\n\n<\/div>\n<div data-type=\"solution\"><details open=\"open\"><summary>Show answer<\/summary>\n<p id=\"fs-id1169146662927\">\\(42.7\\) feet<\/p>\n\n<\/details><\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1550190\">The Pythagorean Theorem tells how the lengths of the three sides of a right triangle relate to each other. It states that in any right triangle, the sum of the squares of the two legs equals the square of the hypotenuse.<\/p>\n\n<div id=\"fs-id1701775\" data-type=\"note\">\n<div data-type=\"title\">\n<div class=\"textbox textbox--key-takeaways\"><header class=\"textbox__header\">\n<p class=\"textbox__title\">The Pythagorean Theorem<\/p>\n\n<\/header>\n<div class=\"textbox__content\">\n<p id=\"fs-id1548386\">In any right triangle \\(\\Delta ABC\\),<\/p>\n\n<div id=\"fs-id1171104156908\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\">\\({a}^{2}+{b}^{2}={c}^{2}\\)<\/div>\n<p id=\"fs-id1422647\">where \\(c\\) is the length of the hypotenuse \\(a\\) and \\(b\\) are the lengths of the legs.<\/p>\n<span id=\"fs-id1723708\" data-type=\"media\" data-alt=\"A right triangle is shown. The right angle is marked with a box. Across from the box is side c. The sides touching the right angle are marked a and b.\"><img class=\"alignnone\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_BMath_Figure_09_03_025.jpg\" alt=\"A right triangle is shown. The right angle is marked with a box. Across from the box is side c. The sides touching the right angle are marked a and b.\" data-media-type=\"image\/jpeg\"><\/span>\n\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1538008\">We will use this definition of square roots to solve for the length of a side in a right triangle.<\/p>\n\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\n<p class=\"textbox__title\">EXAMPLE 10<\/p>\n\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1419626\" data-type=\"problem\">\n<p id=\"fs-id1420675\">Use the Pythagorean Theorem to find the length of the hypotenuse.<\/p>\n<span id=\"fs-id1411841\" data-type=\"media\" data-alt=\"Right triangle with legs labeled as 3 and 4.\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_BMath_Figure_09_03_026_img.jpg\" alt=\"Right triangle with legs labeled as 3 and 4.\" data-media-type=\"image\/jpeg\"><\/span>\n\n<strong>Solution<\/strong>\n\n<\/div>\n<div id=\"fs-id1172599766488\" data-type=\"solution\">\n<table id=\"eip-id1168469450887\" style=\"width: 100%;\" summary=\"Step 1 says, \u201cRead the problem.\u201d The word \u201cread\u201d is in bold. Beside this is a picture of a right triangle. The side across from the right angle is labeled c, the sides touching the right angle are labeled 3 and 4. Step 2 says, \u201cIdentify what you are looking for.\u201d The word \u201cidentify\u201d is in bold. Beside this it says, \u201cthe length of the hypotenuse of the triangle.\u201d Step 3 says, \u201cName. Choose a variable to represent it. \u201c The word \u201cname\u201d is in bold. Beside this it says, \u201cLet c equal the length of the hypotenuse.\u201d Step 4 says, \u201cTranslate. Write the appropriate formula. Substitute.\u201d The word \u201ctranslate\u201d is in bold. Beside this is the equation a squared plus b squared equals c squared, then 3 squared plus 4 squared equals c squared. Step 5 says, \u201cSolve the equation. Simplify. Use the definition of square root. Simplify.\u201d The word \u201csolve\u201d is in bold. Beside this is 9 plus 16 equals c squared, then 25 equals c squared, then square root of 25 equals c, then 5 equals c. Step 6 says \u201cCheck\u201d in bold. Beside this is 3 squared plus 4 squared followed by an equal sign with a question mark, then a red 5 squared. Below this is 9 plus 16 followed by an equal sign with a question mark, then 25. The next line says 25 equals 25 with a check mark. Step 7 says, \u201cAnswer the question.\u201d The word \u201canswer\u201d is in bold. Beside this is the answer, \u201cThe length of the hypotenuse is 5.\u201d\" data-label=\"\">\n<tbody>\n<tr>\n<td>Step 1. <strong data-effect=\"bold\">Read<\/strong> the problem.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Step 2. <strong data-effect=\"bold\">Identify<\/strong> what you are looking for.<\/td>\n<td>the length of the hypotenuse of the triangle<\/td>\n<\/tr>\n<tr>\n<td>Step 3. <strong data-effect=\"bold\">Name.<\/strong> Choose a variable to represent it.<\/td>\n<td>Let \\(c=\\text{the length of the hypotenuse}\\)\n<span id=\"eip-id1168469449113\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_BMath_Figure_09_03_053_img-01.png\" alt=\".\" data-media-type=\"image\/jpeg\"><\/span><\/td>\n<\/tr>\n<tr>\n<td>Step 4. <strong data-effect=\"bold\">Translate.<\/strong>\nWrite the appropriate formula.\nSubstitute.<\/td>\n<td><span id=\"eip-id1168469890812\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_BMath_Figure_09_03_053_img-02.png\" alt=\".\" data-media-type=\"image\/jpeg\"><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\">Step 5. <strong data-effect=\"bold\">Solve<\/strong> the equation.<\/td>\n<td><span id=\"eip-id1168469890834\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_BMath_Figure_09_03_053_img-03.png\" alt=\".\" data-media-type=\"image\/jpeg\"><\/span><\/td>\n<\/tr>\n<tr>\n<td>Step 6. <strong data-effect=\"bold\">Check:<\/strong>\n<span id=\"eip-id1168469481160\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_BMath_Figure_09_03_053_img-04.png\" alt=\".\" data-media-type=\"image\/jpeg\"><\/span><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Step 7. <strong data-effect=\"bold\">Answer<\/strong> the question.<\/td>\n<td>The length of the hypotenuse is 5.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 10<\/p>\n\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1542055\" data-type=\"problem\">\n<p id=\"fs-id1542057\">Use the Pythagorean Theorem to find the length of the hypotenuse.<\/p>\n<span id=\"fs-id1321685\" data-type=\"media\" data-alt=\"A right triangle is shown. The right angle is marked with a box. Across from the box is side c. The sides touching the right angle are marked 6 and 8.\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_BMath_Figure_09_03_029_img.jpg\" alt=\"A right triangle is shown. The right angle is marked with a box. Across from the box is side c. The sides touching the right angle are marked 6 and 8.\" data-media-type=\"image\/jpeg\"><\/span>\n\n<\/div>\n<div id=\"fs-id1727689\" data-type=\"solution\"><details open=\"open\"><summary>Show answer<\/summary>\n<p id=\"fs-id1259642\">10<\/p>\n\n<\/details><\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1169144566399\" class=\"media-2\" data-type=\"note\">\n<p id=\"fs-id1169149329664\">Access these online resources for additional instruction and practice with solving radical equations.<\/p>\n\n<ul id=\"fs-id1169147089449\" data-bullet-style=\"bullet\">\n \t<li><a href=\"https:\/\/openstax.org\/l\/37RadEquat5\">Radical Equation Application<\/a><\/li>\n<\/ul>\n<\/div>\n<h1>Glossary<\/h1>\n<div class=\"textbox shaded\">\n\n<strong>square root notation<\/strong>\n<ul>\n \t<li>\\(\\sqrt{m}\\) is read \u2018the square root of <em data-effect=\"italics\">m<\/em>\u2019<\/li>\n \t<li>If <em data-effect=\"italics\">n<\/em><sup>2<\/sup> = <em data-effect=\"italics\">m<\/em>, then \\(n=\\sqrt{m},\\) for \\(n\\ge 0.\\)<\/li>\n<\/ul>\n<div data-type=\"newline\"><\/div>\n<span id=\"fs-id1169148870699\" data-type=\"media\" data-alt=\"The image shows the variable m inside a square root symbol. The symbol is a line that goes up along the left side and then flat above the variable. The symbol is labeled \u201cradical sign\u201d. The variable m is labeled \u201cradicand\u201d.\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_IntAlg_Figure_08_01_009_img_new.jpg\" alt=\"The image shows the variable m inside a square root symbol. The symbol is a line that goes up along the left side and then flat above the variable. The symbol is labeled \u201cradical sign\u201d. The variable m is labeled \u201cradicand\u201d.\" data-media-type=\"image\/jpeg\"><\/span>\n<ul>\n \t<li>The square root of <em data-effect=\"italics\">m<\/em>, \\(\\sqrt{m},\\) is a positive number whose square is <em data-effect=\"italics\">m<\/em>.<\/li>\n<\/ul>\n&nbsp;\n\n<strong data-effect=\"bold\"><em data-effect=\"italics\">n<\/em><sup>th<\/sup> root of a number<\/strong>\n<ul>\n \t<li>If \\({b}^{n}=a,\\) then <em data-effect=\"italics\">b<\/em> is an <em data-effect=\"italics\">n<sup>th<\/sup><\/em> root of <em data-effect=\"italics\">a<\/em>.<\/li>\n \t<li>The principal <em data-effect=\"italics\">n<sup>th<\/sup><\/em> root of <em data-effect=\"italics\">a<\/em> is written \\(\\sqrt[n]{a}.\\)<\/li>\n \t<li><em data-effect=\"italics\">n<\/em> is called the <em data-effect=\"italics\">index<\/em> of the radical.<\/li>\n<\/ul>\n<\/div>\n<h1>1.6 Exercise Set<\/h1>\n<p id=\"fs-id1169149285863\">In the following exercises, simplify.<\/p>\n\n<ol class=\"twocolumn\">\n \t<li>\n<ol type=\"a\">\n \t<li>\\(\\sqrt{64}\\)<\/li>\n \t<li>\\(-\\sqrt{81}\\)<\/li>\n<\/ol>\n<\/li>\n \t<li>\n<ol type=\"a\">\n \t<li>\\(\\sqrt{196}\\)<\/li>\n \t<li>\\(-\\sqrt{1}\\)<\/li>\n<\/ol>\n<\/li>\n \t<li>\n<ol type=\"a\">\n \t<li>\\(\\sqrt{\\frac{4}{9}}\\)<\/li>\n \t<li>\\(-\\sqrt{0.01}\\)<\/li>\n<\/ol>\n<\/li>\n \t<li>\n<ol type=\"a\">\n \t<li>\\(\\sqrt{-121}\\)<\/li>\n \t<li>\\(-\\sqrt{289}\\)<\/li>\n<\/ol>\n<\/li>\n \t<li>\n<ol type=\"a\">\n \t<li>\\(-\\sqrt{225}\\)<\/li>\n \t<li>\\(\\sqrt{-9}\\)<\/li>\n<\/ol>\n<\/li>\n \t<li>\n<ol type=\"a\">\n \t<li>\\(\\sqrt[3]{216}\\)<\/li>\n \t<li>\\(\\sqrt[4]{256}\\)<\/li>\n<\/ol>\n<\/li>\n \t<li>\n<ol type=\"a\">\n \t<li>\\(\\sqrt[3]{512}\\)<\/li>\n \t<li>\\(\\sqrt[4]{81}\\)<\/li>\n \t<li>\\(\\sqrt[5]{1}\\)<\/li>\n<\/ol>\n<\/li>\n \t<li>\n<ol type=\"a\">\n \t<li>\\(\\sqrt[3]{-8}\\)<\/li>\n \t<li>\\(\\sqrt[4]{-81}\\)<\/li>\n \t<li>\\(\\sqrt[5]{-32}\\)<\/li>\n<\/ol>\n<\/li>\n \t<li>\n<ol type=\"a\">\n \t<li>\\(\\sqrt[3]{-125}\\)<\/li>\n \t<li>\\(\\sqrt[4]{-1296}\\)<\/li>\n \t<li>\\(\\sqrt[5]{-1024}\\)<\/li>\n<\/ol>\n<\/li>\n<\/ol>\n<div id=\"fs-id1169144768977\" data-type=\"exercise\">\n<div id=\"fs-id1169144768979\" data-type=\"problem\">\n<p id=\"fs-id1169144768981\"><span style=\"text-align: initial; font-size: 14pt;\">In the following exercises, estimate each root between two consecutive whole numbers.<\/span><\/p>\n\n<ol class=\"twocolumn\" start=\"10\">\n \t<li>\n<ol type=\"a\">\n \t<li>\\(\\sqrt{70}\\)<\/li>\n \t<li>\\(\\sqrt[3]{71}\\)<\/li>\n<\/ol>\n<\/li>\n \t<li>\n<ol type=\"a\">\n \t<li>\\(\\sqrt{200}\\)<\/li>\n \t<li>\\(\\sqrt[3]{137}\\)<\/li>\n<\/ol>\n<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<div id=\"fs-id1169149087431\" data-type=\"exercise\">\n<div id=\"fs-id1169149087433\" data-type=\"problem\">\n<p id=\"fs-id1169149289896\"><span style=\"font-size: 14pt; text-align: initial;\">In the following exercises, approximate each root and round to two decimal places.<\/span><\/p>\n\n<ol class=\"twocolumn\" start=\"12\">\n \t<li>\n<ol type=\"a\">\n \t<li>\\(\\sqrt{19}\\)<\/li>\n \t<li>\\(\\sqrt[3]{89}\\)<\/li>\n \t<li>\\(\\sqrt[4]{97}\\)<\/li>\n<\/ol>\n<\/li>\n \t<li>\n<ol type=\"a\">\n \t<li>\\(\\sqrt{53}\\)<\/li>\n \t<li>\\(\\sqrt[3]{147}\\)<\/li>\n \t<li>\\(\\sqrt[4]{452}\\)<\/li>\n<\/ol>\n<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<div id=\"fs-id1169144768438\" data-type=\"exercise\">\n<div id=\"fs-id1169144768440\" data-type=\"problem\">\n<p id=\"fs-id1169144768442\"><span style=\"font-size: 14pt; text-align: initial;\">In the following exercises, solve. Round approximations to one decimal place.<\/span><\/p>\n\n<ol start=\"14\">\n \t<li><strong data-effect=\"bold\">Landscaping.<\/strong> Reed wants to have a square garden plot in his backyard. He has enough compost to cover an area of 75 square feet. Use the formula \\(s=\\sqrt{A}\\) to find the length of each side of his garden. Round your answer to the nearest tenth of a foot.<\/li>\n \t<li><strong data-effect=\"bold\">Gravity.<\/strong> A hang glider dropped his cell phone from a height of 350 feet. Use the formula \\(t=\\frac{\\sqrt{h}}{4}\\) to find how many seconds it took for the cell phone to reach the ground.<\/li>\n \t<li><strong data-effect=\"bold\">Accident investigation<\/strong> The skid marks for a car involved in an accident measured 216 feet. Use the formula \\(s=\\sqrt{24d}\\) to find the speed of the car before the brakes were applied. Round your answer to the nearest tenth.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<dl id=\"fs-id1166424761164\">\n \t<dd id=\"fs-id1166422865542\">\n<div id=\"fs-id1169149285867\" data-type=\"exercise\">\n<div id=\"fs-id1169149285869\" data-type=\"problem\">\n<div id=\"fs-id1169149235406\" data-type=\"exercise\">\n<div id=\"fs-id1169149235408\" data-type=\"problem\">\n\nIn the following exercises, use the Pythagorean Theorem to find the length of the hypotenuse.\n<table style=\"border-collapse: collapse; width: 100%; height: 189px;\" border=\"0\">\n<tbody>\n<tr style=\"height: 175px;\">\n<td style=\"width: 50.4345%; height: 175px;\"><span id=\"fs-id1238083\" data-type=\"media\" data-alt=\"A right triangle is shown. The right angle is marked with a box. One of the sides touching the right angle is labeled as 16, the other as 12.\">17. <img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_BMath_Figure_09_03_204_img.jpg\" alt=\"A right triangle is shown. The right angle is marked with a box. One of the sides touching the right angle is labeled as 16, the other as 12.\" data-media-type=\"image\/jpeg\"><\/span><\/td>\n<td style=\"width: 49.5655%; height: 175px;\"><span id=\"fs-id1543755\" data-type=\"media\" data-alt=\"A right triangle is shown. The right angle is marked with a box. One of the sides touching the right angle is labeled as 15, the other as 20.\">18.<span id=\"fs-id1428325\" data-type=\"media\" data-alt=\"A right triangle is shown. The right angle is marked with a box. One of the sides touching the right angle is labeled as 5, the other as 12.\"> <img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_BMath_Figure_09_03_206_img.jpg\" alt=\"A right triangle is shown. The right angle is marked with a box. One of the sides touching the right angle is labeled as 5, the other as 12.\" data-media-type=\"image\/jpeg\"><\/span><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"eip-851\">In the following exercises, use the Pythagorean Theorem to find the length of the missing side. Round to the nearest tenth, if necessary.<\/p>\n\n<table style=\"border-collapse: collapse; width: 100%; height: 152px;\" border=\"0\">\n<tbody>\n<tr style=\"height: 110px;\">\n<td style=\"width: 50%; height: 110px;\"><span id=\"fs-id1576553\" data-type=\"media\" data-alt=\"A right triangle is shown. The right angle is marked with a box. The side across from the right angle is labeled as 10. One of the sides touching the right angle is labeled as 6.\">19.<span id=\"fs-id1547852\" data-type=\"media\" data-alt=\"A right triangle is shown. The right angle is marked with a box. The side across from the right angle is labeled as 17. One of the sides touching the right angle is labeled as 8.\"> <img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_BMath_Figure_09_03_208_img.jpg\" alt=\"A right triangle is shown. The right angle is marked with a box. The side across from the right angle is labeled as 17. One of the sides touching the right angle is labeled as 8.\" data-media-type=\"image\/jpeg\"><\/span><\/span><\/td>\n<td style=\"width: 50%; height: 110px;\"><span id=\"fs-id1701813\" data-type=\"media\" data-alt=\"A right triangle is shown. The right angle is marked with a box. The side across from the right angle is labeled as 13. One of the sides touching the right angle is labeled as 5.\"><span id=\"fs-id1439355\" data-type=\"media\" data-alt=\"A right triangle is shown. The right angle is marked with a box. The side across from the right angle is labeled as 20. One of the sides touching the right angle is labeled as 16.\">20.\u00a0 \u00a0 \u00a0 <img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_BMath_Figure_09_03_210_img.jpg\" alt=\"A right triangle is shown. The right angle is marked with a box. The side across from the right angle is labeled as 20. One of the sides touching the right angle is labeled as 16.\" data-media-type=\"image\/jpeg\"><\/span><\/span><\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"width: 50%; height: 14px;\"><span id=\"fs-id1172599767679\" data-type=\"media\" data-alt=\"A right triangle is shown. The right angle is marked with a box. The side across from the right angle is labeled as 13. One of the sides touching the right angle is labeled as 8.\">21 <span id=\"fs-id1704712\" data-type=\"media\" data-alt=\"A right triangle is shown. The right angle is marked with a box. Both of the sides touching the right angle are labeled as 6.\">. <img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_BMath_Figure_09_03_212_img.jpg\" alt=\"A right triangle is shown. The right angle is marked with a box. Both of the sides touching the right angle are labeled as 6.\" data-media-type=\"image\/jpeg\"><\/span><\/span><\/td>\n<td style=\"width: 50%; height: 14px;\"><span id=\"fs-id1614367\" data-type=\"media\" data-alt=\"A right triangle is shown. The right angle is marked with a box. The side across from the right angle is labeled as 17. One of the sides touching the right angle is labeled as 15.\"><span id=\"fs-id1537416\" data-type=\"media\" data-alt=\"A right triangle is shown. The right angle is marked with a box. The side across from the right angle is labeled as 7. One of the sides touching the right angle is labeled as 5.\">22.<img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_BMath_Figure_09_03_214_img.jpg\" alt=\"A right triangle is shown. The right angle is marked with a box. The side across from the right angle is labeled as 7. One of the sides touching the right angle is labeled as 5.