{"id":2003,"date":"2020-08-21T19:42:00","date_gmt":"2020-08-21T23:42:00","guid":{"rendered":"https:\/\/opentextbc.ca\/introalgebra\/chapter\/simplify-and-use-square-roots-2\/"},"modified":"2025-09-15T18:45:27","modified_gmt":"2025-09-15T22:45:27","slug":"simplify-and-use-square-roots-2","status":"publish","type":"chapter","link":"https:\/\/opentextbc.ca\/introalgebra\/chapter\/simplify-and-use-square-roots-2\/","title":{"raw":"7.4 Simplify and Use Square Roots","rendered":"7.4 Simplify and Use Square Roots"},"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, you will be able to:\n<ul>\n \t<li>Simplify expressions with square roots<\/li>\n \t<li>Estimate square roots<\/li>\n \t<li>Approximate square roots<\/li>\n \t<li>Simplify variable expressions with square roots<\/li>\n \t<li>Use square roots in applications<\/li>\n<\/ul>\n<\/div>\n<\/div>\n<h1>Simplify Expressions with Square Roots<\/h1>\n<p id=\"fs-id2427723\">To start this section, we need to review some important vocabulary and notation.<\/p>\n<p id=\"fs-id1458017\">Remember that when a number \\(n\\) is multiplied by itself, we can write this as \\({n}^{2}\\), which we read aloud as \\(\\text{\u201c}\\mathit{\\text{n}}\\phantom{\\rule{0.2em}{0ex}}\\text{squared.\u201d}\\) For example, \\({8}^{2}\\) is read as \\(\\text{\u201c8}\\phantom{\\rule{0.2em}{0ex}}\\text{squared.\u201d}\\)<\/p>\n<p id=\"fs-id2165684\">We call \\(64\\) the <em data-effect=\"italics\">square<\/em> of \\(8\\) because \\({8}^{2}=64\\). Similarly, \\(121\\) is the square of \\(11\\), because \\({11}^{2}=121\\).<\/p>\n\n<div id=\"fs-id2784022\" 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 \\(m\\) is the square of \\(n\\).\n\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<h1>Modeling Squares<\/h1>\n<p id=\"fs-id2276864\">Do you know why we use the word <em data-effect=\"italics\">square<\/em>? If we construct a square with three tiles on each side, the total number of tiles would be nine.<\/p>\n<span id=\"fs-id4163946\" data-type=\"media\" data-alt=\"A square is shown with 3 tiles on each side. There are a total of 9 tiles in the square.\"><img class=\"aligncenter\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2020\/08\/CNX_BMath_Figure_05_07_001_img.jpg\" alt=\"A square is shown with 3 tiles on each side. There are a total of 9 tiles in the square.\" data-media-type=\"image\/jpeg\"><\/span>\n<p id=\"fs-id2580549\">This is why we say that the square of three is nine.<\/p>\n\n<div id=\"fs-id1701315\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\">\\({3}^{2}=9\\)<\/div>\n<p id=\"fs-id2446968\">The number \\(9\\) is called a <span class=\"no-emphasis\" data-type=\"term\">perfect square<\/span> because it is the square of a whole number.<\/p>\n<p id=\"fs-id1693488\">The chart shows the squares of the counting numbers \\(1\\) through \\(15\\). You can refer to it to help you identify the perfect squares.<\/p>\n<span id=\"eip-id1170322670247\" data-type=\"media\" data-alt=\"A table with two columns is shown. The first column is labeled \u201cNumber\u201d and has the values: n, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, and 15. The second column is labeled \u201cSquare\u201d and has the values: n squared, 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, and 225.\"><img class=\"aligncenter\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_05_07_008_img.jpg\" alt=\"A table with two columns is shown. The first column is labeled \u201cNumber\u201d and has the values: n, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, and 15. The second column is labeled \u201cSquare\u201d and has the values: n squared, 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, and 225.\" data-media-type=\"image\/jpg\"><\/span>\n<div id=\"fs-id1773473\" data-type=\"note\">\n<div data-type=\"title\">\n<div class=\"textbox textbox--key-takeaways\"><header class=\"textbox__header\">\n<p class=\"textbox__title\">Perfect Squares<\/p>\n\n<\/header>\n<div class=\"textbox__content\">\n\nA perfect square is the square of a whole number.\n\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1951839\">What happens when you square a negative number?<\/p>\n\n<div id=\"fs-id2171365\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\">\\(\\begin{array}{cc}\\hfill {\\left(-8\\right)}^{2}&amp; =\\left(-8\\right)\\left(-8\\right)\\\\ &amp; =64\\hfill \\end{array}\\)<\/div>\n<p id=\"fs-id1987814\">When we multiply two negative numbers, the product is always positive. So, the square of a negative number is always positive.<\/p>\n<p id=\"fs-id3415479\">The chart shows the squares of the negative integers from \\(-1\\) to \\(-15\\).<\/p>\n<span id=\"eip-id1170322726953\" data-type=\"media\" data-alt=\"A table is shown with 2 columns. The first column is labeled \u201cNumber\u201d and contains the values: n, negative 1, negative 2, negative 3, negative 4, negative 5, negative 6, negative 7, negative 8, negative 9, negative 10, negative 11, negative 12, negative 13, negative 14, and negative 15. The next column is labeled \u201cSquare\u201d and contains the values: n squared, 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, and 225.\"><img class=\"aligncenter\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_05_07_009_img.jpg\" alt=\"A table is shown with 2 columns. The first column is labeled \u201cNumber\u201d and contains the values: n, negative 1, negative 2, negative 3, negative 4, negative 5, negative 6, negative 7, negative 8, negative 9, negative 10, negative 11, negative 12, negative 13, negative 14, and negative 15. The next column is labeled \u201cSquare\u201d and contains the values: n squared, 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, and 225.\" data-media-type=\"image\/jpg\"><\/span>\n<p id=\"fs-id2483567\">Did you notice that these squares are the same as the squares of the positive numbers?<\/p>\n\n<h1>Square Roots<\/h1>\n<p id=\"fs-id2645660\">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 can also say that \\(10\\) is a square root of \\(100\\).<\/p>\n\n<div id=\"fs-id1596688\" 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\nA number whose square is \\(m\\) is called a <span class=\"no-emphasis\" data-type=\"term\">square root<\/span> of \\(m\\).\n<p id=\"fs-id2211118\">If \\({n}^{2}=m\\), then \\(n\\) is a square root of \\(m\\).<\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1596349\">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=\"eip-901\">So, every positive number has two square roots: one positive and one negative.<\/p>\nWhat if we only want the positive square root of a positive number? The <em data-effect=\"italics\">radical sign,<\/em> \\(\\sqrt{\\phantom{0}}\\), stands for the positive square root. The positive square root is also called the principal square root.\n<div id=\"fs-id2662111\" 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-id2270433\">\\(\\sqrt{m}\\) is read as \u201cthe square root of \\(m\\text{.\u201d}\\)<\/p>\n<p id=\"fs-id3017179\">\\(\\text{If}\\phantom{\\rule{0.2em}{0ex}}m={n}^{2},\\phantom{\\rule{0.2em}{0ex}}\\text{then}\\phantom{\\rule{0.2em}{0ex}}\\sqrt{m}=n\\phantom{\\rule{0.2em}{0ex}}\\text{for}\\phantom{\\rule{0.2em}{0ex}}\\text{n}\\ge 0\\).<\/p>\n<span id=\"fs-id1382029\" data-type=\"media\" data-alt=\"A picture of an m inside a square root sign is shown. The sign is labeled as a radical sign and the m is labeled as the radicand.\"><img class=\"aligncenter\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_05_07_003_img.jpg\" alt=\"A picture of an m inside a square root sign is shown. The sign is labeled as a radical sign and the m is labeled as the radicand.\" data-media-type=\"image\/jpeg\"><\/span>\n\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id2670991\">We can 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<p id=\"fs-id1396522\">The chart shows the square roots of the first \\(15\\) perfect square numbers.<\/p>\n<a href=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/uploads\/sites\/688\/2019\/06\/CNX_BMath_Figure_05_07_010_img.jpg\"><img class=\"aligncenter\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_05_07_010_img.jpg\" alt=\"A table is shown with 2 columns. The first column contains the values: 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 column contains the values: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, and 15.\" width=\"598\" height=\"59\" data-media-type=\"image\/jpg\"><\/a>\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-id2758986\" data-type=\"problem\">\n<p id=\"fs-id2203262\">Simplify: a) \\(\\phantom{\\rule{0.2em}{0ex}}\\sqrt{25}\\phantom{\\rule{0.2em}{0ex}}\\) b) \\(\\phantom{\\rule{0.2em}{0ex}}\\sqrt{121}\\).<\/p>\n\n<\/div>\n<div id=\"fs-id2346035\" data-type=\"solution\">\n<div data-type=\"title\"><\/div>\n<div data-type=\"title\"><strong>Solution<\/strong><\/div>\n<table id=\"eip-idm859956128\" class=\"unnumbered unstyled\" summary=\".\" data-label=\"\">\n<tbody>\n<tr>\n<td>a)<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>\\(\\sqrt{25}\\)<\/td>\n<\/tr>\n<tr>\n<td>Since \\({5}^{2}=25\\)<\/td>\n<td>\\(5\\)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table id=\"eip-idm1481723040\" class=\"unnumbered unstyled\" summary=\".\" data-label=\"\">\n<tbody>\n<tr>\n<td>b)<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>\\(\\sqrt{121}\\)<\/td>\n<\/tr>\n<tr>\n<td>Since \\({11}^{2}=121\\)<\/td>\n<td>\\(-11\\)<\/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 1.1<\/p>\n\n<\/header>\n<div class=\"textbox__content\">\n<div data-type=\"problem\">\n<p id=\"fs-id2940472\">Simplify: a) \\(\\phantom{\\rule{0.2em}{0ex}}\\sqrt{36}\\phantom{\\rule{0.2em}{0ex}}\\) b) \\(\\phantom{\\rule{0.2em}{0ex}}\\sqrt{169}\\).<\/p>\n\n<\/div>\n<div data-type=\"solution\"><details><summary>Show answer<\/summary>\n<ol id=\"eip-id1168468305633\" class=\"circled\" type=\"a\">\n \t<li>6<\/li>\n \t<li>13<\/li>\n<\/ol>\n<\/details><\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id2323788\" class=\"try\" data-type=\"note\">\n<div id=\"fs-id2483689\" data-type=\"exercise\">\n<div data-type=\"problem\">\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 1.2<\/p>\n\n<\/header>\n<div class=\"textbox__content\">\n<div data-type=\"problem\">\n\nSimplify: a) \\(\\phantom{\\rule{0.2em}{0ex}}\\sqrt{16}\\phantom{\\rule{0.2em}{0ex}}\\) b) \\(\\phantom{\\rule{0.2em}{0ex}}\\sqrt{196}\\).\n\n<\/div>\n<div data-type=\"solution\"><details><summary>Show answer<\/summary>\n<ol id=\"eip-id1168469858541\" class=\"circled\" type=\"a\">\n \t<li>4<\/li>\n \t<li>14<\/li>\n<\/ol>\n<\/details><\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1714044\">Every positive number has two <span class=\"no-emphasis\" data-type=\"term\">square root<\/span>s 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, \\(-\\sqrt{100}=-10\\).<\/p>\n\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 data-type=\"problem\">\n<p id=\"fs-id2372735\">Simplify. a) \\(\\phantom{\\rule{0.2em}{0ex}}-\\sqrt{9}\\phantom{\\rule{0.2em}{0ex}}\\) b) \\(\\phantom{\\rule{0.2em}{0ex}}-\\sqrt{144.}\\)<\/p>\n\n<\/div>\n<div data-type=\"solution\">\n<div data-type=\"title\"><\/div>\n<div data-type=\"title\"><strong>Solution<\/strong><\/div>\n<table id=\"eip-id1168469874827\" class=\"unnumbered unstyled\" summary=\".\" data-label=\"\">\n<tbody>\n<tr>\n<td>a)<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>\\(-\\sqrt{9}\\)<\/td>\n<\/tr>\n<tr>\n<td>The negative is in front of the radical sign.<\/td>\n<td>\\(-3\\)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table id=\"eip-id1168466190277\" class=\"unnumbered unstyled\" summary=\".\" data-label=\"\">\n<tbody>\n<tr>\n<td>b)<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>\\(-\\sqrt{144}\\)<\/td>\n<\/tr>\n<tr>\n<td>The negative is in front of the radical sign.<\/td>\n<td>\\(-12\\)<\/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 2.1<\/p>\n\n<\/header>\n<div class=\"textbox__content\">\n<div data-type=\"problem\">\n\nSimplify: a) \\(\\phantom{\\rule{0.2em}{0ex}}-\\sqrt{4}\\phantom{\\rule{0.2em}{0ex}}\\) b) \\(\\phantom{\\rule{0.2em}{0ex}}-\\sqrt{225}\\).\n\n<\/div>\n<div id=\"fs-id1753416\" data-type=\"solution\"><details><summary>Show answer<\/summary>\n<ol id=\"eip-id1168469781883\" class=\"circled\" type=\"a\">\n \t<li>\u22122<\/li>\n \t<li>\u221215<\/li>\n<\/ol>\n<\/details><\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 2.2<\/p>\n\n<\/header>\n<div class=\"textbox__content\">\n<div data-type=\"problem\">\n<p id=\"fs-id1363622\">Simplify: a) \\(\\phantom{\\rule{0.2em}{0ex}}-\\sqrt{81}\\phantom{\\rule{0.2em}{0ex}}\\) b) \\(\\phantom{\\rule{0.2em}{0ex}}-\\sqrt{64}\\).<\/p>\n\n<\/div>\n<div id=\"fs-id1322037\" data-type=\"solution\"><details><summary>Show answer<\/summary>\n<ol id=\"eip-id1168468356466\" class=\"circled\" type=\"a\">\n \t<li>\u22129<\/li>\n \t<li>\u22128<\/li>\n<\/ol>\n<\/details><\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1398123\" class=\"try\" data-type=\"note\">\n<div data-type=\"exercise\"><\/div>\n<\/div>\n<h1>Square Root of a Negative Number<\/h1>\n<p id=\"fs-id1737447\">Can we simplify \\(\\sqrt{-25}?\\) Is there a number whose square is \\(-25?\\)<\/p>\n\n<div id=\"fs-id1231669\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\">\\({\\left(\\phantom{\\rule{1em}{0ex}}\\right)}^{2}=-25?\\)<\/div>\nNone of the numbers that we have dealt with so far have a square that is \\(-25\\). Why? Any positive number squared is positive, and any negative number squared is also positive. In the next chapter we will see that all the numbers we work with are called the real numbers. So we say there is no real number equal to \\(\\sqrt{-25}\\). If we are asked to find the <span class=\"no-emphasis\" data-type=\"term\">square root<\/span> of any negative number, we say that the solution is not a real number.\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-id1518393\" data-type=\"problem\">\n<p id=\"fs-id1613479\">Simplify: a) \\(\\phantom{\\rule{0.2em}{0ex}}\\sqrt{-169}\\phantom{\\rule{0.2em}{0ex}}\\) b) \\(\\phantom{\\rule{0.2em}{0ex}}-\\sqrt{121}\\).<\/p>\n\n<\/div>\n<div id=\"fs-id3015148\" data-type=\"solution\">\n<div data-type=\"title\"><\/div>\n<div data-type=\"title\"><strong>Solution<\/strong><\/div>\n<div data-type=\"title\"><\/div>\n<p id=\"fs-id2788439\">a) There is no real number whose square is \\(-169\\). Therefore, \\(\\sqrt{-169}\\) is not a real number.<\/p>\n<p id=\"fs-id1807537\">b) The negative is in front of the radical sign, so we find the opposite of the square root of \\(121\\).<\/p>\n\n<table id=\"eip-id1168466130843\" class=\"unnumbered unstyled\" summary=\".\" data-label=\"\">\n<tbody>\n<tr>\n<td><\/td>\n<td>\\(-\\sqrt{121}\\)<\/td>\n<\/tr>\n<tr>\n<td>The negative is in front of the radical.<\/td>\n<td>\\(-11\\)<\/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 3.1<\/p>\n\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id2382510\" data-type=\"problem\">\n<p id=\"fs-id1834970\">Simplify: a) \\(\\phantom{\\rule{0.2em}{0ex}}\\sqrt{-196}\\phantom{\\rule{0.2em}{0ex}}\\) b) \\(\\phantom{\\rule{0.2em}{0ex}}-\\sqrt{81}\\).<\/p>\n\n<\/div>\n<div id=\"fs-id2470950\" data-type=\"solution\"><details><summary>Show answer<\/summary>\n<ol id=\"eip-id1168468279641\" class=\"circled\" type=\"a\">\n \t<li>not a real number<\/li>\n \t<li>\u22129<\/li>\n<\/ol>\n<\/details><\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id2170175\" class=\"try\" data-type=\"note\">\n<div id=\"fs-id3000454\" data-type=\"exercise\">\n<div id=\"fs-id2382510\" data-type=\"problem\">\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 3.2<\/p>\n\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id2488039\" data-type=\"problem\">\n<p id=\"fs-id2139146\">Simplify: a) \\(\\phantom{\\rule{0.2em}{0ex}}\\sqrt{-49}\\phantom{\\rule{0.2em}{0ex}}\\) b) \\(\\phantom{\\rule{0.2em}{0ex}}-\\sqrt{121}\\).<\/p>\n\n<\/div>\n<div id=\"fs-id2708177\" data-type=\"solution\"><details><summary>Show answer<\/summary>\n<ol id=\"eip-id1168469427053\" class=\"circled\" type=\"a\">\n \t<li>\u22127<\/li>\n \t<li>not a real number<\/li>\n<\/ol>\n<\/details><\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<h1>Square Roots and the Order of Operations<\/h1>\n<p id=\"fs-id2199152\">When using the order of operations to simplify an expression that has square roots, we treat the radical sign as a grouping symbol. We simplify any expressions under the radical sign before performing other operations.<\/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-id2628173\" data-type=\"problem\">\n<p id=\"fs-id2445144\">Simplify: a) \\(\\phantom{\\rule{0.2em}{0ex}}\\sqrt{25}+\\sqrt{144}\\phantom{\\rule{0.2em}{0ex}}\\) b) \\(\\phantom{\\rule{0.2em}{0ex}}\\sqrt{25+144}\\).<\/p>\n\n<\/div>\n<div id=\"fs-id2202087\" data-type=\"solution\">\n<div data-type=\"title\"><\/div>\n<div data-type=\"title\"><strong>Solution<\/strong><\/div>\n<table id=\"eip-id1168466048150\" class=\"unnumbered unstyled\" summary=\".\" data-label=\"\">\n<tbody>\n<tr>\n<td>a) Use the order of operations.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>\\(\\sqrt{25}+\\sqrt{144}\\)<\/td>\n<\/tr>\n<tr>\n<td>Simplify each radical.<\/td>\n<td>\\(5+12\\)<\/td>\n<\/tr>\n<tr>\n<td>Add.<\/td>\n<td>\\(17\\)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table id=\"eip-id1168469785439\" class=\"unnumbered unstyled\" summary=\".\" data-label=\"\">\n<tbody>\n<tr>\n<td>b) Use the order of operations.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>\\(\\sqrt{25+144}\\)<\/td>\n<\/tr>\n<tr>\n<td>Add under the radical sign.<\/td>\n<td>\\(\\sqrt{169}\\)<\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td>\\(13\\)<\/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 4.1<\/p>\n\n<\/header>\n<div class=\"textbox__content\">\n<div data-type=\"problem\">\n<p id=\"fs-id1829806\">Simplify: a) \\(\\phantom{\\rule{0.2em}{0ex}}\\sqrt{9}+\\sqrt{16}\\phantom{\\rule{0.2em}{0ex}}\\) b) \\(\\phantom{\\rule{0.2em}{0ex}}\\sqrt{9+16}\\).