\" data-media-type=\"image\/jpeg\">\u00a0<\/span><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1428288\"><span style=\"font-family: Helvetica, Arial, 'GFS Neohellenic', sans-serif; font-size: 1.2em; font-weight: bold; text-align: initial;\">Answers:<\/span><\/p>\n\n<ol class=\"twocolumn\">\n \t<li>\n<ol type=\"a\">\n \t<li>8<\/li>\n \t<li><span style=\"font-family: inherit; font-size: inherit;\"> \\(-9\\)<\/span><\/li>\n<\/ol>\n<\/li>\n \t<li>\n<ol type=\"a\">\n \t<li>14<\/li>\n \t<li>\\(-1\\)<\/li>\n<\/ol>\n<\/li>\n \t<li>\n<ol type=\"a\">\n \t<li>\\(\\frac{2}{3}\\)<\/li>\n \t<li>\\(-0.1\\)<\/li>\n<\/ol>\n<\/li>\n \t<li>\n<ol type=\"a\">\n \t<li>not real number<\/li>\n \t<li>\\(-17\\)<\/li>\n<\/ol>\n<\/li>\n \t<li>\n<ol type=\"a\">\n \t<li>\\(-15\\)<\/li>\n \t<li>not real number<\/li>\n<\/ol>\n<\/li>\n \t<li>\n<ol type=\"a\">\n \t<li>6<\/li>\n \t<li>4<\/li>\n<\/ol>\n<\/li>\n \t<li>\n<ol type=\"a\">\n \t<li>8<\/li>\n \t<li>3<\/li>\n \t<li>1<\/li>\n<\/ol>\n<\/li>\n \t<li>\n<ol type=\"a\">\n \t<li>\\(-2\\)<\/li>\n \t<li>\\(\\text{not real}\\)<\/li>\n \t<li>\\(-2\\)<\/li>\n<\/ol>\n<\/li>\n \t<li>\n<ol type=\"a\">\n \t<li>\\(-5\\)<\/li>\n \t<li>\\(\\text{not real}\\)<\/li>\n \t<li>\\(-4\\)<\/li>\n<\/ol>\n<\/li>\n \t<li>\n<ol type=\"a\">\n \t<li>\\(8&lt;\\sqrt{70}&lt;9\\)<\/li>\n \t<li>\\(4&lt;\\sqrt[3]{71}&lt;5\\)<\/li>\n<\/ol>\n<\/li>\n \t<li>\n<ol type=\"a\">\n \t<li>\\(14&lt;\\sqrt{200}&lt;15\\)<\/li>\n \t<li>\\(5&lt;\\sqrt[3]{137}&lt;6\\)<\/li>\n<\/ol>\n<\/li>\n \t<li>\n<ol type=\"a\">\n \t<li>\\(4.36\\)<\/li>\n \t<li>\\(\\approx 4.46\\)<\/li>\n \t<li>\\(\\approx 3.14\\)<\/li>\n<\/ol>\n<\/li>\n \t<li>\n<ol type=\"a\">\n \t<li>\\(7.28\\)<\/li>\n \t<li>\\(\\approx 5.28\\)<\/li>\n \t<li>\\(\\approx 4.61\\)<\/li>\n<\/ol>\n<\/li>\n \t<li>\\(8.7\\) feet<\/li>\n \t<li>\\(4.7\\) seconds<\/li>\n \t<li>72 feet<\/li>\n \t<li>20<\/li>\n \t<li>13<\/li>\n \t<li>15<\/li>\n \t<li>12<\/li>\n \t<li>8.5<\/li>\n \t<li>8.6<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1169144768977\" data-type=\"exercise\">\n<div id=\"fs-id1169148956903\" data-type=\"solution\">\n<p id=\"fs-id1169148956905\"><span style=\"font-family: Helvetica, Arial, 'GFS Neohellenic', sans-serif; font-size: 1.2em; font-weight: bold; text-align: initial;\">Attributions ( imported from my Introductory Algebra)<\/span><\/p>\n\n<\/div>\n<\/div><\/dd>\n<\/dl>\nThis chapter has been adapted from \u201cThe Real Numbers\u201d in <a href=\"https:\/\/openstax.org\/details\/books\/elementary-algebra\"><em>Elementary Algebra<\/em><\/a> (OpenStax) by Lynn Marecek and MaryAnne Anthony-Smith, which is under a <a href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY 4.0 Licence<\/a>. Adapted by Izabela Mazur. See the Copyright page for more information.","rendered":"<div class=\"textbox textbox--learning-objectives\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Learning Objectives<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>By the end of this section it is expected that you will be able to:<\/p>\n<ul>\n<li>Simplify expressions with roots<\/li>\n<li>Estimate and approximate roots<\/li>\n<li>Use radicals in applications<\/li>\n<\/ul>\n<\/div>\n<\/div>\n<h1 data-type=\"title\">Simplify Expressions with Square Roots<\/h1>\n<p id=\"fs-id1170654963188\">Remember that when a number <em data-effect=\"italics\">n<\/em> is multiplied by itself, we write <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-444500c4e43e38956dd248f19f2f7129_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#110;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"18\" style=\"vertical-align: 0px;\" \/> and read it \u201cn squared.\u201d The result is called the <span data-type=\"term\">square<\/span> of <em data-effect=\"italics\">n<\/em>. For example,<\/p>\n<div id=\"fs-id1170654905253\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-b6e1b1316c66553cad374a3591c13a04_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#125;&#123;&#56;&#125;&#94;&#123;&#50;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#114;&#101;&#97;&#100;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#39;&#125;&#56;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#115;&#113;&#117;&#97;&#114;&#101;&#100;&#39;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#54;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#54;&#52;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#105;&#115;&#32;&#99;&#97;&#108;&#108;&#101;&#100;&#32;&#116;&#104;&#101;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#109;&#97;&#116;&#104;&#105;&#116;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#115;&#113;&#117;&#97;&#114;&#101;&#125;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#111;&#102;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#56;&#46;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"40\" width=\"268\" style=\"vertical-align: -14px;\" \/><\/div>\n<p id=\"fs-id1170655353155\">Similarly, 121 is the square of 11, because <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-c98915ee12cb78b8c6e7c5c2b937fb3a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#49;&#49;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"24\" style=\"vertical-align: -1px;\" \/> is 121.<\/p>\n<div id=\"fs-id1170655158009\" data-type=\"note\">\n<div data-type=\"title\">\n<div class=\"textbox textbox--key-takeaways\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Square of a Number<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>If <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-5e29f42b1ff88fd9272ac7dc8bf372c9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#110;&#125;&#94;&#123;&#50;&#125;&#61;&#109;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"62\" style=\"vertical-align: -4px;\" \/> then <em data-effect=\"italics\">m<\/em> is the <strong data-effect=\"bold\">square<\/strong> of <em data-effect=\"italics\">n<\/em>.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1170654957861\">Complete the following table to show the squares of the counting numbers 1 through 15.<\/p>\n<p><span id=\"fs-id1170655354103\" data-type=\"media\" data-alt=\"There is a table with two rows and 17 columns. The first row reads from left to right Number, n, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, and 15. The second row reads from left to right Square, n squared, blank, blank, blank, blank, blank, blank, blank, 64, blank, blank, 121, blank, blank, blank, and blank.\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/06\/CNX_ElemAlg_Figure_01_08_016_img_new.jpg\" alt=\"There is a table with two rows and 17 columns. The first row reads from left to right Number, n, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, and 15. The second row reads from left to right Square, n squared, blank, blank, blank, blank, blank, blank, blank, 64, blank, blank, 121, blank, blank, blank, and blank.\" data-media-type=\"image\/jpeg\" \/><\/span><\/p>\n<p id=\"fs-id1170655105477\">The numbers in the second row are called perfect square numbers. It will be helpful to learn to recognize the perfect square numbers.<\/p>\n<p id=\"fs-id1170655202790\">The squares of the counting numbers are positive numbers. What about the squares of negative numbers? We know that when the signs of two numbers are the same, their product is positive. So the square of any negative number is also positive.<\/p>\n<div id=\"fs-id1170655229994\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-4d55a1cfe36af84a6a322fc6f2edb959_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#125;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#61;&#57;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#56;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#61;&#54;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#61;&#49;&#50;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#61;&#50;&#50;&#53;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"513\" style=\"vertical-align: -5px;\" \/><\/div>\n<p id=\"fs-id1170654940925\">Did you notice that these squares are the same as the squares of the positive numbers?<\/p>\n<p id=\"fs-id1170655007029\">Sometimes we will need to look at the relationship between numbers and their squares in reverse. Because <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-4c465a5fd7db52cdd10635f242e06d84_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#49;&#48;&#125;&#94;&#123;&#50;&#125;&#61;&#49;&#48;&#48;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"79\" style=\"vertical-align: -4px;\" \/> we say 100 is the square of 10. We also say that 10 is a <em data-effect=\"italics\">square root<\/em> of 100. A number whose square is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-6b41df788161942c6f98604d37de8098_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#109;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"15\" style=\"vertical-align: 0px;\" \/> is called a <span data-type=\"term\">square root<\/span> of <em data-effect=\"italics\">m<\/em>.<\/p>\n<div id=\"fs-id1170654966160\" data-type=\"note\">\n<div data-type=\"title\">\n<div class=\"textbox textbox--key-takeaways\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Square Root of a Number<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>If <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-5e29f42b1ff88fd9272ac7dc8bf372c9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#110;&#125;&#94;&#123;&#50;&#125;&#61;&#109;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"62\" style=\"vertical-align: -4px;\" \/> then <em data-effect=\"italics\">n<\/em> is a <strong data-effect=\"bold\">square root<\/strong> of <em data-effect=\"italics\">m<\/em>.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1170655151795\">Notice <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-3bd3ccc26a4a88f5a84134b6afc2774d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#61;&#49;&#48;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"103\" style=\"vertical-align: -4px;\" \/> also, so <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-24329a36a4e10288288979f77a565ca2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"31\" style=\"vertical-align: -1px;\" \/> is also a square root of 100. Therefore, both 10 and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-24329a36a4e10288288979f77a565ca2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"31\" style=\"vertical-align: -1px;\" \/> are square roots of 100.<\/p>\n<p id=\"fs-id1170655227720\">So, every positive number has two square roots\u2014one positive and one negative. What if we only wanted the positive square root of a positive number? The <span class=\"no-emphasis\" data-type=\"term\">radical sign<\/span>, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-2447480eb926a3c6e489624b0a8579e6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#109;&#125;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"33\" style=\"vertical-align: -4px;\" \/> denotes the positive square root. The positive square root is called the <span class=\"no-emphasis\" data-type=\"term\">principal square root<\/span>. When we use the radical sign that always means we want the principal square root.<\/p>\n<p id=\"fs-id1170655228645\">We also use the radical sign for the square root of zero. Because <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-6444a3ad0fa39919f49d0c2cbc284d31_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#48;&#125;&#94;&#123;&#50;&#125;&#61;&#48;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"53\" style=\"vertical-align: -4px;\" \/> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-d6e1b3453375b8d478d6961a0d9c26ad_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#48;&#125;&#61;&#48;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"59\" style=\"vertical-align: -2px;\" \/> Notice that zero has only one square root.<\/p>\n<div id=\"fs-id1170654985840\" data-type=\"note\">\n<div data-type=\"title\">\n<div class=\"textbox textbox--key-takeaways\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Square Root Notation<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p id=\"fs-id1170655111452\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-ff6e04829c3778dcab4453317cfa77ce_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#109;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"30\" style=\"vertical-align: -4px;\" \/> is read \u201cthe square root of <em data-effect=\"italics\">m<\/em>\u201d<\/p>\n<p><span data-type=\"media\" data-alt=\"A square root is given, with an arrow to the radical sign (it looks like a checkmark with a horizontal line extending from its long end) denoted radical sign and an arrow to the number under the radical sign, which is marked radicand.\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/06\/CNX_ElemAlg_Figure_01_08_001_img_new.jpg\" alt=\"A square root is given, with an arrow to the radical sign (it looks like a checkmark with a horizontal line extending from its long end) denoted radical sign and an arrow to the number under the radical sign, which is marked radicand.\" data-media-type=\"image\/jpeg\" \/><\/span><\/p>\n<p id=\"fs-id1170655207964\">If <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-4c7c8991cf54ee68b9a5ca8f8014af5b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#109;&#61;&#123;&#110;&#125;&#94;&#123;&#50;&#125;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"62\" style=\"vertical-align: -4px;\" \/> then <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-1144c0e1f4a31eb91b31a23122c558c4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#109;&#125;&#61;&#110;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"68\" style=\"vertical-align: -4px;\" \/> for <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-de0b3f268c78cbcbba48c42081539c2a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#110;&#92;&#103;&#101;&#32;&#48;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"47\" style=\"vertical-align: -3px;\" \/><\/p>\n<p id=\"fs-id1170654872607\">The square root of <em data-effect=\"italics\">m<\/em>, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-2447480eb926a3c6e489624b0a8579e6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#109;&#125;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"33\" style=\"vertical-align: -4px;\" \/> is the positive number whose square is <em data-effect=\"italics\">m<\/em>.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1170655230646\">Since 10 is the principal square root of 100, we write <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-f5687e7adbc139c31409a60b7c14f986_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#49;&#48;&#48;&#125;&#61;&#49;&#48;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"86\" style=\"vertical-align: -2px;\" \/> You may want to complete the following table to help you recognize square roots.<\/p>\n<p><span id=\"fs-id1170655162081\" data-type=\"media\" data-alt=\"There is a table with two rows and 15 columns. The first row reads from left to right square root of 1, square root of 4, square root of 9, square root of 16, square root of 25, square root of 36, square root of 49, square root of 64, square root of 81, square root of 100, square root of 121, square root of 144, square root of 169, square root of 196, and square root of 225. The second row consists of all blanks except for the tenth cell under the square root of 100, which reads 10.\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/06\/CNX_ElemAlg_Figure_01_08_017_img_new.jpg\" alt=\"There is a table with two rows and 15 columns. The first row reads from left to right square root of 1, square root of 4, square root of 9, square root of 16, square root of 25, square root of 36, square root of 49, square root of 64, square root of 81, square root of 100, square root of 121, square root of 144, square root of 169, square root of 196, and square root of 225. The second row consists of all blanks except for the tenth cell under the square root of 100, which reads 10.\" data-media-type=\"image\/jpeg\" \/><\/span><\/p>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">EXAMPLE 1<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1170655105180\" data-type=\"problem\">\n<p>Simplify: a) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-0c6a36f543576cfd4fa34a645e3eff65_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#50;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"32\" style=\"vertical-align: -2px;\" \/> b) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-33a1775be551b38f9add1619460fc332_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#49;&#50;&#49;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"44\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div id=\"fs-id1170655192917\" data-type=\"solution\">\n<div data-type=\"title\"><\/div>\n<div data-type=\"title\"><strong>Solution<\/strong><\/div>\n<table id=\"eip-105\" style=\"height: 64px; width: 100%;\" summary=\".\">\n<tbody>\n<tr style=\"height: 32px;\">\n<td style=\"height: 32px; width: 145.203px;\"><span class=\"token\">a)<\/span><span data-type=\"newline\"><br \/>\n<\/span>Since <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-fdc53830c9614b4d54be741e3490a7a2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#53;&#125;&#94;&#123;&#50;&#125;&#61;&#50;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"57\" style=\"vertical-align: 0px;\" \/><\/td>\n<td style=\"height: 32px; width: 285.297px;\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-ccaf0de5da17523f6f9e8ebf7bbc191b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#92;&#115;&#113;&#114;&#116;&#123;&#50;&#53;&#125;&#92;&#92;&#32;&#53;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"37\" width=\"32\" style=\"vertical-align: -11px;\" \/><\/td>\n<\/tr>\n<tr style=\"height: 32px;\">\n<td style=\"height: 32px; width: 145.203px;\"><span class=\"token\">b)<\/span><span data-type=\"newline\"><br \/>\n<\/span>Since <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-db18d56395a50dd9fb746a994410cbc7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#49;&#49;&#125;&#94;&#123;&#50;&#125;&#61;&#49;&#50;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"74\" style=\"vertical-align: -1px;\" \/><\/td>\n<td style=\"height: 32px; width: 285.297px;\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-fed4427c907315f7c1c262baa661175d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#92;&#115;&#113;&#114;&#116;&#123;&#49;&#50;&#49;&#125;&#92;&#92;&#32;&#49;&#49;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"41\" style=\"vertical-align: -12px;\" \/><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 1<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1170655222336\" data-type=\"problem\">\n<p id=\"fs-id1170655205656\">Simplify: a) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-98077b985c4d3e263c7a4b085c65329f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#51;&#54;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"32\" style=\"vertical-align: -2px;\" \/> b) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-354fa2b70e38338ad8783e474b3808d7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#49;&#54;&#57;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"44\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div id=\"fs-id1170655113123\" data-type=\"solution\">\n<details>\n<summary>Show answer<\/summary>\n<p id=\"fs-id1170654981435\"><span class=\"token\">a)<\/span> 6 b) 13<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<p>We know that every positive number has two square roots and the radical sign indicates the positive one. We write <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-f5687e7adbc139c31409a60b7c14f986_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#49;&#48;&#48;&#125;&#61;&#49;&#48;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"86\" style=\"vertical-align: -2px;\" \/> If we want to find the negative square root of a number, we place a negative in front of the radical sign. For example, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-bec4c2008437cc1952fc812ab5e36fa1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#45;&#125;&#92;&#115;&#113;&#114;&#116;&#123;&#49;&#48;&#48;&#125;&#61;&#45;&#49;&#48;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"107\" style=\"vertical-align: -2px;\" \/> We read <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-96ab7b235f9f2795073db790cdbb34d5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#45;&#125;&#92;&#115;&#113;&#114;&#116;&#123;&#49;&#48;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"48\" style=\"vertical-align: -2px;\" \/> as \u201cthe opposite of the square root of 100.