<\/p>\n\n<\/div>\n<div id=\"fs-id2583182\" data-type=\"solution\"><details><summary>Show answer<\/summary>\n<ol id=\"eip-id1168468305202\" class=\"circled\" type=\"a\">\n \t<li>7<\/li>\n \t<li>5<\/li>\n<\/ol>\n<\/details><\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id2195519\" class=\"try\" data-type=\"note\">\n<div id=\"fs-id2644747\" data-type=\"exercise\">\n<div id=\"fs-id2213867\" data-type=\"problem\">\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 4.2<\/p>\n\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id2213867\" data-type=\"problem\">\n\nSimplify: a) \\(\\phantom{\\rule{0.2em}{0ex}}\\sqrt{64+225}\\phantom{\\rule{0.2em}{0ex}}\\) b) \\(\\phantom{\\rule{0.2em}{0ex}}\\sqrt{64}+\\sqrt{225}\\).\n\n<\/div>\n<div id=\"fs-id1529759\" data-type=\"solution\"><details><summary>Show answer<\/summary>\n<ol id=\"eip-id1168468526118\" class=\"circled\" type=\"a\">\n \t<li>17<\/li>\n \t<li>23<\/li>\n<\/ol>\n<\/details><\/div>\n<\/div>\n<\/div>\nNotice the different answers in parts a) and b) of <a href=\"#fs-id2628173\">(Example 4)<\/a>. It is important to follow the order of operations correctly. In a), we took each square root first and then added them. In b), we added under the radical sign first and then found the square root.\n\n<\/div>\n<\/div>\n<\/div>\n<h1>Estimate Square Roots<\/h1>\n<p id=\"fs-id1731496\">So far we have only worked with square roots of perfect squares. The square roots of other numbers are not whole numbers.<\/p>\n<span id=\"fs-id1788761\" data-type=\"media\" data-alt=\"A table is shown with 2 columns. The first column is labeled \u201cNumber\u201d and contains the values: 4, 5, 6, 7, 8, 9. The second column is labeled \u201cSquare root\u201d and contains the values: square root of 4 equals 2, square root of 5, square root of 6, square root of 7, square root of 8, square root of 9 equals 3.\"><img class=\"aligncenter\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_05_07_007_img.jpg\" alt=\"A table is shown with 2 columns. The first column is labeled \u201cNumber\u201d and contains the values: 4, 5, 6, 7, 8, 9. The second column is labeled \u201cSquare root\u201d and contains the values: square root of 4 equals 2, square root of 5, square root of 6, square root of 7, square root of 8, square root of 9 equals 3.\" data-media-type=\"image\/jpeg\"><\/span>\n<p id=\"fs-id1569078\">We might conclude that the square roots of numbers between \\(4\\) and \\(9\\) will be between \\(2\\) and \\(3\\), and they will not be whole numbers. Based on the pattern in the table above, we could say that \\(\\sqrt{5}\\) is between \\(2\\) and \\(3\\). Using inequality symbols, we write<\/p>\n\n<div id=\"fs-id3324454\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\">\\(2&lt;\\sqrt{5}&lt;3\\)<\/div>\n<div data-type=\"equation\" data-label=\"\"><\/div>\n<div data-type=\"equation\" data-label=\"\">\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-id2136031\" data-type=\"problem\">\n<p id=\"fs-id2327121\">Estimate \\(\\sqrt{60}\\) between two consecutive whole numbers.<\/p>\n\n<\/div>\n<div id=\"fs-id3014912\" data-type=\"solution\">\n<div data-type=\"title\"><\/div>\n<div data-type=\"title\"><strong>Solution<\/strong><\/div>\n<div data-type=\"title\"><\/div>\n<p id=\"fs-id1385193\">Think of the perfect squares closest to \\(60\\). Make a small table of these perfect squares and their squares roots.<\/p>\n<span id=\"fs-id2419067\" data-type=\"media\" data-alt=\"A table is shown with 2 columns. The first column is labeled \u201cNumber\u201d and contains the values: 36, 49, 64, and 81. There is a balloon coming out of the table between 49 and 64 that says 60. The second column is labeled \u201cSquare root\u201d and contains the values: 6, 7, 8, and 9. There is a balloon coming out of the table between 7 and 8 that says square root of 60.\"><img class=\"aligncenter\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_05_07_006_img.jpg\" alt=\"A table is shown with 2 columns. The first column is labeled \u201cNumber\u201d and contains the values: 36, 49, 64, and 81. There is a balloon coming out of the table between 49 and 64 that says 60. The second column is labeled \u201cSquare root\u201d and contains the values: 6, 7, 8, and 9. There is a balloon coming out of the table between 7 and 8 that says square root of 60.\" data-media-type=\"image\/jpeg\"><\/span>\n<table id=\"eip-id1168467173482\" class=\"unnumbered unstyled\" summary=\".\" data-label=\"\">\n<tbody>\n<tr>\n<td>\\(\\text{Locate 60 between two consecutive perfect squares.}\\)<\/td>\n<td>\\(49&lt;60&lt;64\\)<\/td>\n<\/tr>\n<tr>\n<td>\\(\\sqrt{60}\\phantom{\\rule{0.2em}{0ex}}\\text{is between their square roots.}\\)<\/td>\n<td>\\(7&lt;\\sqrt{60}&lt;8\\)<\/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 5.1<\/p>\n\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1883735\" data-type=\"problem\">\n<p id=\"fs-id2171736\">Estimate \\(\\sqrt{38}\\) between two consecutive whole numbers.<\/p>\n\n<\/div>\n<div id=\"fs-id1883425\" data-type=\"solution\"><details><summary>Show answer<\/summary>\n<p id=\"fs-id2906465\">\\(6&lt;\\sqrt{38}&lt;7\\)<\/p>\n\n<\/details><\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 5.2<\/p>\n\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id2084941\" data-type=\"problem\">\n<p id=\"fs-id1731218\">Estimate \\(\\sqrt{84}\\) between two consecutive whole numbers.<\/p>\n\n<\/div>\n<div id=\"fs-id2139079\" data-type=\"solution\"><details><summary>Show answer<\/summary>\n<p id=\"fs-id1410348\">\\(9&lt;\\sqrt{84}&lt;10\\)<\/p>\n\n<\/details><\/div>\n<\/div>\n<\/div>\n<\/div>\n<h1>Approximate Square Roots with a Calculator<\/h1>\nThe square roots of\u00a0 numbers that are not\u00a0 perfect squares are not whole numbers, they are irrational numbers. Its decimal form does not stop and does not repeat. Are irrational numbers real numbers? Yes, they are. When we put together the irrational numbers and rational numbers, we get the set of real numbers.\n\nLet's see how we can use calculator to find the approximate square roots of those irrational numbers.\n\nThere are mathematical methods to approximate square roots, but it is much more convenient to use a calculator to find square roots. Find the \\(\\sqrt{\\phantom{0}}\\) or \\(\\sqrt{x}\\) key on your calculator. You will to use this key to approximate square roots. When you use your calculator to find the square root of a number that is not a perfect square, the answer that you see is not the exact number. It is an approximation, to the number of digits shown on your calculator\u2019s display. The symbol for an approximation is \\(\\approx \\) and it is read <em data-effect=\"italics\">approximately<\/em>.\n<p id=\"fs-id2669486\">Suppose your calculator has a \\(\\text{10-digit}\\) display. Using it to find the square root of \\(5\\) will give \\(2.236067977\\). This is the approximate square root of \\(5\\). When we report the answer, we should use the \u201capproximately equal to\u201d sign instead of an equal sign.<\/p>\n\n<div id=\"fs-id2168295\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\">\\(\\sqrt{5}\\approx 2.236067978\\). The square root of \\(5\\) is the example of irrational number and its approximation displays nine digits after the decimal place.<\/div>\n<p id=\"fs-id2137210\">You will seldom use this many digits for applications in algebra. So, if you wanted to round \\(\\sqrt{5}\\) to two decimal places, you would write<\/p>\n\n<div id=\"fs-id1337932\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\">\\(\\sqrt{5}\\approx 2.24\\)<\/div>\n<p id=\"fs-id2174920\">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-id1816562\" class=\"unnumbered\" data-type=\"equation\">\\(\\begin{array}{ccc}\\hfill {2.236067978}^{2}&amp; =&amp; 5.000000002\\hfill \\\\ \\hfill {2.24}^{2}&amp; =&amp; 5.0176\\hfill \\end{array}\\)<\/div>\n<p id=\"fs-id1811799\">The squares are close, but not exactly equal, to \\(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 data-type=\"problem\">\n<p id=\"fs-id3961155\">Round \\(\\sqrt{17}\\) to two decimal places using a calculator.<\/p>\n\n<\/div>\n<div id=\"fs-id1862325\" data-type=\"solution\">\n<div data-type=\"title\"><\/div>\n<div data-type=\"title\"><strong>Solution<\/strong><\/div>\n<table id=\"eip-id1168468614270\" class=\"unnumbered unstyled\" summary=\".\" data-label=\"\">\n<tbody>\n<tr>\n<td><\/td>\n<td>\\(\\sqrt{17}\\)<\/td>\n<\/tr>\n<tr>\n<td>Use the calculator square root key.<\/td>\n<td>\\(4.123105626\\)<\/td>\n<\/tr>\n<tr>\n<td>Round to two decimal places.<\/td>\n<td>\\(4.12\\)<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>\\(\\sqrt{17}\\approx 4.12\\)<\/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 6.1<\/p>\n\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id2472223\" data-type=\"problem\">\n\nRound \\(\\sqrt{11}\\) to two decimal places.\n\n<\/div>\n<div id=\"fs-id1166761303640\" data-type=\"solution\"><details><summary>Show answer<\/summary>\u2248 3.32\n\n<\/details><\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 6.2<\/p>\n\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1613047\" data-type=\"problem\">\n<p id=\"fs-id1362232\">Round \\(\\sqrt{13}\\) to two decimal places.<\/p>\n\n<\/div>\n<div id=\"fs-id4326999\" data-type=\"solution\"><details><summary>Show answer<\/summary>\n<p id=\"fs-id1863006\">\u2248 3.61<\/p>\n\n<\/details><\/div>\n<\/div>\n<\/div>\n<h1>Simplify Variable Expressions with Square Roots<\/h1>\n<p id=\"fs-id1619406\">Expressions with square root that we have looked at so far have not had any variables. What happens when we have to find a square root of a variable expression?<\/p>\n<p id=\"fs-id2244406\">Consider \\(\\sqrt{9{x}^{2}}\\), where \\(x\\ge 0\\). Can you think of an expression whose square is \\(9{x}^{2}?\\)<\/p>\n\n<div id=\"fs-id2281080\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\">\\(\\begin{array}{ccc}\\hfill {\\left(?\\right)}^{2}&amp; =&amp; 9{x}^{2}\\hfill \\\\ \\hfill {\\left(3x\\right)}^{2}&amp; =&amp; 9{x}^{2}\\phantom{\\rule{2em}{0ex}}\\text{so}\\phantom{\\rule{0.2em}{0ex}}\\sqrt{9{x}^{2}}=3x\\hfill \\end{array}\\)<\/div>\n<p id=\"fs-id2279868\">When we use a variable in a square root expression, for our work, we will assume that the variable represents a non-negative number. In every example and exercise that follows, each variable in a square root expression is greater than or equal to zero.<\/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-id3016846\" data-type=\"problem\">\n<p id=\"fs-id1518785\">Simplify: \\(\\sqrt{{x}^{2}}\\).<\/p>\n\n<\/div>\n<div id=\"fs-id1969210\" data-type=\"solution\">\n<div data-type=\"title\"><\/div>\n<div data-type=\"title\"><strong>Solution<\/strong><\/div>\n<div data-type=\"title\"><\/div>\n<p id=\"eip-id1170368510506\">Think about what we would have to square to get \\({x}^{2}\\). Algebraically, \\({\\left(?\\right)}^{2}={x}^{2}\\)<\/p>\n\n<table id=\"eip-id1168467419284\" class=\"unnumbered unstyled\" summary=\".\" data-label=\"\">\n<tbody>\n<tr>\n<td><\/td>\n<td>\\(\\sqrt{{x}^{2}}\\)<\/td>\n<\/tr>\n<tr>\n<td>Since \\({\\left(x\\right)}^{2}={x}^{2}\\)<\/td>\n<td>\\(x\\)<\/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 7.1<\/p>\n\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id2260926\" data-type=\"problem\">\n<p id=\"fs-id2784470\">Simplify: \\(\\sqrt{{y}^{2}}\\).<\/p>\n\n<\/div>\n<div id=\"fs-id1568264\" data-type=\"solution\"><details><summary>Show answer<\/summary>\n<p id=\"fs-id1789141\"><em data-effect=\"italics\">y<\/em><\/p>\n\n<\/details><\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 7.2<\/p>\n\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1733267\" data-type=\"problem\">\n<p id=\"fs-id1171104390264\">Simplify: \\(\\sqrt{{m}^{2}}\\).<\/p>\n\n<\/div>\n<div id=\"fs-id2268730\" data-type=\"solution\"><details><summary>Show answer<\/summary>\n<p id=\"fs-id2900277\"><em data-effect=\"italics\">m<\/em><\/p>\n\n<\/details><\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1171103937955\" class=\"try\" data-type=\"note\">\n<div id=\"fs-id1202617\" data-type=\"exercise\">\n<div id=\"fs-id1733267\" data-type=\"problem\">\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-id1968764\" data-type=\"problem\">\n<p id=\"fs-id2208836\">Simplify: \\(\\sqrt{16{x}^{2}}\\).<\/p>\n\n<\/div>\n<div id=\"fs-id2376108\" data-type=\"solution\">\n<div data-type=\"title\"><\/div>\n<div data-type=\"title\"><strong>Solution<\/strong><\/div>\n<table id=\"eip-id1168466092245\" class=\"unnumbered unstyled\" summary=\".\" data-label=\"\">\n<tbody>\n<tr>\n<td><\/td>\n<td>\\(\\sqrt{16{x}^{2}}\\)<\/td>\n<\/tr>\n<tr>\n<td>\\(\\text{Since}\\phantom{\\rule{0.2em}{0ex}}{\\left(4x\\right)}^{2}=16{x}^{2}\\)<\/td>\n<td>\\(4x\\)<\/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 8.1<\/p>\n\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id2170992\" data-type=\"problem\">\n<p id=\"fs-id2597350\">Simplify: \\(\\sqrt{64{x}^{2}}\\).<\/p>\n\n<\/div>\n<div id=\"fs-id1752622\" data-type=\"solution\"><details><summary>Show answer<\/summary>\n<p id=\"fs-id2359180\">8<em data-effect=\"italics\">x<\/em><\/p>\n\n<\/details><\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 8.2<\/p>\n\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id2660664\" data-type=\"problem\">\n<p id=\"fs-id3408056\">Simplify: \\(\\sqrt{169{y}^{2}}\\).<\/p>\n\n<\/div>\n<div id=\"fs-id2686481\" data-type=\"solution\"><details><summary>Show answer<\/summary>\n<p id=\"fs-id2675196\">13<em data-effect=\"italics\">y<\/em><\/p>\n\n<\/details><\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\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-id1714033\" data-type=\"problem\">\n<p id=\"fs-id2793037\">Simplify: \\(-\\sqrt{81{y}^{2}}\\).<\/p>\n\n<\/div>\n<div id=\"fs-id1528886\" data-type=\"solution\">\n<div data-type=\"title\"><\/div>\n<div data-type=\"title\"><strong>Solution<\/strong><\/div>\n<table id=\"eip-id1168468389692\" class=\"unnumbered unstyled\" summary=\".\" data-label=\"\">\n<tbody>\n<tr>\n<td><\/td>\n<td>\\(-\\sqrt{81{y}^{2}}\\)<\/td>\n<\/tr>\n<tr>\n<td>\\(\\text{Since}\\phantom{\\rule{0.2em}{0ex}}{\\left(9y\\right)}^{2}=81{y}^{2}\\)<\/td>\n<td>\\(-9y\\)<\/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 9.1<\/p>\n\n<\/header>\n<div class=\"textbox__content\">\n<div data-type=\"problem\">\n<p id=\"fs-id2140225\">Simplify: \\(-\\sqrt{121{y}^{2}}\\).<\/p>\n\n<\/div>\n<div id=\"fs-id1724861\" data-type=\"solution\"><details><summary>Show answer<\/summary>\n<p id=\"fs-id2907510\">\u221211<em data-effect=\"italics\">y<\/em><\/p>\n\n<\/details><\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 9.2<\/p>\n\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1773567\" data-type=\"problem\">\n<p id=\"fs-id1773569\">Simplify: \\(-\\sqrt{100{p}^{2}}\\).<\/p>\n\n<\/div>\n<div id=\"fs-id3961816\" data-type=\"solution\"><details><summary>Show answer<\/summary>\u221210<em data-effect=\"italics\">p<\/em>\n\n<\/details><\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1546320\" class=\"try\" data-type=\"note\">\n<div id=\"fs-id2345827\" data-type=\"exercise\">\n<div id=\"fs-id3961816\" data-type=\"solution\">\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-id1943547\" data-type=\"problem\">\n<p id=\"fs-id1588623\">Simplify: \\(\\sqrt{36{x}^{2}{y}^{2}}\\).<\/p>\n\n<\/div>\n<div id=\"fs-id2684921\" data-type=\"solution\">\n<div data-type=\"title\"><\/div>\n<div data-type=\"title\"><strong>Solution<\/strong><\/div>\n<table id=\"eip-id1168466004603\" class=\"unnumbered unstyled\" summary=\".\" data-label=\"\">\n<tbody>\n<tr>\n<td><\/td>\n<td>\\(\\sqrt{36{x}^{2}{y}^{2}}\\)<\/td>\n<\/tr>\n<tr>\n<td>\\(\\text{Since}\\phantom{\\rule{0.2em}{0ex}}{\\left(6xy\\right)}^{2}=36{x}^{2}{y}^{2}\\)<\/td>\n<td>\\(6xy\\)<\/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 10.1<\/p>\n\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1883085\" data-type=\"problem\">\n<p id=\"fs-id13852320\">Simplify: \\(\\sqrt{100{a}^{2}{b}^{2}}\\).<\/p>\n\n<\/div>\n<div id=\"fs-id1502705\" data-type=\"solution\"><details><summary>Show answer<\/summary>\n<p id=\"fs-id1502707\">10<em data-effect=\"italics\">ab<\/em><\/p>\n\n<\/details><\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 10.2<\/p>\n\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id2135180\" data-type=\"problem\">\n<p id=\"fs-id1385232\">Simplify: \\(\\sqrt{225{m}^{2}{n}^{2}}\\).<\/p>\n\n<\/div>\n<div id=\"fs-id1735566\" data-type=\"solution\"><details><summary>Show answer<\/summary>\n<p id=\"fs-id2608595\">15<em data-effect=\"italics\">mn<\/em><\/p>\n\n<\/details><\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<h1>Use Square Roots in Applications<\/h1>\n<p id=\"fs-id2451417\">As you progress through your college courses, you\u2019ll encounter several applications of square roots. Once again, if we use our strategy for applications, it will give us a plan for finding the answer!<\/p>\n\n<div id=\"fs-id2754333\" class=\"howto\" data-type=\"note\">\n<div data-type=\"title\">\n<div class=\"textbox textbox--key-takeaways\"><header class=\"textbox__header\">HOW TO: Use a strategy for applications with square roots.<\/header>\n<div class=\"textbox__content\">\n<ol id=\"eip-id1168466250274\" class=\"stepwise\" type=\"1\">\n \t<li>Identify what you are asked to find.<\/li>\n \t<li>Write a phrase that gives the information to find it.<\/li>\n \t<li>Translate the phrase to an expression.<\/li>\n \t<li>Simplify the expression.<\/li>\n \t<li>Write a complete sentence that answers the question.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<h1>Square Roots and Area<\/h1>\n<p id=\"fs-id1956366\">We have solved applications with <span class=\"no-emphasis\" data-type=\"term\">area<\/span> before. If we were given the length of the sides of a square, we could find its area by squaring the length of its sides. Now we can find the length of the sides of a square if we are given the area, by finding the square root of the area.