\u201d<\/p>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">EXAMPLE 2<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1170654988401\" data-type=\"problem\">\n<p id=\"fs-id1170655112232\">Simplify: a) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-bf0fdb4aa93dceec27fe6786e9651b6c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#115;&#113;&#114;&#116;&#123;&#52;&#57;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"46\" style=\"vertical-align: -2px;\" \/> b) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-91953ab59eda1f37919c53e5fa975d00_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#115;&#113;&#114;&#116;&#123;&#49;&#52;&#52;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"58\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div id=\"fs-id1170655155062\" data-type=\"solution\">\n<div data-type=\"title\"><\/div>\n<div data-type=\"title\"><strong>Solution<\/strong><\/div>\n<table id=\"eip-995\" style=\"width: 100%;\" summary=\".\">\n<tbody>\n<tr>\n<td><span class=\"token\">a)<\/span><span data-type=\"newline\"><br \/>\n<\/span>The negative is in front of the radical sign.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-0de70aaa354173a09af4630c82d80c09_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#45;&#92;&#115;&#113;&#114;&#116;&#123;&#52;&#57;&#125;&#92;&#92;&#32;&#45;&#55;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"37\" width=\"46\" style=\"vertical-align: -11px;\" \/><\/td>\n<\/tr>\n<tr>\n<td><span class=\"token\">b)<\/span><span data-type=\"newline\"><br \/>\n<\/span>The negative is in front of the radical sign.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-75c48a6e574f54828b4415c5ef24cf12_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#45;&#92;&#115;&#113;&#114;&#116;&#123;&#49;&#52;&#52;&#125;&#92;&#92;&#32;&#45;&#49;&#50;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"55\" style=\"vertical-align: -12px;\" \/><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 2<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div data-type=\"problem\">\n<p id=\"fs-id1170654901222\">Simplify: a) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-876c68a6c3bfecd495ad3ed598360667_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#115;&#113;&#114;&#116;&#123;&#49;&#54;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"46\" style=\"vertical-align: -2px;\" \/> b) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-d7aa44ee6bd4314979d39540ed26ca87_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#115;&#113;&#114;&#116;&#123;&#50;&#50;&#53;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"58\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div id=\"fs-id1170655125652\" data-type=\"solution\">\n<details>\n<summary>Show answer<\/summary>\n<p><span class=\"token\">a) <\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-00b9cce9021441b203ec0271d72e6ba2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"22\" style=\"vertical-align: -1px;\" \/> b) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-0329d19b932005c996d83dbb9e09ffed_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"30\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1169148889441\" class=\"bc-section section\" data-depth=\"1\">\n<p id=\"fs-id1169149123590\">Can we simplify <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-9a0bed49fd0ad7849df3fd3768080c7e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#45;&#52;&#57;&#125;&#63;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"53\" style=\"vertical-align: -3px;\" \/> Is there a number whose square is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-5ae54e9bac1840e4405f42c3831370b3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#52;&#57;&#63;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"39\" style=\"vertical-align: 0px;\" \/><\/p>\n<div id=\"fs-id1169149004881\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-9cc0386bd0b48118bb441236470e78e9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#61;&#45;&#52;&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"80\" style=\"vertical-align: -5px;\" \/><\/div>\n<p id=\"fs-id1169148861223\">Any positive number squared is positive. Any negative number squared is positive. There is no real number equal to <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-4d39a9428d46e6afc02655d0a63c943c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#45;&#52;&#57;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"49\" style=\"vertical-align: -3px;\" \/> The square root of a negative number is not a real number.<\/p>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">EXAMPLE 3<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1169148820580\" data-type=\"problem\">\n<p id=\"fs-id1169148859897\">Simplify: a) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-1154b34d2f8ec68cebe7a9283e374a45_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#45;&#49;&#57;&#54;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"55\" style=\"vertical-align: -3px;\" \/> b) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-380ba2aaa11f3e763467e451b5eaa0c3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#115;&#113;&#114;&#116;&#123;&#54;&#52;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"49\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div id=\"fs-id1169144560004\" data-type=\"solution\">\n<p id=\"fs-id1169148836684\"><strong><span class=\"token\">Solution<\/span><\/strong><\/p>\n<p><span class=\"token\">a)<\/span><\/p>\n<table style=\"border-collapse: collapse; width: 100%; height: 48px;\">\n<tbody>\n<tr style=\"height: 16px;\">\n<td style=\"width: 50%; height: 16px;\"><\/td>\n<td style=\"width: 50%; height: 16px;\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-1154b34d2f8ec68cebe7a9283e374a45_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#45;&#49;&#57;&#54;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"55\" style=\"vertical-align: -3px;\" \/><\/td>\n<\/tr>\n<tr style=\"height: 32px;\">\n<td style=\"width: 50%; height: 32px;\">There is no real number whose square is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-26b977db705619fb60f28fe3c9f79206_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;&#57;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"40\" style=\"vertical-align: 0px;\" \/>.<\/td>\n<td style=\"width: 50%; height: 32px;\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-1154b34d2f8ec68cebe7a9283e374a45_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#45;&#49;&#57;&#54;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"55\" style=\"vertical-align: -3px;\" \/>.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><span class=\"token\">b)<\/span><\/p>\n<div data-type=\"newline\">\n<table style=\"border-collapse: collapse; width: 100%;\">\n<tbody>\n<tr>\n<td style=\"width: 50%;\"><\/td>\n<td style=\"width: 50%;\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-3b6616169b7490db4d681c4f04663aa8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#115;&#113;&#114;&#116;&#123;&#54;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"46\" style=\"vertical-align: -2px;\" \/><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 50%;\">The negative is in front of the radical.<\/td>\n<td style=\"width: 50%;\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-ca32393b1b5af7c55a95d89cf9d610f2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#56;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"22\" style=\"vertical-align: 0px;\" \/><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 3<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1169146638717\" data-type=\"problem\">\n<p id=\"fs-id1169148929622\">Simplify: a) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-8c3b9ea68a1a2eb8f9f1cb89bf27632d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#45;&#49;&#54;&#57;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"55\" style=\"vertical-align: -3px;\" \/> b) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-b8ac7f0c2949653b991cd7c9e84ebaab_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#115;&#113;&#114;&#116;&#123;&#56;&#49;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"49\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div id=\"fs-id1169144491036\" data-type=\"solution\">\n<details open=\"open\">\n<summary>Show answer<\/summary>\n<p id=\"fs-id1169146823304\"><span class=\"token\">a)<\/span> not a real number b) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-9672181aec15f8334b80ada7de4e4fc0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"22\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1169148859228\">So far we have only talked about squares and square roots. Let\u2019s now extend our work to include higher powers and higher roots.<\/p>\n<p id=\"fs-id1169149014185\">Let\u2019s review some vocabulary first.<\/p>\n<p id=\"fs-id1169148861279\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-ffb78f9d56946d8f4fc7490aa5199ba9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#99;&#99;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#87;&#101;&#32;&#119;&#114;&#105;&#116;&#101;&#58;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#87;&#101;&#32;&#115;&#97;&#121;&#58;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#123;&#110;&#125;&#94;&#123;&#50;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#110;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#115;&#113;&#117;&#97;&#114;&#101;&#100;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#123;&#110;&#125;&#94;&#123;&#51;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#110;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#99;&#117;&#98;&#101;&#100;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#123;&#110;&#125;&#94;&#123;&#52;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#110;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#116;&#111;&#32;&#116;&#104;&#101;&#32;&#102;&#111;&#117;&#114;&#116;&#104;&#32;&#112;&#111;&#119;&#101;&#114;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#123;&#110;&#125;&#94;&#123;&#53;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#110;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#116;&#111;&#32;&#116;&#104;&#101;&#32;&#102;&#105;&#102;&#116;&#104;&#32;&#112;&#111;&#119;&#101;&#114;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"104\" width=\"322\" style=\"vertical-align: -48px;\" \/><\/p>\n<p id=\"fs-id1169148823401\">The terms \u2018squared\u2019 and \u2018cubed\u2019 come from the formulas for area of a square and volume of a cube.<\/p>\n<p id=\"fs-id1169149038030\">It will be helpful to have a table of the powers of the integers from \u22125 to 5. See <a class=\"autogenerated-content\" href=\"#CNX_IntAlg_Figure_08_01_002\">(Table 1)<\/a>.<\/p>\n<div id=\"CNX_IntAlg_Figure_08_01_002\" class=\"bc-figure figure\">\n<figure style=\"width: 806px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/06\/CNX_IntAlg_Figure_08_01_002_img_new.jpg\" alt=\"The figure contains two tables. The first table has 9 rows and 5 columns. The first row is a header row with the headers \u201cNumber\u201d, \u201cSquare\u201d, \u201cCube\u201d, \u201cFourth power\u201d, and \u201cFifth power\u201d. The second row contains the expressions n, n squared, n cubed, n to the fourth power, and n to the fifth power. The third row contains the number 1 in each column. The fourth row contains the numbers 2, 4, 8, 16, 32. The fifth row contains the numbers 3, 9, 27, 81, 243. The sixth row contains the numbers 4, 16, 64, 256, 1024. The seventh row contains the numbers 5, 25, 125 625, 3125. The eighth row contains the expressions x, x squared, x cubed, x to the fourth power, and x to the fifth power. The last row contains the expressions x squared, x to the fourth power, x to the sixth power, x to the eighth power, and x to the tenth power. The second table has 7 rows and 5 columns. The first row is a header row with the headers \u201cNumber\u201d, \u201cSquare\u201d, \u201cCube\u201d, \u201cFourth power\u201d, and \u201cFifth power\u201d. The second row contains the expressions n, n squared, n cubed, n to the fourth power, and n to the fifth power. The third row contains the numbers negative 1, 1 negative 1, 1, negative 1. The fourth row contains the numbers negative 2, 4, negative 8, 16, negative 32. The fifth row contains the numbers negative 3, 9, negative 27, 81, negative 243. The sixth row contains the numbers negative 4, 16, negative 64, 256, negative 1024. The last row contains the numbers negative 5, 25, negative 125, 625, negative 3125.\" width=\"806\" height=\"189\" data-media-type=\"image\/jpeg\" \/><figcaption class=\"wp-caption-text\">Table 1<\/figcaption><\/figure>\n<\/div>\n<p id=\"fs-id1169148994337\">Notice the signs in the table. All powers of positive numbers are positive, of course. But when we have a negative number, the <em data-effect=\"italics\">even<\/em> powers are positive and the <em data-effect=\"italics\">odd<\/em> powers are negative. We\u2019ll copy the row with the powers of \u22122 to help you see this.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_IntAlg_Figure_08_01_003_img_new.jpg\" alt=\"The image contains a table with 2 rows and 5 columns. The first row contains the expressions n, n squared, n cubed, n to the fourth power, and n to the fifth power. The second row contains the numbers negative 2, 4, negative 8, 16, negative 32. Arrows point to the second and fourth columns with the label \u201cEven power Positive result\u201d. Arrows point to the first third and fifth columns with the label \u201cOdd power Negative result\u201d.\" width=\"375\" height=\"103\" data-media-type=\"image\/jpeg\" \/><\/p>\n<p id=\"fs-id1169149155980\">We will now extend the square root definition to higher roots.<\/p>\n<div id=\"fs-id1169149108455\" data-type=\"note\">\n<div data-type=\"title\"><\/div>\n<div data-type=\"title\">\n<div class=\"textbox textbox--key-takeaways\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\"><em data-effect=\"italics\">n<\/em><sup>th<\/sup> Root of a Number<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-bce73c1457e00d949ea81a709b17a46c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#125;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#73;&#102;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#98;&#125;&#94;&#123;&#110;&#125;&#61;&#97;&#44;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#116;&#104;&#101;&#110;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#98;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#105;&#115;&#32;&#97;&#110;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#110;&#125;&#94;&#123;&#116;&#104;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#114;&#111;&#111;&#116;&#32;&#111;&#102;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#97;&#46;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#84;&#104;&#101;&#32;&#112;&#114;&#105;&#110;&#99;&#105;&#112;&#97;&#108;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#110;&#125;&#94;&#123;&#116;&#104;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#114;&#111;&#111;&#116;&#32;&#111;&#102;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#97;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#105;&#115;&#32;&#119;&#114;&#105;&#116;&#116;&#101;&#110;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#115;&#113;&#114;&#116;&#091;&#110;&#093;&#123;&#97;&#125;&#46;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#110;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#105;&#115;&#32;&#99;&#97;&#108;&#108;&#101;&#100;&#32;&#116;&#104;&#101;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#105;&#110;&#100;&#101;&#120;&#125;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#111;&#102;&#32;&#116;&#104;&#101;&#32;&#114;&#97;&#100;&#105;&#99;&#97;&#108;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"60\" width=\"312\" style=\"vertical-align: -45px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1169149368114\">Just like we use the word \u2018cubed\u2019 for <em data-effect=\"italics\">b<\/em><sup>3<\/sup>, we use the term \u2018cube root\u2019 for <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-d9b8fcd747ef7e614330ec91c1c0787d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#091;&#51;&#093;&#123;&#97;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"27\" style=\"vertical-align: -4px;\" \/><\/p>\n<p id=\"fs-id1169149310383\">We can refer to <a class=\"autogenerated-content\" href=\"#CNX_IntAlg_Figure_08_01_002\">(Table 1)<\/a> to help find higher roots.<\/p>\n<div id=\"fs-id1169146874884\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-ae49f158b7d5f39bdd200b083c538371_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#123;&#52;&#125;&#94;&#123;&#51;&#125;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#54;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#123;&#51;&#125;&#94;&#123;&#52;&#125;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#56;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#53;&#125;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#45;&#51;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#54;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#115;&#113;&#114;&#116;&#091;&#51;&#093;&#123;&#54;&#52;&#125;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#115;&#113;&#114;&#116;&#091;&#52;&#093;&#123;&#56;&#49;&#125;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#115;&#113;&#114;&#116;&#091;&#53;&#093;&#123;&#45;&#51;&#50;&#125;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#45;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"66\" width=\"359\" style=\"vertical-align: -28px;\" \/><\/div>\n<p id=\"fs-id1169148827466\">Could we have an even root of a negative number? We know that the square root of a negative number is not a real number. The same is true for any even root. <em data-effect=\"italics\">Even<\/em> roots of negative numbers are not real numbers. <em data-effect=\"italics\">Odd<\/em> roots of negative numbers are real numbers.