<\/p>\n<p id=\"fs-id2430870\">If the area of the square is \\(A\\) square units, the length of a side is \\(\\sqrt{A}\\) units. See the table below.<\/p>\n\n<table id=\"fs-id1973397\" class=\"aligncenter\" summary=\"A table is shown with 2 columns and 4 rows. The first column is labeled \u201cArea (square units)\u201d and contains the values: 9, 144, A. The second column is labeled \u201cLength of side (units)\u201d and contains the values: square root of 9 equals 3, square root of 144 equals 12, square root of A.\" data-frame=\"none\">\n<thead>\n<tr valign=\"top\">\n<th data-align=\"left\">Area (square units)<\/th>\n<th data-align=\"left\">Length of side (units)<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr valign=\"top\">\n<td data-align=\"center\">\\(9\\)<\/td>\n<td data-align=\"center\">\\(\\sqrt{9}=3\\)<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-align=\"center\">\\(144\\)<\/td>\n<td data-align=\"center\">\\(\\sqrt{144}=12\\)<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-align=\"center\">\\(A\\)<\/td>\n<td data-align=\"center\">\\(\\sqrt{A}\\)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div id=\"fs-id1736567\" class=\"try\" data-type=\"note\">\n<div id=\"fs-id1736570\" data-type=\"exercise\">\n<div id=\"fs-id1736572\" data-type=\"problem\">\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\n<p class=\"textbox__title\">EXAMPLE 11<\/p>\n\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1736581\" data-type=\"problem\">\n<p id=\"fs-id1736583\">Mike and Lychelle want to make a square patio. They have enough concrete for an area of \\(200\\) square feet. To the nearest tenth of a foot, how long can a side of their square patio be?<\/p>\n\n<\/div>\n<div id=\"fs-id1979887\" data-type=\"solution\">\n<div data-type=\"title\"><\/div>\n<div data-type=\"title\"><strong>Solution<\/strong><\/div>\n<div data-type=\"title\"><\/div>\n<p id=\"fs-id2686507\">We know the area of the square is \\(200\\) square feet and want to find the length of the side. If the area of the square is \\(A\\) square units, the length of a side is \\(\\sqrt{A}\\) units.<\/p>\n\n<table id=\"eip-id1168468768448\" class=\"unnumbered unstyled\" summary=\".\" data-label=\"\">\n<tbody>\n<tr>\n<td>What are you asked to find?<\/td>\n<td>The length of each side of a square patio<\/td>\n<\/tr>\n<tr>\n<td>Write a phrase.<\/td>\n<td>The length of a side<\/td>\n<\/tr>\n<tr>\n<td>Translate to an expression.<\/td>\n<td>\\(\\sqrt{A}\\)<\/td>\n<\/tr>\n<tr>\n<td>Evaluate \\(\\sqrt{A}\\) when \\(A=200\\).<\/td>\n<td>\\(\\sqrt{200}\\)<\/td>\n<\/tr>\n<tr>\n<td>Use your calculator.<\/td>\n<td>\\(14.142135..\\).<\/td>\n<\/tr>\n<tr>\n<td>Round to one decimal place.<\/td>\n<td>\\(\\text{14.1 feet}\\)<\/td>\n<\/tr>\n<tr>\n<td>Write a sentence.<\/td>\n<td>Each side of the patio should be \\(14.1\\) feet.<\/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 11.1<\/p>\n\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1736572\" data-type=\"problem\">\n<p id=\"fs-id1966485\">Katie wants to plant a square lawn in her front yard. She has enough sod to cover an area of \\(370\\) square feet. To the nearest tenth of a foot, how long can a side of her square lawn be?<\/p>\n\n<\/div>\n<div id=\"fs-id1588636\" data-type=\"solution\"><details open=\"open\"><summary>Show answer<\/summary>\n<p id=\"fs-id2708152\">19.2 feet<\/p>\n\n<\/details><\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 11.2<\/p>\n\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id2283966\" data-type=\"problem\">\n<p id=\"fs-id2707825\">Sergio wants to make a square mosaic as an inlay for a table he is building. He has enough tile to cover an area of \\(2704\\) square centimetres. How long can a side of his mosaic be?<\/p>\n\n<\/div>\n<div id=\"fs-id1377070\" data-type=\"solution\"><details open=\"open\"><summary>Show answer<\/summary>\n<p id=\"fs-id1460987\">52 centimetres<\/p>\n\n<\/details><\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<h1>Square Roots and Gravity<\/h1>\n<p id=\"fs-id1530631\">Another application of <span class=\"no-emphasis\" data-type=\"term\">square root<\/span>s involves <span class=\"no-emphasis\" data-type=\"term\">gravity<\/span>. On Earth, if an object is dropped from a height of \\(h\\) feet, the time in seconds it will take to reach the ground is found by evaluating the expression \\(\\frac{\\sqrt{h}}{4}\\). 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 evaluating \\(\\frac{\\sqrt{64}}{4}\\).<\/p>\n\n<table id=\"eip-id1168466722027\" class=\"unnumbered unstyled\" summary=\".\" data-label=\"\">\n<tbody>\n<tr>\n<td><\/td>\n<td>\\(\\frac{\\sqrt{64}}{4}\\)<\/td>\n<\/tr>\n<tr>\n<td>Take the square root of 64.<\/td>\n<td>\\(\\frac{8}{4}\\)<\/td>\n<\/tr>\n<tr>\n<td>Simplify the fraction.<\/td>\n<td>\\(2\\)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id2946358\">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 12<\/p>\n\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id3415154\" data-type=\"problem\">\n<p id=\"fs-id3415156\">Christy dropped her sunglasses from a bridge \\(400\\) feet above a river. How many seconds does it take for the sunglasses to reach the river?<\/p>\n\n<\/div>\n<div id=\"fs-id4210836\" data-type=\"solution\">\n<div data-type=\"title\"><\/div>\n<div data-type=\"title\"><strong>Solution<\/strong><\/div>\n<table id=\"eip-id1168468767850\" class=\"unnumbered unstyled\" summary=\".\" data-label=\"\">\n<tbody>\n<tr>\n<td>What are you asked to find?<\/td>\n<td>The number of seconds it takes for the sunglasses to reach the river<\/td>\n<\/tr>\n<tr>\n<td>Write a phrase.<\/td>\n<td>The time it will take to reach the river<\/td>\n<\/tr>\n<tr>\n<td>Translate to an expression.<\/td>\n<td>\\(\\frac{\\sqrt{h}}{4}\\)<\/td>\n<\/tr>\n<tr>\n<td>Evaluate \\(\\frac{\\sqrt{h}}{4}\\) when \\(h=400\\).<\/td>\n<td>\\(\\frac{\\sqrt{400}}{4}\\)<\/td>\n<\/tr>\n<tr>\n<td>Find the square root of 400.<\/td>\n<td>\\(\\frac{20}{4}\\)<\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td>\\(5\\)<\/td>\n<\/tr>\n<tr>\n<td>Write a sentence.<\/td>\n<td>It will take 5 seconds for the sunglasses 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 12.1<\/p>\n\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1960973\" data-type=\"problem\">\n<p id=\"fs-id1960975\">A helicopter drops a rescue package from a height of \\(1296\\) feet. How many seconds does it take for the package to reach the ground?<\/p>\n\n<\/div>\n<div id=\"fs-id3326094\" data-type=\"solution\"><details open=\"open\"><summary>Show answer<\/summary>\n<p id=\"fs-id3326096\">9 seconds<\/p>\n\n<\/details><\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 12.2<\/p>\n\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id2742055\" data-type=\"problem\">\n<p id=\"fs-id2742057\">A window washer drops a squeegee from a platform \\(196\\) feet above the sidewalk. How many seconds does it take for the squeegee to reach the sidewalk?<\/p>\n\n<\/div>\n<div id=\"fs-id1884553\" data-type=\"solution\"><details open=\"open\"><summary>Show answer<\/summary>\n<p id=\"fs-id1884555\">3.5 seconds<\/p>\n\n<\/details><\/div>\n<\/div>\n<\/div>\n<h1>Square Roots and Accident Investigations<\/h1>\n<p id=\"fs-id1933926\">Police officers investigating car accidents measure the length of the skid marks on the pavement. Then they use square roots to determine the speed, in miles per hour, a car was going before applying the brakes. According to some formulas, if the length of the skid marks is \\(d\\) feet, then the speed of the car can be found by evaluating \\(\\sqrt{24d}\\).<\/p>\n\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\n<p class=\"textbox__title\">EXAMPLE 13<\/p>\n\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id3327796\" data-type=\"problem\">\n<p id=\"fs-id3327798\">After a car accident, the skid marks for one car measured \\(190\\) feet. To the nearest tenth, what was the speed of the car (in mph) before the brakes were applied?<\/p>\n\n<\/div>\n<div id=\"fs-id2134026\" data-type=\"solution\">\n<div data-type=\"title\"><\/div>\n<div data-type=\"title\"><strong>Solution<\/strong><\/div>\n<table id=\"eip-id1168468505344\" class=\"unnumbered unstyled\" summary=\".\" data-label=\"\">\n<tbody>\n<tr>\n<td>What are you asked to find?<\/td>\n<td>The speed of the car before the brakes were applied<\/td>\n<\/tr>\n<tr>\n<td>Write a phrase.<\/td>\n<td>The speed of the car<\/td>\n<\/tr>\n<tr>\n<td>Translate to an expression.<\/td>\n<td>\\(\\sqrt{24d}\\)<\/td>\n<\/tr>\n<tr>\n<td>Evaluate\\(\\phantom{\\rule{0.2em}{0ex}}\\sqrt{24d}\\phantom{\\rule{0.2em}{0ex}}\\)when\\(\\phantom{\\rule{0.2em}{0ex}}d=190\\).<\/td>\n<td>\\(\\sqrt{24\\cdot190}\\)<\/td>\n<\/tr>\n<tr>\n<td>Multiply.<\/td>\n<td>\\(\\sqrt{4,560}\\)<\/td>\n<\/tr>\n<tr>\n<td>Use your calculator.<\/td>\n<td>\\(67.527772..\\).<\/td>\n<\/tr>\n<tr>\n<td>Round to tenths.<\/td>\n<td>\\(67.5\\)<\/td>\n<\/tr>\n<tr>\n<td>Write a sentence.<\/td>\n<td>The speed of the car was approximately 67.5 miles per hour.<\/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 13.1<\/p>\n\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1828365\" data-type=\"problem\">\n<p id=\"fs-id1828367\">An accident investigator measured the skid marks of a car and found their length was \\(76\\) feet. To the nearest tenth, what was the speed of the car before the brakes were applied?<\/p>\n\n<\/div>\n<div id=\"fs-id1321756\" data-type=\"solution\"><details open=\"open\"><summary>Show answer<\/summary>\n<p id=\"fs-id1321759\">42.7 mph<\/p>\n\n<\/details><\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 13.2<\/p>\n\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1804208\" data-type=\"problem\">\n<p id=\"fs-id2431428\">The skid marks of a vehicle involved in an accident were \\(122\\) feet long. To the nearest tenth, how fast had the vehicle been going before the brakes were applied?<\/p>\n\n<\/div>\n<div id=\"fs-id3414964\" data-type=\"solution\"><details open=\"open\"><summary>Show answer<\/summary>\n<p id=\"fs-id1985261\">54.1 mph<\/p>\n\n<\/details><\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1168468515639\" class=\"links-to-literacy\" data-type=\"note\">The <em data-effect=\"italics\">Links to Literacy<\/em> activity \"Sea Squares\" will provide you with another view of the topics covered in this section.<\/div>\n<div id=\"fs-id2150426\" class=\"media-2\" data-type=\"note\">\n<div data-type=\"title\">ACCESS ADDITIONAL ONLINE RESOURCES<\/div>\n<ul id=\"fs-id1728151\" data-display=\"block\">\n \t<li><a href=\"http:\/\/www.openstax.org\/l\/24introsqroots\">Introduction to Square Roots<\/a><\/li>\n \t<li><a href=\"http:\/\/www.openstax.org\/l\/24estsqrtcalc\">Estimating Square Roots with a Calculator<\/a><\/li>\n<\/ul>\n<\/div>\nKey Concepts\n<ul id=\"eip-88\">\n \t<li><strong data-effect=\"bold\">Square Root Notation<\/strong>\\(\\sqrt{m}\\) is read \u2018the square root of \\(m\\)\u2019\nIf \\(m={n}^{2}\\), then \\(\\sqrt{m}=n\\), for \\(n\\ge 0\\). <span id=\"eip-id1170320621963\" data-type=\"media\" data-alt=\".\"><img class=\"aligncenter\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_05_07_011_img.png\" alt=\".\" data-media-type=\"image\/jpg\"><\/span><\/li>\n \t<li><strong data-effect=\"bold\">Use a strategy for applications with square roots.<\/strong>\n<ul id=\"eip-id1170325410660\">\n \t<li>Identify what you are asked to find.<\/li>\n \t<li>Write a phrase that gives the information to find it.<\/li>\n \t<li>Translate the phrase to an expression.<\/li>\n \t<li>Simplify the expression.<\/li>\n \t<li>Write a complete sentence that answers the question.<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<h1>Practice Makes Perfect<\/h1>\n<h2 id=\"fs-id2629399\">Simplify Expressions with Square Roots<\/h2>\n<p id=\"eip-673\">In the following exercises, simplify.<\/p>\n\n<table style=\"border-collapse: collapse; width: 100%; height: 126px;\" border=\"0\">\n<tbody>\n<tr style=\"height: 14px;\">\n<td style=\"width: 50%; height: 14px;\">1. \\(\\sqrt{36}\\)<\/td>\n<td style=\"width: 50%; height: 14px;\">2. \\(\\sqrt{4}\\)<\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"width: 50%; height: 14px;\">3. \\(\\sqrt{64}\\)<\/td>\n<td style=\"width: 50%; height: 14px;\">4. \\(\\sqrt{144}\\)<\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"width: 50%; height: 14px;\">5. \\(-\\sqrt{4}\\)<\/td>\n<td style=\"width: 50%; height: 14px;\">6. \\(-\\sqrt{100}\\)<\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"width: 50%; height: 14px;\">7. \\(-\\sqrt{1}\\)<\/td>\n<td style=\"width: 50%; height: 14px;\">8. \\(-\\sqrt{121}\\)<\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"width: 50%; height: 14px;\">9. \\(\\sqrt{-121}\\)<\/td>\n<td style=\"width: 50%; height: 14px;\">10. \\(\\sqrt{-36}\\)<\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"width: 50%; height: 14px;\">11. \\(\\sqrt{-9}\\)<\/td>\n<td style=\"width: 50%; height: 14px;\">12. \\(\\sqrt{-49}\\)<\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"width: 50%; height: 14px;\">13. \\(\\sqrt{9+16}\\)<\/td>\n<td style=\"width: 50%; height: 14px;\">14. \\(\\sqrt{25+144}\\)<\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"width: 50%; height: 14px;\">15. \\(\\sqrt{9}+\\sqrt{16}\\)<\/td>\n<td style=\"width: 50%; height: 14px;\">16. \\(\\sqrt{25}+\\sqrt{144}\\)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h2 id=\"fs-id1618239\">Estimate Square Roots<\/h2>\n<p id=\"eip-681\">In the following exercises, estimate each square root between two consecutive whole numbers.<\/p>\n\n<table style=\"border-collapse: collapse; width: 100%; height: 28px;\" border=\"0\">\n<tbody>\n<tr style=\"height: 14px;\">\n<td style=\"width: 50%; height: 14px;\">17. \\(\\sqrt{70}\\)<\/td>\n<td style=\"width: 50%; height: 14px;\">18. \\(\\sqrt{55}\\)<\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"width: 50%; height: 14px;\">19. \\(\\sqrt{200}\\)<\/td>\n<td style=\"width: 50%; height: 14px;\">20. \\(\\sqrt{172}\\)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h2 id=\"fs-id1390408\"><strong data-effect=\"bold\">\n<\/strong>Approximate Square Roots with a Calculator<\/h2>\n<p id=\"eip-83\">In the following exercises, use a calculator to approximate each square root and round to two decimal places.<\/p>\n\n<div id=\"fs-id1724408\" data-type=\"exercise\">\n<div id=\"fs-id1724410\" data-type=\"problem\">\n<table style=\"border-collapse: collapse; width: 100%;\" border=\"0\">\n<tbody>\n<tr>\n<td style=\"width: 50%;\">21. \\(\\sqrt{19}\\)<\/td>\n<td style=\"width: 50%;\">22. \\(\\sqrt{21}\\)<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 50%;\">23. \\(\\sqrt{53}\\)<\/td>\n<td style=\"width: 50%;\">24. \\(\\sqrt{47}\\)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h2 id=\"fs-id1724412\">Simplify Variable Expressions with Square Roots<\/h2>\n<\/div>\n<\/div>\n<p id=\"eip-591\">In the following exercises, simplify. (Assume all variables are greater than or equal to zero.)<\/p>\n\n<table style=\"border-collapse: collapse; width: 100%;\" border=\"0\">\n<tbody>\n<tr>\n<td style=\"width: 50%;\">25. \\(\\sqrt{{y}^{2}}\\)<\/td>\n<td style=\"width: 50%;\">26. \\(\\sqrt{{b}^{2}}\\)<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 50%;\">27. \\(\\sqrt{49{x}^{2}}\\)<\/td>\n<td style=\"width: 50%;\">28. \\(\\sqrt{100{y}^{2}}\\)<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 50%;\">29. \\(-\\sqrt{64{a}^{2}}\\)<\/td>\n<td style=\"width: 50%;\">30. \\(-\\sqrt{25{x}^{2}}\\)<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 50%;\">31. \\(\\sqrt{144{x}^{2}{y}^{2}}\\)<\/td>\n<td style=\"width: 50%;\">32. \\(\\sqrt{196{a}^{2}{b}^{2}}\\)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h2 id=\"fs-id1362921\"><strong data-effect=\"bold\">\n<\/strong>Use Square Roots in Applications<\/h2>\n<p id=\"eip-412\">In the following exercises, solve. Round to one decimal place.<\/p>\n\n<table style=\"border-collapse: collapse; width: 100%;\" border=\"0\">\n<tbody>\n<tr>\n<td style=\"width: 50%;\">33. <strong data-effect=\"bold\">Landscaping<\/strong> Reid wants to have a square garden plot in his backyard. He has enough compost to cover an area of \\(75\\) square feet. How long can a side of his garden be?<\/td>\n<td style=\"width: 50%;\">34.<strong data-effect=\"bold\"> Landscaping<\/strong> Tasha wants to make a square patio in her yard. She has enough concrete to pave an area of \\(130\\) square feet. How long can a side of her patio be?<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 50%;\">35.<strong data-effect=\"bold\"> Gravity<\/strong> An airplane dropped a flare from a height of \\(1,024\\) feet above a lake. How many seconds did it take for the flare to reach the water?<\/td>\n<td style=\"width: 50%;\">36. <strong data-effect=\"bold\">Gravity<\/strong> A hang glider dropped his cell phone from a height of \\(350\\) feet. How many seconds did it take for the cell phone to reach the ground?<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 50%;\">37. <strong data-effect=\"bold\">Gravity<\/strong> A construction worker dropped a hammer while building the Grand Canyon skywalk, \\(4,000\\) feet above the Colorado River. How many seconds did it take for the hammer to reach the river?<\/td>\n<td style=\"width: 50%;\">38. <strong data-effect=\"bold\">Accident investigation<\/strong> The skid marks from a car involved in an accident measured \\(54\\) feet. What was the speed of the car before the brakes were applied?<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 50%;\">39.<strong data-effect=\"bold\"> Accident investigation<\/strong> The skid marks from a car involved in an accident measured \\(216\\) feet. What was the speed of the car before the brakes were applied?