<\/p>\n<div id=\"fs-id1169147027485\" data-type=\"note\">\n<div data-type=\"title\">\n<div class=\"textbox textbox--key-takeaways\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Properties of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-1c6b7326bec5fc086e8335bb7e382f30_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#091;&#110;&#093;&#123;&#97;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"23\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p id=\"fs-id1169149326612\">When <em data-effect=\"italics\">n<\/em> is an even number and<\/p>\n<ul id=\"fs-id1169149002308\" data-bullet-style=\"bullet\">\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-f75133c2b77e2f59b82bc9d582b4e840_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#92;&#103;&#101;&#32;&#48;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"46\" style=\"vertical-align: -4px;\" \/> then <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-1c6b7326bec5fc086e8335bb7e382f30_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#091;&#110;&#093;&#123;&#97;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"23\" style=\"vertical-align: -4px;\" \/> is a real number.<\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-36c26ea6de309fb94494920502960413_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#60;&#48;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"46\" style=\"vertical-align: -4px;\" \/> then <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-1c6b7326bec5fc086e8335bb7e382f30_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#091;&#110;&#093;&#123;&#97;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"23\" style=\"vertical-align: -4px;\" \/> is not a real number.<\/li>\n<\/ul>\n<p id=\"fs-id1169148898687\">When <em data-effect=\"italics\">n<\/em> is an odd number, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-1c6b7326bec5fc086e8335bb7e382f30_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#091;&#110;&#093;&#123;&#97;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"23\" style=\"vertical-align: -4px;\" \/> is a real number for all values of <em data-effect=\"italics\">a<\/em>.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1169149096279\">We will apply these properties in the next two examples.<\/p>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">EXAMPLE 4<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1169146654571\" data-type=\"problem\">\n<p id=\"fs-id1169148924022\">Simplify: a) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-206d8539b1c6adeef121d7010ed1c6fc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#091;&#51;&#093;&#123;&#54;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"32\" style=\"vertical-align: -2px;\" \/> b) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-fe9b91eb5b6fd4a4bcb7545fa44d8399_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#091;&#52;&#093;&#123;&#56;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"32\" style=\"vertical-align: -2px;\" \/> c) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-7059889d78c7615fdfb2465801eb9153_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#091;&#53;&#093;&#123;&#51;&#50;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"36\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div id=\"fs-id1169146648492\" data-type=\"solution\">\n<p><strong><span class=\"token\">Solution<\/span><\/strong><\/p>\n<p id=\"fs-id1169149329984\"><span class=\"token\">a)<\/span><\/p>\n<table style=\"border-collapse: collapse; width: 100%; height: 32px;\">\n<tbody>\n<tr style=\"height: 16px;\">\n<td style=\"width: 50%; height: 16px;\"><\/td>\n<td style=\"width: 50%; height: 16px;\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-206d8539b1c6adeef121d7010ed1c6fc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#091;&#51;&#093;&#123;&#54;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"32\" style=\"vertical-align: -2px;\" \/><\/td>\n<\/tr>\n<tr style=\"height: 16px;\">\n<td style=\"width: 50%; height: 16px;\">Since <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-1d0c9e2d24bb4bbfe20dd2637c6573fe_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#52;&#125;&#94;&#123;&#51;&#125;&#61;&#54;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"58\" style=\"vertical-align: -1px;\" \/>.<\/td>\n<td style=\"width: 50%; height: 16px;\">\u00a04<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><span class=\"token\">b)<\/span><\/p>\n<table style=\"border-collapse: collapse; width: 100%;\">\n<tbody>\n<tr>\n<td style=\"width: 50%;\"><\/td>\n<td style=\"width: 50%;\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-fe9b91eb5b6fd4a4bcb7545fa44d8399_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#091;&#52;&#093;&#123;&#56;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"32\" style=\"vertical-align: -2px;\" \/><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 50%;\">Since <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-0ae8fcbb71a09f3293cc8d869af359ca_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#52;&#125;&#61;&#56;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"70\" style=\"vertical-align: -5px;\" \/><\/td>\n<td style=\"width: 50%;\">\u00a03<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><span class=\"token\">c)<\/span><\/p>\n<table style=\"border-collapse: collapse; width: 100%;\">\n<tbody>\n<tr>\n<td style=\"width: 50%;\"><\/td>\n<td style=\"width: 50%;\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-448871fbd9d51dcf7f05494cf8ff32f7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#091;&#53;&#093;&#123;&#51;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"32\" style=\"vertical-align: -2px;\" \/><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 50%;\">Since <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-97810bcede5c3755d6c61eada243f091_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#53;&#125;&#61;&#51;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"70\" style=\"vertical-align: -5px;\" \/><\/td>\n<td style=\"width: 50%;\">2<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 4<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1169148843404\" data-type=\"problem\">\n<p id=\"fs-id1169148929680\">Simplify: a) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-4a034c4aef4e5bf9fe610952f7ab4cd3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#091;&#51;&#093;&#123;&#50;&#55;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"32\" style=\"vertical-align: -2px;\" \/> b) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-e8fc6228ef8608f4a53018ae413a46f5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#091;&#52;&#093;&#123;&#50;&#53;&#54;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"41\" style=\"vertical-align: -2px;\" \/> c) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-e4de30de6d2cdd366183a4b2a3fef339_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#091;&#53;&#093;&#123;&#50;&#52;&#51;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"44\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div id=\"fs-id1169146658975\" data-type=\"solution\">\n<details open=\"open\">\n<summary>Show answer<\/summary>\n<p id=\"fs-id1169149157689\"><span class=\"token\">a)<\/span> 3 b) 4 c) 3<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1165927757740\">In this example be alert for the negative signs as well as even and odd powers.<\/p>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">EXAMPLE 5<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1169149204958\" data-type=\"problem\">\n<p id=\"fs-id1169144560227\">Simplify: a) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-5767974107a2c199d59ce16c113b6869_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#091;&#51;&#093;&#123;&#45;&#49;&#50;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"55\" style=\"vertical-align: -3px;\" \/> b) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-6c7687fd0928c89a07eeb730c18c1039_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#091;&#52;&#093;&#123;&#45;&#49;&#54;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"46\" style=\"vertical-align: -3px;\" \/> c) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-0c60a777483598b3cf552cc9e713c7e2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#091;&#53;&#093;&#123;&#45;&#50;&#52;&#51;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"58\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/div>\n<div id=\"fs-id1169149065347\" data-type=\"solution\">\n<p><strong><span class=\"token\">Solution<\/span><\/strong><\/p>\n<p id=\"fs-id1169146662763\"><span class=\"token\">a)<\/span><\/p>\n<table style=\"border-collapse: collapse; width: 100%;\">\n<tbody>\n<tr>\n<td style=\"width: 50%;\"><\/td>\n<td style=\"width: 50%;\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-5767974107a2c199d59ce16c113b6869_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#091;&#51;&#093;&#123;&#45;&#49;&#50;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"55\" style=\"vertical-align: -3px;\" \/><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 50%;\">Since <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-256fefe92530789b72fd65731e3c116e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#51;&#125;&#61;&#45;&#49;&#50;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"107\" style=\"vertical-align: -5px;\" \/><\/td>\n<td style=\"width: 50%;\">\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-7b5b9d9f382b11767d19f257afca0019_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"21\" style=\"vertical-align: 0px;\" \/><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><span class=\"token\">b)<\/span><\/p>\n<table style=\"border-collapse: collapse; width: 100%;\">\n<tbody>\n<tr>\n<td style=\"width: 50%;\"><\/td>\n<td style=\"width: 50%;\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-6c7687fd0928c89a07eeb730c18c1039_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#091;&#52;&#093;&#123;&#45;&#49;&#54;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"46\" style=\"vertical-align: -3px;\" \/><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 50%;\">Think, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-2b84826df777a2c36f071fd148796525_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#63;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#52;&#125;&#61;&#45;&#49;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"84\" style=\"vertical-align: -5px;\" \/>. No real number raised to the fourth power is negative.<\/td>\n<td style=\"width: 50%;\">Not a real number.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><span class=\"token\">c)<\/span><\/p>\n<table style=\"border-collapse: collapse; width: 100%;\">\n<tbody>\n<tr>\n<td style=\"width: 50%;\"><\/td>\n<td style=\"width: 50%;\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-5e7a0aba2027796da64ab5676358198d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#091;&#53;&#093;&#123;&#45;&#50;&#52;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"55\" style=\"vertical-align: -3px;\" \/><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 50%;\">Since <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-79f49350bfc94657e5744a63a7ee6a06_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#108;&#101;&#102;&#116;&#40;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#53;&#125;&#61;&#45;&#50;&#52;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"133\" style=\"vertical-align: -5px;\" \/>.<\/td>\n<td style=\"width: 50%;\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-470cb162cf92c55d139f4f69216225e5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"22\" style=\"vertical-align: 0px;\" \/><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 5<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1169144873604\" data-type=\"problem\">\n<p id=\"fs-id1169149349661\">Simplify: a) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-5bcdc974578f76a36b8f5554ea50a1e6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#091;&#51;&#093;&#123;&#45;&#50;&#55;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"46\" style=\"vertical-align: -3px;\" \/> b) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-b3d7f314c676fa57d90fb40fb048af5b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#091;&#52;&#093;&#123;&#45;&#50;&#53;&#54;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"55\" style=\"vertical-align: -3px;\" \/> c) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-7a15d8df382b390432fbc8e66ddb55a6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#091;&#53;&#093;&#123;&#45;&#51;&#50;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"49\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/div>\n<div id=\"fs-id1169144379098\" data-type=\"solution\">\n<details open=\"open\">\n<summary>Show answer<\/summary>\n<p id=\"fs-id1169148990626\"><span class=\"token\">a) <\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-470cb162cf92c55d139f4f69216225e5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"22\" style=\"vertical-align: 0px;\" \/> b)\u00a0 not real c)\u00a0 <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-17c33e2329e29a62a80ad2b547b4753d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"21\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<p><span style=\"font-family: Helvetica, Arial, 'GFS Neohellenic', sans-serif; font-size: 1.2em; font-weight: bold;\">Estimate and Approximate Roots<\/span><\/p>\n<\/div>\n<div id=\"fs-id1169148831408\" class=\"bc-section section\" data-depth=\"1\">\n<p id=\"fs-id1169149288530\">When we see a number with a radical sign, we often don\u2019t think about its numerical value. While we probably know that the <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-978264354badcbc9000159bce3be8cc9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#52;&#125;&#61;&#50;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"59\" style=\"vertical-align: -4px;\" \/> what is the value of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-4b4e70cc33e57ada16f421cbe82312bc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#091;&#093;&#123;&#50;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"32\" style=\"vertical-align: -2px;\" \/> or <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-ee11d625c08012bbe22beb83f20954a6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#091;&#51;&#093;&#123;&#53;&#48;&#125;&#63;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"40\" style=\"vertical-align: -2px;\" \/> In some situations a quick estimate is meaningful and in others it is convenient to have a decimal approximation.<\/p>\n<p id=\"fs-id1169149092400\">To get a numerical estimate of a square root, we look for <span class=\"no-emphasis\" data-type=\"term\">perfect square<\/span> numbers closest to the radicand. To find an estimate of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-62ef991bc3215dc1cc90e79aaa7a0b42_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#49;&#49;&#125;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"36\" style=\"vertical-align: -4px;\" \/> we see 11 is between perfect square numbers 9 and 16, <em data-effect=\"italics\">closer<\/em> to 9. Its square root then will be between 3 and 4, but closer to 3.<\/p>\n<p><span id=\"fs-id1169147066070\" data-type=\"media\" data-alt=\"The figure contains two tables. The first table has 5 rows and 2 columns. The first row is a header row with the headers \u201cNumber\u201d and \u201cSquare Root\u201d. The second row has the numbers 4 and 2. The third row is 9 and 3. The fourth row is 16 and 4. The last row is 25 and 5. A callout containing the number 11 is directed between the 9 and 16 in the first column. Another callout containing the number square root of 11 is directed between the 3 and 4 of the second column. Below the table are the inequalities 9 is less than 11 is less than 16 and 3 is less than square root of 11 is less than 4. The second table has 5 rows and 2 columns. The first row is a header row with the headers \u201cNumber\u201d and \u201cCube Root\u201d. The second row has the numbers 8 and 2. The third row is 27 and 3. The fourth row is 64 and 4. The last row is 125 and 5. A callout containing the number 91 is directed between the 64 and 125 in the first column. Another callout containing the number cube root of 91 is directed between the 4 and 5 of the second column. Below the table are the inequalities 64 is less than 91 is less than 125 and 4 is less than cube root of 91 is less than 5.\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_IntAlg_Figure_08_01_004_img_new.jpg\" alt=\"The figure contains two tables. The first table has 5 rows and 2 columns. The first row is a header row with the headers \u201cNumber\u201d and \u201cSquare Root\u201d. The second row has the numbers 4 and 2. The third row is 9 and 3. The fourth row is 16 and 4. The last row is 25 and 5. A callout containing the number 11 is directed between the 9 and 16 in the first column. Another callout containing the number square root of 11 is directed between the 3 and 4 of the second column. Below the table are the inequalities 9 is less than 11 is less than 16 and 3 is less than square root of 11 is less than 4. The second table has 5 rows and 2 columns. The first row is a header row with the headers \u201cNumber\u201d and \u201cCube Root\u201d. The second row has the numbers 8 and 2. The third row is 27 and 3. The fourth row is 64 and 4. The last row is 125 and 5. A callout containing the number 91 is directed between the 64 and 125 in the first column. Another callout containing the number cube root of 91 is directed between the 4 and 5 of the second column. Below the table are the inequalities 64 is less than 91 is less than 125 and 4 is less than cube root of 91 is less than 5.\" data-media-type=\"image\/jpeg\" \/><\/span><\/p>\n<p id=\"fs-id1169148932974\">Similarly, to estimate <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-3ffb4e6d51d3901a900818f17f8974d2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#091;&#51;&#093;&#123;&#57;&#49;&#125;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"36\" style=\"vertical-align: -4px;\" \/> we see 91 is between perfect cube numbers 64 and 125. The cube root then will be between 4 and 5.<\/p>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">EXAMPLE 6<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1169149031465\" data-type=\"problem\">\n<p id=\"fs-id1169149037560\">Estimate each root between two consecutive whole numbers: a) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-8a3cfe68431f3d872dc27abae39559db_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#49;&#48;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"41\" style=\"vertical-align: -2px;\" \/> b) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-233a14ee1d23c09b4b2a851a9ab7d4b2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#091;&#51;&#093;&#123;&#52;&#51;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"36\" style=\"vertical-align: -2px;\" \/><\/p>\n<p><strong><span class=\"token\">Solution<\/span><\/strong><\/p>\n<\/div>\n<div id=\"fs-id1169149109449\" data-type=\"solution\">\n<p id=\"fs-id1169149029219\"><span class=\"token\">a)<\/span> Think of the perfect square numbers closest to 105. Make a small table of these perfect squares and their squares roots.