<\/td>\n<td style=\"width: 50%;\">40. <strong data-effect=\"bold\">Accident investigation<\/strong> An accident investigator measured the skid marks of one of the vehicles involved in an accident. The length of the skid marks was \\(175\\) feet. What was the speed of the vehicle before the brakes were applied?<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 50%;\">41. <strong data-effect=\"bold\">Accident investigation<\/strong> An accident investigator measured the skid marks of one of the vehicles involved in an accident. The length of the skid marks was \\(117\\) feet. What was the speed of the vehicle before the brakes were applied?<\/td>\n<td style=\"width: 50%;\"><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h2 style=\"text-align: left;\" data-type=\"title\">Everyday Math<\/h2>\n<table style=\"border-collapse: collapse; width: 100%;\" border=\"0\">\n<tbody>\n<tr>\n<td style=\"width: 50%;\">42. <strong data-effect=\"bold\">Decorating<\/strong> Denise wants to install a square accent of designer tiles in her new shower. She can afford to buy \\(625\\) square centimetres of the designer tiles. How long can a side of the accent be?<\/td>\n<td style=\"width: 50%;\">43. <strong data-effect=\"bold\">Decorating<\/strong> Morris wants to have a square mosaic inlaid in his new patio. His budget allows for \\(2,025\\) tiles. Each tile is square with an area of one square inch. How long can a side of the mosaic be?<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h2 style=\"text-align: left;\" data-type=\"title\">Writing Exercises<\/h2>\n<table style=\"border-collapse: collapse; width: 100%;\" border=\"0\">\n<tbody>\n<tr>\n<td style=\"width: 50%;\">44. Why is there no real number equal to \\(\\sqrt{-64}?\\)<\/td>\n<td style=\"width: 50%;\">45. What is the difference between \\({9}^{2}\\) and \\(\\sqrt{9}?\\)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h1>Answers<\/h1>\n<table style=\"border-collapse: collapse; width: 100%;\" border=\"0\">\n<tbody>\n<tr>\n<td style=\"width: 33.3333%;\">1. 6<\/td>\n<td style=\"width: 33.3333%;\">3. 8<\/td>\n<td style=\"width: 33.3333%;\">5. -2<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 33.3333%;\">7. -1<\/td>\n<td style=\"width: 33.3333%;\">9. not a real number<\/td>\n<td style=\"width: 33.3333%;\">11. not a real number<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 33.3333%;\">13. 5<\/td>\n<td style=\"width: 33.3333%;\">15. 7<\/td>\n<td style=\"width: 33.3333%;\">17. \\(8&lt;\\sqrt{70}&lt;9\\)<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 33.3333%;\">19. \\(14&lt;\\sqrt{200}&lt;15\\)<\/td>\n<td style=\"width: 33.3333%;\">21. 4.36<\/td>\n<td style=\"width: 33.3333%;\">23. 7.28<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 33.3333%;\">25. <em data-effect=\"italics\">y<\/em><\/td>\n<td style=\"width: 33.3333%;\">27. 7<em data-effect=\"italics\">x<\/em><\/td>\n<td style=\"width: 33.3333%;\">29. \u22128<em data-effect=\"italics\">a<\/em><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 33.3333%;\">31. 12<em data-effect=\"italics\">xy<\/em><\/td>\n<td style=\"width: 33.3333%;\">33. 8.7 feet<\/td>\n<td style=\"width: 33.3333%;\">35. 8 seconds<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 33.3333%;\">37. 15.8 seconds<\/td>\n<td style=\"width: 33.3333%;\">39. 72 mph<\/td>\n<td style=\"width: 33.3333%;\">41. 53.0 mph<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 33.3333%;\">43. 45 inches<\/td>\n<td style=\"width: 33.3333%;\">45.\u00a0Answers will vary. 9<sup>2<\/sup> reads: \u201cnine squared\u201d and means nine times itself. The expression \\(\\sqrt{9}\\) reads: \u201cthe square root of nine\u201d which gives us the number such that if it were multiplied by itself would give you the number inside of the square root.<\/td>\n<td style=\"width: 33.3333%;\"><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h1>Attributions<\/h1>\nThis chapter has been adapted from \u201cSimplify and Use Square Roots\u201d in <a href=\"https:\/\/openstax.org\/details\/books\/elementary-algebra\"><em>Elementary Algebra<\/em> (OpenStax)<\/a> 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, you will be able to:<\/p>\n<ul>\n<li>Simplify expressions with square roots<\/li>\n<li>Estimate square roots<\/li>\n<li>Approximate square roots<\/li>\n<li>Simplify variable expressions with square roots<\/li>\n<li>Use square roots in applications<\/li>\n<\/ul>\n<\/div>\n<\/div>\n<h1>Simplify Expressions with Square Roots<\/h1>\n<p id=\"fs-id2427723\">To start this section, we need to review some important vocabulary and notation.<\/p>\n<p id=\"fs-id1458017\">Remember that when a number <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-b170995d512c659d8668b4e42e1fef6b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#110;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"11\" style=\"vertical-align: 0px;\" \/> is multiplied by itself, we can write this as <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/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;\" \/>, which we read aloud as <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-b87fd520bfafb76efe37779f79da9308_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#96;&#96;&#125;&#92;&#109;&#97;&#116;&#104;&#105;&#116;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#110;&#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;&#115;&#113;&#117;&#97;&#114;&#101;&#100;&#46;&#39;&#39;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"92\" style=\"vertical-align: -3px;\" \/> For example, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-c07e84fa47c71df6d6c4bc1c2208e8e4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#56;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"16\" style=\"vertical-align: 0px;\" \/> is read as <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-c67ec635d766ede090810d6f158e5931_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#96;&#96;&#56;&#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;&#115;&#113;&#117;&#97;&#114;&#101;&#100;&#46;&#39;&#39;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"91\" style=\"vertical-align: -3px;\" \/><\/p>\n<p id=\"fs-id2165684\">We call <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-de25eb06760758d2c59eb63fca270413_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#54;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"18\" style=\"vertical-align: -1px;\" \/> the <em data-effect=\"italics\">square<\/em> of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-23212a120eb92d226d44696a0b80bead_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#56;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: 0px;\" \/> because <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-84f2ba2e5a7dc9961d0c767ea5868df3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#56;&#125;&#94;&#123;&#50;&#125;&#61;&#54;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"58\" style=\"vertical-align: 0px;\" \/>. Similarly, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-c71eeef0468cd1464a1c1d6ade7e3f1e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#50;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"25\" style=\"vertical-align: -1px;\" \/> is the square of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-cba90047eec6a05e400b16dfb86d7f29_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"16\" style=\"vertical-align: -1px;\" \/>, because <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/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;\" \/>.<\/p>\n<div id=\"fs-id2784022\" 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\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-3f34b9f1699f7ceabd49661750a0a472_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#110;&#125;&#94;&#123;&#50;&#125;&#61;&#109;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"57\" style=\"vertical-align: 0px;\" \/>, then <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/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 the square of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-b170995d512c659d8668b4e42e1fef6b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#110;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"11\" style=\"vertical-align: 0px;\" \/>.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<h1>Modeling Squares<\/h1>\n<p id=\"fs-id2276864\">Do you know why we use the word <em data-effect=\"italics\">square<\/em>? If we construct a square with three tiles on each side, the total number of tiles would be nine.<\/p>\n<p><span id=\"fs-id4163946\" data-type=\"media\" data-alt=\"A square is shown with 3 tiles on each side. There are a total of 9 tiles in the square.\"><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2020\/08\/CNX_BMath_Figure_05_07_001_img.jpg\" alt=\"A square is shown with 3 tiles on each side. There are a total of 9 tiles in the square.\" data-media-type=\"image\/jpeg\" \/><\/span><\/p>\n<p id=\"fs-id2580549\">This is why we say that the square of three is nine.