<\/p>\n<table id=\"fs-id1169149097063\" style=\"width: 100%;\" summary=\"The table has 5 rows and 2 columns. The first row is a header row with the headers \u201cNumber\u201d and \u201cSquare Root\u201d. The second row has the numbers 81and 9. The third row is 100 and 10. The fourth row is 121 and 11. The last row is 144 and 12. A callout containing the number 105 is directed between the 100 and 121 in the first column. Another callout containing the number square root of 105 is directed between the 10 and 11 of the second column. Below the table are the inequalities 100 is less than 105 is less than 121 and 10 is less than square root of 105 is less than 11.\" data-label=\"\">\n<tbody>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span id=\"fs-id1169149014939\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_IntAlg_Figure_08_01_005a_img.jpg\" alt=\".\" data-media-type=\"image\/jpeg\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span id=\"fs-id1169148870813\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_IntAlg_Figure_08_01_005b_img.jpg\" alt=\".\" data-media-type=\"image\/jpeg\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Locate 105 between two consecutive perfect squares.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span id=\"fs-id1169144379992\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_IntAlg_Figure_08_01_005c_img.jpg\" alt=\".\" data-media-type=\"image\/jpeg\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-8a3cfe68431f3d872dc27abae39559db_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#49;&#48;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"41\" style=\"vertical-align: -2px;\" \/> is between their square roots.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span id=\"fs-id1169149101962\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_IntAlg_Figure_08_01_005d_img.jpg\" alt=\".\" data-media-type=\"image\/jpeg\" \/><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1169148993521\"><span class=\"token\">b)<\/span> Similarly we locate 43 between two perfect cube numbers.<\/p>\n<table id=\"fs-id1169148970427\" style=\"width: 100%;\" summary=\"The table has 5 rows and 2 columns. The first row is a header row with the headers \u201cNumber\u201d and \u201cCube Root\u201d. The second row has the numbers 8 and 2. The third row is 27 and 3. The fourth row is 64 and 4. The last row is 125 and 5. A callout containing the number 43 is directed between the 27 and 64 in the first column. Another callout containing the number cube root of 43 is directed between the 3 and 4 of the second column. Below the table are the inequalities 27 is less than 43 is less than 64 and 3 is less than cube root of 43 is less than 4.\" data-label=\"\">\n<tbody>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span id=\"fs-id1169149115401\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_IntAlg_Figure_08_01_006a_img.jpg\" alt=\".\" data-media-type=\"image\/jpeg\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span id=\"fs-id1169149005295\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_IntAlg_Figure_08_01_006b_img.jpg\" alt=\".\" data-media-type=\"image\/jpeg\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Locate 43 between two consecutive perfect cubes.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span id=\"fs-id1169149108611\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_IntAlg_Figure_08_01_006c_img.jpg\" alt=\".\" data-media-type=\"image\/jpeg\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-ea2a1a75ad206e1bf11b3db3d90f3db2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#091;&#51;&#093;&#123;&#52;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"32\" style=\"vertical-align: -2px;\" \/> is between their cube roots.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span id=\"fs-id1169148992327\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_IntAlg_Figure_08_01_006d_img.jpg\" alt=\".\" data-media-type=\"image\/jpeg\" \/><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 6<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1169148915764\" data-type=\"problem\">\n<p id=\"fs-id1169149105146\">Estimate each root between two consecutive whole numbers:<\/p>\n<p id=\"fs-id1169149144160\"><span class=\"token\">a) <\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-f91bfc6e14c44c3d4fe6b0d8ed08a4ab_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#51;&#56;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"32\" style=\"vertical-align: -2px;\" \/>b)\u00a0 <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-800504f340bdfd3007b45a314f114c6a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#091;&#51;&#093;&#123;&#57;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"32\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div id=\"fs-id1169148898667\" data-type=\"solution\">\n<details open=\"open\">\n<summary>Show answer<\/summary>\n<p id=\"fs-id1169149117873\"><span class=\"token\">a) <\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-86734a9e43fa4ac31d10cf66bf78cfe5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#54;&#60;&#92;&#115;&#113;&#114;&#116;&#123;&#51;&#56;&#125;&#60;&#55;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"98\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/details>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">b) <\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-18f31454b3f084dc4aef2e3250fdf02c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#60;&#92;&#115;&#113;&#114;&#116;&#091;&#51;&#093;&#123;&#57;&#51;&#125;&#60;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"98\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1169148871159\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-6857\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/06\/business-color_calculator_icon-icons.com_53466.png\" alt=\"\" width=\"18\" height=\"18\" \/> There are mathematical methods to approximate square roots, but nowadays most people use a calculator to find square roots. To find a square root you will use the <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-b9f7bc96a9743d361c5f388ee3a20b06_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#120;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"25\" style=\"vertical-align: -4px;\" \/> key on your calculator. To find a cube root, or any root with higher index, you will use the <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-20fb7e6bd15ecd2fe82c65707eaa5481_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#091;&#121;&#093;&#123;&#120;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"24\" style=\"vertical-align: -4px;\" \/> key.<\/p>\n<p id=\"fs-id1169146646664\">When you use these keys, you get an approximate value. It is an approximation, accurate to the number of digits shown on your calculator\u2019s display. The symbol for an approximation is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-4c5427ea5fb7ebae9eb00565bcb39791_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#97;&#112;&#112;&#114;&#111;&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"9\" width=\"12\" style=\"vertical-align: 0px;\" \/> and it is read \u2018approximately\u2019.<\/p>\n<p id=\"fs-id1169144382250\">Suppose your calculator has a 10 digit display. You would see that<\/p>\n<div id=\"fs-id1169149014395\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-166ef812bc578c50c3411e58278c28ca_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#115;&#113;&#114;&#116;&#123;&#53;&#125;&#92;&#97;&#112;&#112;&#114;&#111;&#120;&#32;&#50;&#46;&#50;&#51;&#54;&#48;&#54;&#55;&#57;&#55;&#56;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#114;&#111;&#117;&#110;&#100;&#101;&#100;&#32;&#116;&#111;&#32;&#116;&#119;&#111;&#32;&#100;&#101;&#99;&#105;&#109;&#97;&#108;&#32;&#112;&#108;&#97;&#99;&#101;&#115;&#32;&#105;&#115;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#115;&#113;&#114;&#116;&#123;&#53;&#125;&#92;&#97;&#112;&#112;&#114;&#111;&#120;&#32;&#50;&#46;&#50;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#115;&#113;&#114;&#116;&#091;&#52;&#093;&#123;&#57;&#51;&#125;&#92;&#97;&#112;&#112;&#114;&#111;&#120;&#32;&#51;&#46;&#49;&#48;&#53;&#52;&#50;&#50;&#55;&#57;&#57;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#114;&#111;&#117;&#110;&#100;&#101;&#100;&#32;&#116;&#111;&#32;&#116;&#119;&#111;&#32;&#100;&#101;&#99;&#105;&#109;&#97;&#108;&#32;&#112;&#108;&#97;&#99;&#101;&#115;&#32;&#105;&#115;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#115;&#113;&#114;&#116;&#091;&#52;&#093;&#123;&#57;&#51;&#125;&#92;&#97;&#112;&#112;&#114;&#111;&#120;&#32;&#51;&#46;&#49;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"497\" style=\"vertical-align: -16px;\" \/><\/div>\n<p id=\"fs-id1169149303690\">How do we know these values are approximations and not the exact values? Look at what happens when we square them:<\/p>\n<div id=\"fs-id1169149121409\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-7c97fe0148c6fa141f7713f3938f291b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#46;&#50;&#51;&#54;&#48;&#54;&#55;&#57;&#55;&#56;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#53;&#46;&#48;&#48;&#48;&#48;&#48;&#48;&#48;&#48;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#46;&#50;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#53;&#46;&#48;&#49;&#55;&#54;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#52;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#46;&#49;&#48;&#53;&#52;&#50;&#50;&#55;&#57;&#57;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#52;&#125;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#57;&#50;&#46;&#57;&#57;&#57;&#57;&#57;&#57;&#57;&#57;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#46;&#49;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#52;&#125;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#57;&#51;&#46;&#53;&#52;&#57;&#53;&#49;&#56;&#52;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"45\" width=\"605\" style=\"vertical-align: -17px;\" \/><\/div>\n<p id=\"fs-id1169148926161\">Their squares are close to 5, but are not exactly equal to 5. The fourth powers are close to 93, but not equal to 93.<\/p>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">EXAMPLE 7<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1169149294811\" data-type=\"problem\">\n<p id=\"fs-id1169149344377\">Round to two decimal places: a) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-cd14f5d843f634160162c2af6617f5d0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#49;&#55;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"32\" style=\"vertical-align: -2px;\" \/> b) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-260000d3c73d83447cfbc213342665ae_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#091;&#51;&#093;&#123;&#52;&#57;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"32\" style=\"vertical-align: -2px;\" \/> c) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-ad3bcc052654357f79c7a50af43388fa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#091;&#52;&#093;&#123;&#53;&#49;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"36\" style=\"vertical-align: -2px;\" \/><\/p>\n<p><strong><span class=\"token\">Solution<\/span><\/strong><\/p>\n<\/div>\n<div id=\"fs-id1169149360529\" data-type=\"solution\">\n<p id=\"fs-id1169149305330\"><span class=\"token\">a)<\/span><\/p>\n<table style=\"border-collapse: collapse; width: 100%;\">\n<tbody>\n<tr>\n<td style=\"width: 50%;\"><\/td>\n<td style=\"width: 50%;\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-cd14f5d843f634160162c2af6617f5d0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#49;&#55;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"32\" style=\"vertical-align: -2px;\" \/><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 50%;\">Use the calculator square root key.<\/td>\n<td style=\"width: 50%;\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-5b593ffba0dd304dad1ab0ad19ecdf3f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#46;&#49;&#50;&#51;&#49;&#48;&#53;&#54;&#50;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"95\" style=\"vertical-align: -1px;\" \/>&#8230;<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 50%;\">Round to two decimal places.<\/td>\n<td style=\"width: 50%;\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-584a17025bc80f8cf31fe466e32a36e2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#46;&#49;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"31\" style=\"vertical-align: -1px;\" \/><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 50%;\"><\/td>\n<td style=\"width: 50%;\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-a4199d9adaeb7314ba824aad88919f1f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#49;&#55;&#125;&#92;&#97;&#112;&#112;&#114;&#111;&#120;&#32;&#52;&#46;&#49;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"86\" style=\"vertical-align: -2px;\" \/><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><span class=\"token\">b)<\/span><\/p>\n<table style=\"border-collapse: collapse; width: 100%; height: 64px;\">\n<tbody>\n<tr style=\"height: 16px;\">\n<td style=\"width: 50%; height: 16px;\"><\/td>\n<td style=\"width: 50%; height: 16px;\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-260000d3c73d83447cfbc213342665ae_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#091;&#51;&#093;&#123;&#52;&#57;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"32\" style=\"vertical-align: -2px;\" \/><\/td>\n<\/tr>\n<tr style=\"height: 16px;\">\n<td style=\"width: 50%; height: 16px;\">Use the calculator <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-20fb7e6bd15ecd2fe82c65707eaa5481_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#091;&#121;&#093;&#123;&#120;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"24\" style=\"vertical-align: -4px;\" \/> key.<\/td>\n<td style=\"width: 50%; height: 16px;\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-eb536016586ff325a85c5098305f89b1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#46;&#54;&#53;&#57;&#51;&#48;&#53;&#55;&#49;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"95\" style=\"vertical-align: 0px;\" \/>&#8230;<\/td>\n<\/tr>\n<tr style=\"height: 16px;\">\n<td style=\"width: 50%; height: 16px;\">Round to two decimal places.<\/td>\n<td style=\"width: 50%; height: 16px;\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-a208bdac0bd3413bdfe3b18ff0f200fc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#46;&#54;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"32\" style=\"vertical-align: 0px;\" \/><\/td>\n<\/tr>\n<tr style=\"height: 16px;\">\n<td style=\"width: 50%; height: 16px;\"><\/td>\n<td style=\"width: 50%; height: 16px;\">\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-7fc6a1b61f5b329eb75c9590cfe1f7ac_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#091;&#51;&#093;&#123;&#52;&#57;&#125;&#92;&#97;&#112;&#112;&#114;&#111;&#120;&#32;&#51;&#46;&#54;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"87\" style=\"vertical-align: -2px;\" \/><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><span class=\"token\">c)<\/span><\/p>\n<\/div>\n<table style=\"border-collapse: collapse; width: 100%;\">\n<tbody>\n<tr>\n<td style=\"width: 50%;\"><\/td>\n<td style=\"width: 50%;\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-bf442e00f8c5fbd8184711a3aaf6fddb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#091;&#52;&#093;&#123;&#53;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"32\" style=\"vertical-align: -2px;\" \/><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 50%;\">Use the calculator <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-20fb7e6bd15ecd2fe82c65707eaa5481_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#091;&#121;&#093;&#123;&#120;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"24\" style=\"vertical-align: -4px;\" \/> key.<\/td>\n<td style=\"width: 50%;\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-8db6011f70833af441e7a3368c180e73_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#46;&#54;&#55;&#50;&#51;&#52;&#53;&#49;&#49;&#55;&#55;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"104\" style=\"vertical-align: 0px;\" \/><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 50%;\">Round to two decimal places.<\/td>\n<td style=\"width: 50%;\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-7af2ab8e9b0ebd881aa6199c3ac62452_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#46;&#54;&#55;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"32\" style=\"vertical-align: 0px;\" \/><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 50%;\"><\/td>\n<td style=\"width: 50%;\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-c188775658bd62e01f957c3c74a25dba_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#091;&#52;&#093;&#123;&#53;&#49;&#125;&#92;&#97;&#112;&#112;&#114;&#111;&#120;&#32;&#50;&#46;&#54;&#55;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"87\" style=\"vertical-align: -2px;\" \/><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 7<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1169144874512\" data-type=\"problem\">\n<p id=\"fs-id1169146731256\">Round to two decimal places:<\/p>\n<p id=\"fs-id1169144645848\"><span class=\"token\">a) <\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-ae895326687430992c6663c58c30e362_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#49;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"32\" style=\"vertical-align: -2px;\" \/> b) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-03e62f8d6f7922a1c53813727829c8f0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#091;&#51;&#093;&#123;&#55;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"32\" style=\"vertical-align: -2px;\" \/> c) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-176a3571c4783bead27b5f1b5704e92a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#091;&#52;&#093;&#123;&#49;&#50;&#55;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"44\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div id=\"fs-id1169149348031\" data-type=\"solution\">\n<details open=\"open\">\n<summary>Show answer<\/summary>\n<p id=\"fs-id1169149097346\"><span class=\"token\">a) <\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-5a61369e469b8e0c9dae6ec57683ed81_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#97;&#112;&#112;&#114;&#111;&#120;&#32;&#51;&#46;&#51;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"49\" style=\"vertical-align: 0px;\" \/> b) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-4113058c5321e4291865f764d1c6483b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#97;&#112;&#112;&#114;&#111;&#120;&#32;&#52;&#46;&#49;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"51\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/details>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">c) <\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-f72274b45dddb98e738797bdb99ad636_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#97;&#112;&#112;&#114;&#111;&#120;&#32;&#51;&#46;&#51;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"51\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<h1 data-type=\"title\">Use Radicals in Applications<\/h1>\n<p id=\"fs-id1169148985032\">As you progress through your college or university courses, you\u2019ll encounter formulas that include radicals in many disciplines. We will modify our Problem Solving Strategy for Geometry Applications slightly to give us a plan for solving applications with formulas from any discipline.<\/p>\n<div id=\"fs-id1169149040822\" class=\"howto\" data-type=\"note\">\n<div data-type=\"title\">\n<div class=\"textbox textbox--key-takeaways\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Use a problem solving strategy for applications with formulas.<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<ol id=\"fs-id1169144543890\" class=\"stepwise\" type=\"1\">\n<li><strong data-effect=\"bold\">Read<\/strong> the problem and make sure all the words and ideas are understood. When appropriate, draw a figure and label it with the given information.<\/li>\n<li><strong data-effect=\"bold\">Identify<\/strong> what we are looking for.<\/li>\n<li><strong data-effect=\"bold\">Name<\/strong> what we are looking for by choosing a variable to represent it.<\/li>\n<li><strong data-effect=\"bold\">Translate<\/strong> into an equation by writing the appropriate formula or model for the situation. Substitute in the given information.<\/li>\n<li><strong data-effect=\"bold\">Solve the equation<\/strong> using good algebra techniques.<\/li>\n<li><strong data-effect=\"bold\">Check<\/strong> the answer in the problem and make sure it makes sense.<\/li>\n<li><strong data-effect=\"bold\">Answer<\/strong> the question with a complete sentence.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1169149118428\">One application of radicals has to do with the effect of <span class=\"no-emphasis\" data-type=\"term\">gravity<\/span> on falling objects. The formula allows us to determine how long it will take a fallen object to hit the gound.<\/p>\n<div id=\"fs-id1169148957158\" data-type=\"note\">\n<div data-type=\"title\">\n<div class=\"textbox textbox--key-takeaways\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Falling Objects<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p id=\"fs-id1169144729653\">On Earth, if an object is dropped from a height of <em data-effect=\"italics\">h<\/em> feet, the time in seconds it will take to reach the ground is found by using the formula<\/p>\n<div id=\"fs-id1169149103897\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-e3490b72d80ae0f6ca49756cb4339450_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#116;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#115;&#113;&#114;&#116;&#123;&#104;&#125;&#125;&#123;&#52;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"57\" style=\"vertical-align: -6px;\" \/><\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1169146628740\">For example, if an object is dropped from a height of 64 feet, we can find the time it takes to reach the ground by substituting <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-e48c241955bc35e28678a07830ddd788_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#104;&#61;&#54;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"52\" style=\"vertical-align: -1px;\" \/> into the formula.<\/p>\n<table class=\"unnumbered unstyled aligncenter\" summary=\"Since h equals 64 we rewrite the formula, replacing h with the number 64. The formula then becomes t equals square root of 64 divided by 4. Taking the square root of 64 we get t equals 8 divided by 4. Simplifying the fraction we get t equals 2. It would take 2 seconds for an object dropped from a height of 64 feet to reach the ground.\" data-label=\"\">\n<tbody>\n<tr valign=\"top\">\n<td style=\"text-align: center;\" data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span id=\"fs-id1169149065270\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_IntAlg_Figure_08_06_010a_img.jpg\" alt=\".\" data-media-type=\"image\/jpeg\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span id=\"fs-id1169149032044\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_IntAlg_Figure_08_06_010b_img.jpg\" alt=\".\" data-media-type=\"image\/jpeg\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Take the square root of 64.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span id=\"fs-id1169149342082\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_IntAlg_Figure_08_06_010c_img.jpg\" alt=\".\" data-media-type=\"image\/jpeg\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Simplify the fraction.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_IntAlg_Figure_08_06_010d_img.jpg\" alt=\".\" data-media-type=\"image\/jpeg\" \/><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1169149190917\" style=\"text-align: center;\">It would take 2 seconds for an object dropped from a height of 64 feet to reach the ground.<\/p>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">EXAMPLE 8<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1169149120935\" data-type=\"problem\">\n<p id=\"fs-id1169148973896\">Marissa dropped her sunglasses from a bridge 400 feet above a river. Use the formula <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-61154b9c5a29ce4c47540f7df2fdc4c4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#116;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#115;&#113;&#114;&#116;&#123;&#104;&#125;&#125;&#123;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"52\" style=\"vertical-align: -6px;\" \/> to find how many seconds it took for the sunglasses to reach the river.<\/p>\n<p><strong><span class=\"token\">Solution<\/span><\/strong><\/p>\n<\/div>\n<div id=\"fs-id1169148890736\" data-type=\"solution\">\n<table id=\"fs-id1169149376727\" style=\"width: 100%;\" summary=\"The first step in the process is to read the problem. Step 2 is to identify what we are looking for. We are looking for the time it takes the sunglasses to reach the river. Step 3 is to name what we are looking for. Let t equal the time. Step 4 is to translate into an equation by writing the appropriate formula and substitute in the given information. t equals the square root of h divided by 4 and h equals 400. So t equals the square root of 400 divided by 4. Step 5 is to solve the equation. So t equals 20 divided by 4. So t equals 5. Step 6 is to check the answer in the problem and make sure it makes sense. Does 5 equal the square root of 400 divided 4. Since 5 equals 20 divided by 4, the answer is a solution to the equation. Does 5 seconds seem like a reasonable length of time? Yes. Step 7 is to answer the question. It will take 5 seconds for the sunglasses to reach the river.\" data-label=\"\">\n<tbody>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Step 1. Read<\/strong> the problem.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Step 2. Identify<\/strong> what we are looking for.<\/td>\n<td data-valign=\"top\" data-align=\"left\">the time it takes for the<\/p>\n<div data-type=\"newline\"><\/div>\n<p>sunglasses to reach the river<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Step 3. Name<\/strong> what we are looking.<\/td>\n<td data-valign=\"top\" data-align=\"left\">Let <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-be6dc1d69e7f8de3302461f289c68554_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#116;&#61;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"24\" style=\"vertical-align: 0px;\" \/> time.<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Step 4. Translate<\/strong> into an equation by writing the<\/p>\n<div data-type=\"newline\"><\/div>\n<p>appropriate formula. Substitute in the given<\/p>\n<div data-type=\"newline\"><\/div>\n<p>information.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span id=\"fs-id1169148938836\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_IntAlg_Figure_08_06_011b_img.jpg\" alt=\".\" data-media-type=\"image\/jpeg\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Step 5. Solve the equation.<\/strong><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span id=\"fs-id1169146744203\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_IntAlg_Figure_08_06_011c_img.jpg\" alt=\".\" data-media-type=\"image\/jpeg\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span id=\"fs-id1169144561652\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_IntAlg_Figure_08_06_011d_img.jpg\" alt=\".\" data-media-type=\"image\/jpeg\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Step 6. Check<\/strong> the answer in the problem and make<\/p>\n<div data-type=\"newline\"><\/div>\n<p>sure it makes sense.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span id=\"fs-id1169149157046\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_IntAlg_Figure_08_06_011a_img.jpg\" alt=\".\" data-media-type=\"image\/jpeg\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Does 5 seconds seem like a reasonable length of<\/p>\n<div data-type=\"newline\"><\/div>\n<p>time?<\/td>\n<td data-valign=\"top\" data-align=\"left\">Yes.<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Step 7. Answer<\/strong> the question.<\/td>\n<td data-valign=\"top\" data-align=\"left\">It will take 5 seconds for the<\/p>\n<div data-type=\"newline\"><\/div>\n<p>sunglasses to reach the river.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 8<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1169149115845\" data-type=\"problem\">\n<p id=\"fs-id1169149089351\">A helicopter dropped a rescue package from a height of 1,296 feet. Use the formula <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-61154b9c5a29ce4c47540f7df2fdc4c4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#116;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#115;&#113;&#114;&#116;&#123;&#104;&#125;&#125;&#123;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"52\" style=\"vertical-align: -6px;\" \/> to find how many seconds it took for the package to reach the ground.<\/p>\n<\/div>\n<div id=\"fs-id1169144450967\" data-type=\"solution\">\n<details open=\"open\">\n<summary>Show answer<\/summary>\n<p id=\"fs-id1169138945430\">9 seconds<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1169148957152\">Police officers investigating car accidents measure the length of the skid marks on the pavement. Then they use square roots to determine the <span class=\"no-emphasis\" data-type=\"term\">speed<\/span>, in miles per hour, a car was going before applying the brakes.<\/p>\n<div id=\"fs-id1169149012194\" data-type=\"note\">\n<div data-type=\"title\">\n<div class=\"textbox textbox--key-takeaways\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Skid Marks and Speed of a Car<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p id=\"fs-id1169144555978\">If the length of the skid marks is <em data-effect=\"italics\">d<\/em> feet, then the speed, <em data-effect=\"italics\">s<\/em>, of the car before the brakes were applied can be found by using the formula<\/p>\n<div id=\"fs-id1169149092734\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-5a9a9c32372c90dc6afdb389799a1856_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#115;&#61;&#92;&#115;&#113;&#114;&#116;&#123;&#50;&#52;&#100;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"75\" style=\"vertical-align: -2px;\" \/><\/div>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"equation\" data-label=\"\">\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">EXAMPLE 9<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1169149311099\" data-type=\"problem\">\n<p id=\"fs-id1169147133499\">After a car accident, the skid marks for one car measured 190 feet. Use the formula <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-5a9a9c32372c90dc6afdb389799a1856_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#115;&#61;&#92;&#115;&#113;&#114;&#116;&#123;&#50;&#52;&#100;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"75\" style=\"vertical-align: -2px;\" \/> to find the speed of the car before the brakes were applied. Round your answer to the nearest tenth.<\/p>\n<p><strong><span class=\"token\">Solution<\/span><\/strong><\/p>\n<\/div>\n<div id=\"fs-id1169148993220\" data-type=\"solution\">\n<table id=\"fs-id1169144381624\" class=\"unnumbered unstyled can-break\" style=\"height: 350px;\" summary=\"The first step in the process is to read the problem. Step 2 is to identify what we are looking for. We are looking for the speed of the car. Step 3 is to name what we are looking for. Let s equal the speed. Step 4 is to translate into an equation by writing the appropriate formula and substitute in the given information. s equals the square root of the quantity 24 d in parentheses, and d equals 190. So s equals the square root of the quantity 24 times 190 in parentheses. Step 5 is to solve the equation. So s equals the square root of 4560. So s is approximately equal to 67.52777. Rounding to 1 decimal place we et s equal to 67.5. Step 6 is to check the answer in the problem and make sure it makes sense. Does the square root of 4560 equal the square root of the quantity 24 times 190 in parentheses? It does. Does 67.5 mph seem like a reasonable speed? Yes. Step 7 is to answer the question. The car was traveling approximately 67.5 mph before the brakes were applied.\" data-label=\"\">\n<tbody>\n<tr style=\"height: 15px;\" valign=\"top\">\n<td style=\"height: 15px; width: 349.757px;\" data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Step 1. Read<\/strong> the problem<\/td>\n<td style=\"height: 15px; width: 396.424px;\" data-valign=\"top\" data-align=\"left\"><\/td>\n<\/tr>\n<tr style=\"height: 15px;\" valign=\"top\">\n<td style=\"height: 15px; width: 349.757px;\" data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Step 2. Identify<\/strong> what we are looking for.<\/td>\n<td style=\"height: 15px; width: 396.424px;\" data-valign=\"top\" data-align=\"left\">the speed of a car<\/td>\n<\/tr>\n<tr style=\"height: 15px;\" valign=\"top\">\n<td style=\"height: 15px; width: 349.757px;\" data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Step 3. Name<\/strong> what we are looking for,<\/td>\n<td style=\"height: 15px; width: 396.424px;\" data-valign=\"top\" data-align=\"left\">Let <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-12ac9f5dee0c0d9acdb247ec52211d05_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#115;&#61;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"26\" style=\"vertical-align: 0px;\" \/> the speed.<\/td>\n<\/tr>\n<tr style=\"height: 92px;\" valign=\"top\">\n<td style=\"height: 92px; width: 349.757px;\" data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Step 4. Translate<\/strong> into an equation by writing<\/p>\n<div data-type=\"newline\"><\/div>\n<p>the appropriate formula. Substitute in the<\/p>\n<div data-type=\"newline\"><\/div>\n<p>given information.<\/td>\n<td style=\"height: 92px; width: 396.424px;\" data-valign=\"top\" data-align=\"left\"><span id=\"fs-id1169147109998\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_IntAlg_Figure_08_06_012a_img.jpg\" alt=\".\" data-media-type=\"image\/jpeg\" \/><\/span><\/td>\n<\/tr>\n<tr style=\"height: 21px;\" valign=\"top\">\n<td style=\"height: 21px; width: 349.757px;\" data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Step 5. Solve the equation.<\/strong><\/td>\n<td style=\"height: 21px; width: 396.424px;\" data-valign=\"top\" data-align=\"left\"><span id=\"fs-id1169148971507\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_IntAlg_Figure_08_06_012b_img.jpg\" alt=\".\" data-media-type=\"image\/jpeg\" \/><\/span><\/td>\n<\/tr>\n<tr style=\"height: 17px;\" valign=\"top\">\n<td style=\"height: 17px; width: 349.757px;\" data-valign=\"top\" data-align=\"left\"><\/td>\n<td style=\"height: 17px; width: 396.424px;\" data-valign=\"top\" data-align=\"left\"><span id=\"fs-id1169149219635\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_IntAlg_Figure_08_06_012c_img.jpg\" alt=\".\" data-media-type=\"image\/jpeg\" \/><\/span><\/td>\n<\/tr>\n<tr style=\"height: 16px;\" valign=\"top\">\n<td style=\"height: 16px; width: 349.757px;\" data-valign=\"top\" data-align=\"left\">Round to 1 decimal place.<\/td>\n<td style=\"height: 16px; width: 396.424px;\" data-valign=\"top\" data-align=\"left\"><span id=\"fs-id1169144558132\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_IntAlg_Figure_08_06_012d_img.jpg\" alt=\".\" data-media-type=\"image\/jpeg\" \/><\/span><\/td>\n<\/tr>\n<tr style=\"height: 98px;\" valign=\"top\">\n<td style=\"height: 98px; width: 349.757px;\" data-valign=\"top\" data-align=\"left\"><\/td>\n<td style=\"height: 98px; width: 396.424px;\" data-valign=\"top\" data-align=\"left\"><span id=\"fs-id1169146594591\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_IntAlg_Figure_08_06_012e_img.jpg\" alt=\".\" data-media-type=\"image\/jpeg\" \/><\/span><\/td>\n<\/tr>\n<tr style=\"height: 61px;\" valign=\"top\">\n<td style=\"height: 61px; width: 349.757px;\" data-valign=\"top\" data-align=\"left\"><\/td>\n<td style=\"height: 61px; width: 396.424px;\" data-valign=\"top\" data-align=\"left\">The speed of the car before the brakes were applied<\/p>\n<div data-type=\"newline\"><\/div>\n<p>was 67.5 miles per hour.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1169144382040\" class=\"try\" data-type=\"note\">\n<div id=\"fs-id1169144382958\" data-type=\"exercise\">\n<div id=\"fs-id1169149330072\" data-type=\"problem\">\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 9<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1169149330072\" data-type=\"problem\">\n<p id=\"fs-id1169149354829\">An accident investigator measured the skid marks of the car. The length of the skid marks was 76 feet. Use the formula <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-5a9a9c32372c90dc6afdb389799a1856_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#115;&#61;&#92;&#115;&#113;&#114;&#116;&#123;&#50;&#52;&#100;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"75\" style=\"vertical-align: -2px;\" \/> to find the speed of the car before the brakes were applied. Round your answer to the nearest tenth.<\/p>\n<\/div>\n<div data-type=\"solution\">\n<details open=\"open\">\n<summary>Show answer<\/summary>\n<p id=\"fs-id1169146662927\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-bba56402cf93c2d584d5a5151045aad9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#50;&#46;&#55;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"32\" style=\"vertical-align: 0px;\" \/> feet<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1550190\">The Pythagorean Theorem tells how the lengths of the three sides of a right triangle relate to each other. It states that in any right triangle, the sum of the squares of the two legs equals the square of the hypotenuse.<\/p>\n<div id=\"fs-id1701775\" data-type=\"note\">\n<div data-type=\"title\">\n<div class=\"textbox textbox--key-takeaways\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">The Pythagorean Theorem<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p id=\"fs-id1548386\">In any right triangle <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-23f00feef91cb9ba3057f9e15d82ff39_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#65;&#66;&#67;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"56\" style=\"vertical-align: 0px;\" \/>,<\/p>\n<div id=\"fs-id1171104156908\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-7ea02600b9972a334e42686c1b0b4cf4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#43;&#123;&#98;&#125;&#94;&#123;&#50;&#125;&#61;&#123;&#99;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"93\" style=\"vertical-align: -2px;\" \/><\/div>\n<p id=\"fs-id1422647\">where <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-41a04eeea923a1a0c28094a8a4680525_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#99;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"8\" style=\"vertical-align: 0px;\" \/> is the length of the hypotenuse <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-5c53d6ebabdbcfa4e107550ea60b1b19_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"9\" style=\"vertical-align: 0px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-f56d50c26583f9a035ff6b4e3c0ca5c0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#98;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"8\" style=\"vertical-align: 0px;\" \/> are the lengths of the legs.<\/p>\n<p><span id=\"fs-id1723708\" data-type=\"media\" data-alt=\"A right triangle is shown. The right angle is marked with a box. Across from the box is side c. The sides touching the right angle are marked a and b.\"><img decoding=\"async\" class=\"alignnone\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_BMath_Figure_09_03_025.jpg\" alt=\"A right triangle is shown. The right angle is marked with a box. Across from the box is side c. The sides touching the right angle are marked a and b.\" data-media-type=\"image\/jpeg\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1538008\">We will use this definition of square roots to solve for the length of a side in a right triangle.<\/p>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">EXAMPLE 10<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1419626\" data-type=\"problem\">\n<p id=\"fs-id1420675\">Use the Pythagorean Theorem to find the length of the hypotenuse.<\/p>\n<p><span id=\"fs-id1411841\" data-type=\"media\" data-alt=\"Right triangle with legs labeled as 3 and 4.\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_BMath_Figure_09_03_026_img.jpg\" alt=\"Right triangle with legs labeled as 3 and 4.\" data-media-type=\"image\/jpeg\" \/><\/span><\/p>\n<p><strong>Solution<\/strong><\/p>\n<\/div>\n<div id=\"fs-id1172599766488\" data-type=\"solution\">\n<table id=\"eip-id1168469450887\" style=\"width: 100%;\" summary=\"Step 1 says, \u201cRead the problem.\u201d The word \u201cread\u201d is in bold. Beside this is a picture of a right triangle. The side across from the right angle is labeled c, the sides touching the right angle are labeled 3 and 4. Step 2 says, \u201cIdentify what you are looking for.\u201d The word \u201cidentify\u201d is in bold. Beside this it says, \u201cthe length of the hypotenuse of the triangle.\u201d Step 3 says, \u201cName. Choose a variable to represent it. \u201c The word \u201cname\u201d is in bold. Beside this it says, \u201cLet c equal the length of the hypotenuse.\u201d Step 4 says, \u201cTranslate. Write the appropriate formula. Substitute.\u201d The word \u201ctranslate\u201d is in bold. Beside this is the equation a squared plus b squared equals c squared, then 3 squared plus 4 squared equals c squared. Step 5 says, \u201cSolve the equation. Simplify. Use the definition of square root. Simplify.\u201d The word \u201csolve\u201d is in bold. Beside this is 9 plus 16 equals c squared, then 25 equals c squared, then square root of 25 equals c, then 5 equals c. Step 6 says \u201cCheck\u201d in bold. Beside this is 3 squared plus 4 squared followed by an equal sign with a question mark, then a red 5 squared. Below this is 9 plus 16 followed by an equal sign with a question mark, then 25. The next line says 25 equals 25 with a check mark. Step 7 says, \u201cAnswer the question.\u201d The word \u201canswer\u201d is in bold. Beside this is the answer, \u201cThe length of the hypotenuse is 5.\u201d\" data-label=\"\">\n<tbody>\n<tr>\n<td>Step 1. <strong data-effect=\"bold\">Read<\/strong> the problem.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Step 2. <strong data-effect=\"bold\">Identify<\/strong> what you are looking for.<\/td>\n<td>the length of the hypotenuse of the triangle<\/td>\n<\/tr>\n<tr>\n<td>Step 3. <strong data-effect=\"bold\">Name.<\/strong> Choose a variable to represent it.<\/td>\n<td>Let <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-710f083184f6961892c6e045dda64ee8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#99;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#116;&#104;&#101;&#32;&#108;&#101;&#110;&#103;&#116;&#104;&#32;&#111;&#102;&#32;&#116;&#104;&#101;&#32;&#104;&#121;&#112;&#111;&#116;&#101;&#110;&#117;&#115;&#101;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"254\" style=\"vertical-align: -3px;\" \/><br \/>\n<span id=\"eip-id1168469449113\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_BMath_Figure_09_03_053_img-01.png\" alt=\".\" data-media-type=\"image\/jpeg\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td>Step 4. <strong data-effect=\"bold\">Translate.<\/strong><br \/>\nWrite the appropriate formula.<br \/>\nSubstitute.<\/td>\n<td><span id=\"eip-id1168469890812\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_BMath_Figure_09_03_053_img-02.png\" alt=\".\" data-media-type=\"image\/jpeg\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\">Step 5. <strong data-effect=\"bold\">Solve<\/strong> the equation.<\/td>\n<td><span id=\"eip-id1168469890834\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_BMath_Figure_09_03_053_img-03.png\" alt=\".\" data-media-type=\"image\/jpeg\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td>Step 6. <strong data-effect=\"bold\">Check:<\/strong><br \/>\n<span id=\"eip-id1168469481160\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_BMath_Figure_09_03_053_img-04.png\" alt=\".\" data-media-type=\"image\/jpeg\" \/><\/span><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Step 7. <strong data-effect=\"bold\">Answer<\/strong> the question.<\/td>\n<td>The length of the hypotenuse is 5.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 10<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1542055\" data-type=\"problem\">\n<p id=\"fs-id1542057\">Use the Pythagorean Theorem to find the length of the hypotenuse.<\/p>\n<p><span id=\"fs-id1321685\" data-type=\"media\" data-alt=\"A right triangle is shown. The right angle is marked with a box. Across from the box is side c. The sides touching the right angle are marked 6 and 8.\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_BMath_Figure_09_03_029_img.jpg\" alt=\"A right triangle is shown. The right angle is marked with a box. Across from the box is side c. The sides touching the right angle are marked 6 and 8.\" data-media-type=\"image\/jpeg\" \/><\/span><\/p>\n<\/div>\n<div id=\"fs-id1727689\" data-type=\"solution\">\n<details open=\"open\">\n<summary>Show answer<\/summary>\n<p id=\"fs-id1259642\">10<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1169144566399\" class=\"media-2\" data-type=\"note\">\n<p id=\"fs-id1169149329664\">Access these online resources for additional instruction and practice with solving radical equations.<\/p>\n<ul id=\"fs-id1169147089449\" data-bullet-style=\"bullet\">\n<li><a href=\"https:\/\/openstax.org\/l\/37RadEquat5\">Radical Equation Application<\/a><\/li>\n<\/ul>\n<\/div>\n<h1>Glossary<\/h1>\n<div class=\"textbox shaded\">\n<p><strong>square root notation<\/strong><\/p>\n<ul>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-ff6e04829c3778dcab4453317cfa77ce_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#109;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"30\" style=\"vertical-align: -4px;\" \/> is read \u2018the square root of <em data-effect=\"italics\">m<\/em>\u2019<\/li>\n<li>If <em data-effect=\"italics\">n<\/em><sup>2<\/sup> = <em data-effect=\"italics\">m<\/em>, then <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-30dea5cfbc072e5165a53a45049a7464_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#110;&#61;&#92;&#115;&#113;&#114;&#116;&#123;&#109;&#125;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"69\" style=\"vertical-align: -4px;\" \/> for <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-de0b3f268c78cbcbba48c42081539c2a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#110;&#92;&#103;&#101;&#32;&#48;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"47\" style=\"vertical-align: -3px;\" \/><\/li>\n<\/ul>\n<div data-type=\"newline\"><\/div>\n<p><span id=\"fs-id1169148870699\" data-type=\"media\" data-alt=\"The image shows the variable m inside a square root symbol. The symbol is a line that goes up along the left side and then flat above the variable. The symbol is labeled \u201cradical sign\u201d. The variable m is labeled \u201cradicand\u201d.\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_IntAlg_Figure_08_01_009_img_new.jpg\" alt=\"The image shows the variable m inside a square root symbol. The symbol is a line that goes up along the left side and then flat above the variable. The symbol is labeled \u201cradical sign\u201d. The variable m is labeled \u201cradicand\u201d.\" data-media-type=\"image\/jpeg\" \/><\/span><\/p>\n<ul>\n<li>The square root of <em data-effect=\"italics\">m<\/em>, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-2447480eb926a3c6e489624b0a8579e6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#109;&#125;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"33\" style=\"vertical-align: -4px;\" \/> is a positive number whose square is <em data-effect=\"italics\">m<\/em>.<\/li>\n<\/ul>\n<p>&nbsp;<\/p>\n<p><strong data-effect=\"bold\"><em data-effect=\"italics\">n<\/em><sup>th<\/sup> root of a number<\/strong><\/p>\n<ul>\n<li>If <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-f6278d46de14949078e82254eccf0f0b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#98;&#125;&#94;&#123;&#110;&#125;&#61;&#97;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"53\" style=\"vertical-align: -4px;\" \/> then <em data-effect=\"italics\">b<\/em> is an <em data-effect=\"italics\">n<sup>th<\/sup><\/em> root of <em data-effect=\"italics\">a<\/em>.<\/li>\n<li>The principal <em data-effect=\"italics\">n<sup>th<\/sup><\/em> root of <em data-effect=\"italics\">a<\/em> is written <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-8926ea991916fb34abf938092c039c24_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#091;&#110;&#093;&#123;&#97;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"27\" style=\"vertical-align: -4px;\" \/><\/li>\n<li><em data-effect=\"italics\">n<\/em> is called the <em data-effect=\"italics\">index<\/em> of the radical.<\/li>\n<\/ul>\n<\/div>\n<h1>1.6 Exercise Set<\/h1>\n<p id=\"fs-id1169149285863\">In the following exercises, simplify.<\/p>\n<ol class=\"twocolumn\">\n<li>\n<ol type=\"a\">\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-0d5fdaeda697f10142dbe9236e272283_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#54;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"32\" style=\"vertical-align: -2px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-8ccf6d622eb18d101b7d435fa62da03c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#115;&#113;&#114;&#116;&#123;&#56;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"46\" style=\"vertical-align: -2px;\" \/><\/li>\n<\/ol>\n<\/li>\n<li>\n<ol type=\"a\">\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-326c826152ee55e8686d89ebc6a491e8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#49;&#57;&#54;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"41\" style=\"vertical-align: -2px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-22d1c3dbf006acb0cf0859ab5bda2e1c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#115;&#113;&#114;&#116;&#123;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"37\" style=\"vertical-align: -2px;\" \/><\/li>\n<\/ol>\n<\/li>\n<li>\n<ol type=\"a\">\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-741e27b01b5ef1cd80f9b51091dbfee4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#125;&#123;&#57;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"32\" width=\"28\" style=\"vertical-align: -10px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-2ea7e4f1e4a030bd68ccff1e90cd5955_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#115;&#113;&#114;&#116;&#123;&#48;&#46;&#48;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"60\" style=\"vertical-align: -2px;\" \/><\/li>\n<\/ol>\n<\/li>\n<li>\n<ol type=\"a\">\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-8428c95cb8eee09f094a7824f1a7d5b1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#45;&#49;&#50;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"55\" style=\"vertical-align: -3px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-acee37943855aa8586d68a80eb13039c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#115;&#113;&#114;&#116;&#123;&#50;&#56;&#57;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"55\" style=\"vertical-align: -2px;\" \/><\/li>\n<\/ol>\n<\/li>\n<li>\n<ol type=\"a\">\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-38c0f8874cf0ae17e83898264c4c7111_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#115;&#113;&#114;&#116;&#123;&#50;&#50;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"55\" style=\"vertical-align: -2px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-c72edf921f549cc7435d7f64e770969a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#45;&#57;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"37\" style=\"vertical-align: -3px;\" \/><\/li>\n<\/ol>\n<\/li>\n<li>\n<ol type=\"a\">\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-efbfa56aba2754d8cc60b1b6bdda3e0a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#091;&#51;&#093;&#123;&#50;&#49;&#54;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"41\" style=\"vertical-align: -2px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-e8fc6228ef8608f4a53018ae413a46f5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#091;&#52;&#093;&#123;&#50;&#53;&#54;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"41\" style=\"vertical-align: -2px;\" \/><\/li>\n<\/ol>\n<\/li>\n<li>\n<ol type=\"a\">\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-07742bd0b9c49b651cd1aa956d22b337_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#091;&#51;&#093;&#123;&#53;&#49;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"41\" style=\"vertical-align: -2px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-fe9b91eb5b6fd4a4bcb7545fa44d8399_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#091;&#52;&#093;&#123;&#56;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"32\" style=\"vertical-align: -2px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-4992394fe501f21188fb1f6e06580d7e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#091;&#53;&#093;&#123;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"23\" style=\"vertical-align: -2px;\" \/><\/li>\n<\/ol>\n<\/li>\n<li>\n<ol type=\"a\">\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-fb90d1635e301b33e9886edf473d0421_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#091;&#51;&#093;&#123;&#45;&#56;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"37\" style=\"vertical-align: -3px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-d0eaa386750f44685e20d3af445e7d74_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#091;&#52;&#093;&#123;&#45;&#56;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"46\" style=\"vertical-align: -3px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-f021507ab9fd854a0b93956ab083ddd7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#091;&#53;&#093;&#123;&#45;&#51;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"46\" style=\"vertical-align: -3px;\" \/><\/li>\n<\/ol>\n<\/li>\n<li>\n<ol type=\"a\">\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-5767974107a2c199d59ce16c113b6869_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#091;&#51;&#093;&#123;&#45;&#49;&#50;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"55\" style=\"vertical-align: -3px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-cfc290b649051857476163d7480d4038_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#091;&#52;&#093;&#123;&#45;&#49;&#50;&#57;&#54;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"64\" style=\"vertical-align: -3px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-dea9d55818f3811029affd8e4c7f24d9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#091;&#53;&#093;&#123;&#45;&#49;&#48;&#50;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"64\" style=\"vertical-align: -3px;\" \/><\/li>\n<\/ol>\n<\/li>\n<\/ol>\n<div id=\"fs-id1169144768977\" data-type=\"exercise\">\n<div id=\"fs-id1169144768979\" data-type=\"problem\">\n<p id=\"fs-id1169144768981\"><span style=\"text-align: initial; font-size: 14pt;\">In the following exercises, estimate each root between two consecutive whole numbers.<\/span><\/p>\n<ol class=\"twocolumn\" start=\"10\">\n<li>\n<ol type=\"a\">\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-38d715eb1e6c8cbdf44c2fec764141ff_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#55;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"32\" style=\"vertical-align: -2px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-03e62f8d6f7922a1c53813727829c8f0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#091;&#51;&#093;&#123;&#55;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"32\" style=\"vertical-align: -2px;\" \/><\/li>\n<\/ol>\n<\/li>\n<li>\n<ol type=\"a\">\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-5d721d31489db78a6cecd84196de173e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#50;&#48;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"41\" style=\"vertical-align: -2px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-91a34f2593c101daa08a03c24921bcae_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#091;&#51;&#093;&#123;&#49;&#51;&#55;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"41\" style=\"vertical-align: -2px;\" \/><\/li>\n<\/ol>\n<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<div id=\"fs-id1169149087431\" data-type=\"exercise\">\n<div id=\"fs-id1169149087433\" data-type=\"problem\">\n<p id=\"fs-id1169149289896\"><span style=\"font-size: 14pt; text-align: initial;\">In the following exercises, approximate each root and round to two decimal places.<\/span><\/p>\n<ol class=\"twocolumn\" start=\"12\">\n<li>\n<ol type=\"a\">\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-adeb83899c57c75eca4704e21eeccc17_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#49;&#57;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"32\" style=\"vertical-align: -2px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-b4731b3de51b3a0cc63f746dcaf64be5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#091;&#51;&#093;&#123;&#56;&#57;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"32\" style=\"vertical-align: -2px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-418a1f6c4db896fdbb8f4d7567e1cb48_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#091;&#52;&#093;&#123;&#57;&#55;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"32\" style=\"vertical-align: -2px;\" \/><\/li>\n<\/ol>\n<\/li>\n<li>\n<ol type=\"a\">\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-6309be23126ff6b16de87943ad81cdb5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#53;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"32\" style=\"vertical-align: -2px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-46d50e7721a8aea1066efe812dc7b901_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#091;&#51;&#093;&#123;&#49;&#52;&#55;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"41\" style=\"vertical-align: -2px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-76c416edced5a30e96c915a92810bed8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#091;&#52;&#093;&#123;&#52;&#53;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"41\" style=\"vertical-align: -2px;\" \/><\/li>\n<\/ol>\n<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<div id=\"fs-id1169144768438\" data-type=\"exercise\">\n<div id=\"fs-id1169144768440\" data-type=\"problem\">\n<p id=\"fs-id1169144768442\"><span style=\"font-size: 14pt; text-align: initial;\">In the following exercises, solve. Round approximations to one decimal place.<\/span><\/p>\n<ol start=\"14\">\n<li><strong data-effect=\"bold\">Landscaping.<\/strong> Reed wants to have a square garden plot in his backyard. He has enough compost to cover an area of 75 square feet. Use the formula <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-710aedf0b9cfbdd26e3c31b132a4e7a2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#115;&#61;&#92;&#115;&#113;&#114;&#116;&#123;&#65;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"61\" style=\"vertical-align: -2px;\" \/> to find the length of each side of his garden. Round your answer to the nearest tenth of a foot.<\/li>\n<li><strong data-effect=\"bold\">Gravity.<\/strong> A hang glider dropped his cell phone from a height of 350 feet. Use the formula <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-61154b9c5a29ce4c47540f7df2fdc4c4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#116;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#115;&#113;&#114;&#116;&#123;&#104;&#125;&#125;&#123;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"52\" style=\"vertical-align: -6px;\" \/> to find how many seconds it took for the cell phone to reach the ground.<\/li>\n<li><strong data-effect=\"bold\">Accident investigation<\/strong> The skid marks for a car involved in an accident measured 216 feet. Use the formula <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-5a9a9c32372c90dc6afdb389799a1856_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#115;&#61;&#92;&#115;&#113;&#114;&#116;&#123;&#50;&#52;&#100;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"75\" style=\"vertical-align: -2px;\" \/> to find the speed of the car before the brakes were applied. Round your answer to the nearest tenth.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<dl id=\"fs-id1166424761164\">\n<dd id=\"fs-id1166422865542\">\n<div id=\"fs-id1169149285867\" data-type=\"exercise\">\n<div id=\"fs-id1169149285869\" data-type=\"problem\">\n<div id=\"fs-id1169149235406\" data-type=\"exercise\">\n<div id=\"fs-id1169149235408\" data-type=\"problem\">\n<p>In the following exercises, use the Pythagorean Theorem to find the length of the hypotenuse.<\/p>\n<table style=\"border-collapse: collapse; width: 100%; height: 189px;\">\n<tbody>\n<tr style=\"height: 175px;\">\n<td style=\"width: 50.4345%; height: 175px;\"><span id=\"fs-id1238083\" data-type=\"media\" data-alt=\"A right triangle is shown. The right angle is marked with a box. One of the sides touching the right angle is labeled as 16, the other as 12.\">17. <img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_BMath_Figure_09_03_204_img.jpg\" alt=\"A right triangle is shown. The right angle is marked with a box. One of the sides touching the right angle is labeled as 16, the other as 12.\" data-media-type=\"image\/jpeg\" \/><\/span><\/td>\n<td style=\"width: 49.5655%; height: 175px;\"><span id=\"fs-id1543755\" data-type=\"media\" data-alt=\"A right triangle is shown. The right angle is marked with a box. One of the sides touching the right angle is labeled as 15, the other as 20.\">18.<span id=\"fs-id1428325\" data-type=\"media\" data-alt=\"A right triangle is shown. The right angle is marked with a box. One of the sides touching the right angle is labeled as 5, the other as 12.\"> <img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_BMath_Figure_09_03_206_img.jpg\" alt=\"A right triangle is shown. The right angle is marked with a box. One of the sides touching the right angle is labeled as 5, the other as 12.\" data-media-type=\"image\/jpeg\" \/><\/span><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"eip-851\">In the following exercises, use the Pythagorean Theorem to find the length of the missing side. Round to the nearest tenth, if necessary.<\/p>\n<table style=\"border-collapse: collapse; width: 100%; height: 152px;\">\n<tbody>\n<tr style=\"height: 110px;\">\n<td style=\"width: 50%; height: 110px;\"><span id=\"fs-id1576553\" data-type=\"media\" data-alt=\"A right triangle is shown. The right angle is marked with a box. The side across from the right angle is labeled as 10. One of the sides touching the right angle is labeled as 6.\">19.<span id=\"fs-id1547852\" data-type=\"media\" data-alt=\"A right triangle is shown. The right angle is marked with a box. The side across from the right angle is labeled as 17. One of the sides touching the right angle is labeled as 8.\"> <img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_BMath_Figure_09_03_208_img.jpg\" alt=\"A right triangle is shown. The right angle is marked with a box. The side across from the right angle is labeled as 17. One of the sides touching the right angle is labeled as 8.\" data-media-type=\"image\/jpeg\" \/><\/span><\/span><\/td>\n<td style=\"width: 50%; height: 110px;\"><span id=\"fs-id1701813\" data-type=\"media\" data-alt=\"A right triangle is shown. The right angle is marked with a box. The side across from the right angle is labeled as 13. One of the sides touching the right angle is labeled as 5.\"><span id=\"fs-id1439355\" data-type=\"media\" data-alt=\"A right triangle is shown. The right angle is marked with a box. The side across from the right angle is labeled as 20. One of the sides touching the right angle is labeled as 16.\">20.\u00a0 \u00a0 \u00a0 <img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_BMath_Figure_09_03_210_img.jpg\" alt=\"A right triangle is shown. The right angle is marked with a box. The side across from the right angle is labeled as 20. One of the sides touching the right angle is labeled as 16.\" data-media-type=\"image\/jpeg\" \/><\/span><\/span><\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"width: 50%; height: 14px;\"><span id=\"fs-id1172599767679\" data-type=\"media\" data-alt=\"A right triangle is shown. The right angle is marked with a box. The side across from the right angle is labeled as 13. One of the sides touching the right angle is labeled as 8.\">21 <span id=\"fs-id1704712\" data-type=\"media\" data-alt=\"A right triangle is shown. The right angle is marked with a box. Both of the sides touching the right angle are labeled as 6.\">. <img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_BMath_Figure_09_03_212_img.jpg\" alt=\"A right triangle is shown. The right angle is marked with a box. Both of the sides touching the right angle are labeled as 6.\" data-media-type=\"image\/jpeg\" \/><\/span><\/span><\/td>\n<td style=\"width: 50%; height: 14px;\"><span id=\"fs-id1614367\" data-type=\"media\" data-alt=\"A right triangle is shown. The right angle is marked with a box. The side across from the right angle is labeled as 17. One of the sides touching the right angle is labeled as 15.\"><span id=\"fs-id1537416\" data-type=\"media\" data-alt=\"A right triangle is shown. The right angle is marked with a box. The side across from the right angle is labeled as 7. One of the sides touching the right angle is labeled as 5.\">22.<img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_BMath_Figure_09_03_214_img.jpg\" alt=\"A right triangle is shown. The right angle is marked with a box. The side across from the right angle is labeled as 7. One of the sides touching the right angle is labeled as 5.\" data-media-type=\"image\/jpeg\" \/>\u00a0<\/span><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1428288\"><span style=\"font-family: Helvetica, Arial, 'GFS Neohellenic', sans-serif; font-size: 1.2em; font-weight: bold; text-align: initial;\">Answers:<\/span><\/p>\n<ol class=\"twocolumn\">\n<li>\n<ol type=\"a\">\n<li>8<\/li>\n<li><span style=\"font-family: inherit; font-size: inherit;\"> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-9672181aec15f8334b80ada7de4e4fc0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"22\" style=\"vertical-align: 0px;\" \/><\/span><\/li>\n<\/ol>\n<\/li>\n<li>\n<ol type=\"a\">\n<li>14<\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-7b34c01098c83fa602de54e9d74d63a9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"21\" style=\"vertical-align: -1px;\" \/><\/li>\n<\/ol>\n<\/li>\n<li>\n<ol type=\"a\">\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-6e89c4f69688b0dd6ab75a55841059de_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"7\" style=\"vertical-align: -6px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-65193683874a338a58c1e613cb1beafb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#48;&#46;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"35\" style=\"vertical-align: -1px;\" \/><\/li>\n<\/ol>\n<\/li>\n<li>\n<ol type=\"a\">\n<li>not real number<\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-0a516826583d9ebf41e251f45bc98b74_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;&#55;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"31\" style=\"vertical-align: -1px;\" \/><\/li>\n<\/ol>\n<\/li>\n<li>\n<ol type=\"a\">\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-0329d19b932005c996d83dbb9e09ffed_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"30\" style=\"vertical-align: -1px;\" \/><\/li>\n<li>not real number<\/li>\n<\/ol>\n<\/li>\n<li>\n<ol type=\"a\">\n<li>6<\/li>\n<li>4<\/li>\n<\/ol>\n<\/li>\n<li>\n<ol type=\"a\">\n<li>8<\/li>\n<li>3<\/li>\n<li>1<\/li>\n<\/ol>\n<\/li>\n<li>\n<ol type=\"a\">\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-17c33e2329e29a62a80ad2b547b4753d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"21\" style=\"vertical-align: 0px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-339f426bc6ae4ca4c98c939c65a9f66c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#110;&#111;&#116;&#32;&#114;&#101;&#97;&#108;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"61\" style=\"vertical-align: -1px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-17c33e2329e29a62a80ad2b547b4753d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"21\" style=\"vertical-align: 0px;\" \/><\/li>\n<\/ol>\n<\/li>\n<li>\n<ol type=\"a\">\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-7b5b9d9f382b11767d19f257afca0019_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"21\" style=\"vertical-align: 0px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-339f426bc6ae4ca4c98c939c65a9f66c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#110;&#111;&#116;&#32;&#114;&#101;&#97;&#108;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"61\" style=\"vertical-align: -1px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-00b9cce9021441b203ec0271d72e6ba2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"22\" style=\"vertical-align: -1px;\" \/><\/li>\n<\/ol>\n<\/li>\n<li>\n<ol type=\"a\">\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-78b1df95f7e187f2bbf272ac8d446cb2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#56;&#60;&#92;&#115;&#113;&#114;&#116;&#123;&#55;&#48;&#125;&#60;&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"98\" style=\"vertical-align: -2px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-e4a0944446debf100c16d7273ea42e95_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#60;&#92;&#115;&#113;&#114;&#116;&#091;&#51;&#093;&#123;&#55;&#49;&#125;&#60;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"98\" style=\"vertical-align: -2px;\" \/><\/li>\n<\/ol>\n<\/li>\n<li>\n<ol type=\"a\">\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-a471d0fdec4fa17765382c0c1b413986_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#52;&#60;&#92;&#115;&#113;&#114;&#116;&#123;&#50;&#48;&#48;&#125;&#60;&#49;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"122\" style=\"vertical-align: -2px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-cb38b66aba392cb7ec17451c78da0777_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;&#60;&#92;&#115;&#113;&#114;&#116;&#091;&#51;&#093;&#123;&#49;&#51;&#55;&#125;&#60;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"108\" style=\"vertical-align: -2px;\" \/><\/li>\n<\/ol>\n<\/li>\n<li>\n<ol type=\"a\">\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-7f19d16a2551faba4c1e42c5a9e4c540_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#46;&#51;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"32\" style=\"vertical-align: 0px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-68d512cc41264f61229ea04549ef79f5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#97;&#112;&#112;&#114;&#111;&#120;&#32;&#52;&#46;&#52;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"51\" style=\"vertical-align: 0px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-40e74bf1912143de131304a98dd0ec9b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#97;&#112;&#112;&#114;&#111;&#120;&#32;&#51;&#46;&#49;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"51\" style=\"vertical-align: 0px;\" \/><\/li>\n<\/ol>\n<\/li>\n<li>\n<ol type=\"a\">\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-e98a3b2f507757334e4ed0e47362f3d6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#55;&#46;&#50;&#56;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"32\" style=\"vertical-align: 0px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-3a8f2f26aa741f77c0cdfa305260b2d7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#97;&#112;&#112;&#114;&#111;&#120;&#32;&#53;&#46;&#50;&#56;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"51\" style=\"vertical-align: 0px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-a5e646a53fba19da5459771ae43425a3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#97;&#112;&#112;&#114;&#111;&#120;&#32;&#52;&#46;&#54;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"50\" style=\"vertical-align: 0px;\" \/><\/li>\n<\/ol>\n<\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-8c3502f62f80d6ebc29154bf5cd04bc5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#56;&#46;&#55;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"23\" style=\"vertical-align: 0px;\" \/> feet<\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-fa8e39683e761eafc77f9d9648a9e402_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#46;&#55;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"23\" style=\"vertical-align: -1px;\" \/> seconds<\/li>\n<li>72 feet<\/li>\n<li>20<\/li>\n<li>13<\/li>\n<li>15<\/li>\n<li>12<\/li>\n<li>8.5<\/li>\n<li>8.6<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1169144768977\" data-type=\"exercise\">\n<div id=\"fs-id1169148956903\" data-type=\"solution\">\n<p id=\"fs-id1169148956905\"><span style=\"font-family: Helvetica, Arial, 'GFS Neohellenic', sans-serif; font-size: 1.2em; font-weight: bold; text-align: initial;\">Attributions ( imported from my Introductory Algebra)<\/span><\/p>\n<\/div>\n<\/div>\n<\/dd>\n<\/dl>\n<p>This chapter has been adapted from \u201cThe Real Numbers\u201d in <a href=\"https:\/\/openstax.org\/details\/books\/elementary-algebra\"><em>Elementary Algebra<\/em><\/a> (OpenStax) by Lynn Marecek and MaryAnne Anthony-Smith, which is under a <a href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY 4.0 Licence<\/a>. Adapted by Izabela Mazur. See the Copyright page for more information.<\/p>\n","protected":false},"author":125,"menu_order":6,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-248","chapter","type-chapter","status-publish","hentry"],"part":21,"_links":{"self":[{"href":"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-json\/pressbooks\/v2\/chapters\/248","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-json\/wp\/v2\/users\/125"}],"version-history":[{"count":1,"href":"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-json\/pressbooks\/v2\/chapters\/248\/revisions"}],"predecessor-version":[{"id":249,"href":"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-json\/pressbooks\/v2\/chapters\/248\/revisions\/249"}],"part":[{"href":"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-json\/pressbooks\/v2\/parts\/21"}],"metadata":[{"href":"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-json\/pressbooks\/v2\/chapters\/248\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-json\/wp\/v2\/media?parent=248"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-json\/pressbooks\/v2\/chapter-type?post=248"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-json\/wp\/v2\/contributor?post=248"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-json\/wp\/v2\/license?post=248"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}