<\/p>\n<div id=\"fs-id1701315\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-d8c34ef5b882fda3c6d2c5efa31a54c2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#51;&#125;&#94;&#123;&#50;&#125;&#61;&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"49\" style=\"vertical-align: 0px;\" \/><\/div>\n<p id=\"fs-id2446968\">The number <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-e99b7c7ddd421682faf67e5b5f865882_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: 0px;\" \/> is called a <span class=\"no-emphasis\" data-type=\"term\">perfect square<\/span> because it is the square of a whole number.<\/p>\n<p id=\"fs-id1693488\">The chart shows the squares of the counting numbers <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-4868771cbc422b5818f85500909ce433_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"7\" style=\"vertical-align: -1px;\" \/> through <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-b59223d2c162a0fdd163af8597f0845e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"16\" style=\"vertical-align: -1px;\" \/>. You can refer to it to help you identify the perfect squares.<\/p>\n<p><span id=\"eip-id1170322670247\" data-type=\"media\" data-alt=\"A table with two columns is shown. The first column is labeled \u201cNumber\u201d and has the values: n, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, and 15. The second column is labeled \u201cSquare\u201d and has the values: n squared, 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, and 225.\"><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_05_07_008_img.jpg\" alt=\"A table with two columns is shown. The first column is labeled \u201cNumber\u201d and has the values: n, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, and 15. The second column is labeled \u201cSquare\u201d and has the values: n squared, 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, and 225.\" data-media-type=\"image\/jpg\" \/><\/span><\/p>\n<div id=\"fs-id1773473\" 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\">Perfect Squares<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>A perfect square is the square of a whole number.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1951839\">What happens when you square a negative number?<\/p>\n<div id=\"fs-id2171365\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-e524a77bac29d034db5e21e5f985ceca_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;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#56;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#38;&#32;&#61;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#56;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#56;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#92;&#32;&#38;&#32;&#61;&#54;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"41\" width=\"153\" style=\"vertical-align: -13px;\" \/><\/div>\n<p id=\"fs-id1987814\">When we multiply two negative numbers, the product is always positive. So, the square of a negative number is always positive.<\/p>\n<p id=\"fs-id3415479\">The chart shows the squares of the negative integers from <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/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;\" \/> to <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/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<p><span id=\"eip-id1170322726953\" data-type=\"media\" data-alt=\"A table is shown with 2 columns. The first column is labeled \u201cNumber\u201d and contains the values: n, negative 1, negative 2, negative 3, negative 4, negative 5, negative 6, negative 7, negative 8, negative 9, negative 10, negative 11, negative 12, negative 13, negative 14, and negative 15. The next column is labeled \u201cSquare\u201d and contains the values: n squared, 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, and 225.\"><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_05_07_009_img.jpg\" alt=\"A table is shown with 2 columns. The first column is labeled \u201cNumber\u201d and contains the values: n, negative 1, negative 2, negative 3, negative 4, negative 5, negative 6, negative 7, negative 8, negative 9, negative 10, negative 11, negative 12, negative 13, negative 14, and negative 15. The next column is labeled \u201cSquare\u201d and contains the values: n squared, 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, and 225.\" data-media-type=\"image\/jpg\" \/><\/span><\/p>\n<p id=\"fs-id2483567\">Did you notice that these squares are the same as the squares of the positive numbers?<\/p>\n<h1>Square Roots<\/h1>\n<p id=\"fs-id2645660\">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\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-e42e8f3a59a7b8fc13140ac6c0891318_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#49;&#48;&#125;&#94;&#123;&#50;&#125;&#61;&#49;&#48;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"75\" style=\"vertical-align: 0px;\" \/>, we say <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-2c4b0a8a79b7bd0e6e72a5fca0be9982_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#48;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"26\" style=\"vertical-align: -1px;\" \/> is the square of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-5450913cc453faf132dc56e5965ca797_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"17\" style=\"vertical-align: -1px;\" \/>. We can also say that <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-5450913cc453faf132dc56e5965ca797_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"17\" style=\"vertical-align: -1px;\" \/> is a square root of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-2c4b0a8a79b7bd0e6e72a5fca0be9982_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#48;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"26\" style=\"vertical-align: -1px;\" \/>.<\/p>\n<div id=\"fs-id1596688\" 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>A number whose square is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/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 class=\"no-emphasis\" data-type=\"term\">square root<\/span> of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/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;\" \/>.<\/p>\n<p id=\"fs-id2211118\">If <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-3f34b9f1699f7ceabd49661750a0a472_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#110;&#125;&#94;&#123;&#50;&#125;&#61;&#109;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"57\" style=\"vertical-align: 0px;\" \/>, then <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-b170995d512c659d8668b4e42e1fef6b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#110;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"11\" style=\"vertical-align: 0px;\" \/> is a square root of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/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;\" \/>.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1596349\">Notice <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/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\/introalgebra\/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 <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-2c4b0a8a79b7bd0e6e72a5fca0be9982_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#48;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"26\" style=\"vertical-align: -1px;\" \/>. Therefore, both <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-5450913cc453faf132dc56e5965ca797_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"17\" style=\"vertical-align: -1px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/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 <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-2c4b0a8a79b7bd0e6e72a5fca0be9982_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#48;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"26\" style=\"vertical-align: -1px;\" \/>.<\/p>\n<p id=\"eip-901\">So, every positive number has two square roots: one positive and one negative.<\/p>\n<p>What if we only want the positive square root of a positive number? The <em data-effect=\"italics\">radical sign,<\/em> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-c0ade7e4ffb73323c9e50a615c59f501_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#48;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"23\" style=\"vertical-align: -2px;\" \/>, stands for the positive square root. The positive square root is also called the principal square root.<\/p>\n<div id=\"fs-id2662111\" 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-id2270433\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/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 as \u201cthe square root of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-063fd0c239970220fe4234532cc58f67_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#109;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#39;&#39;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"28\" style=\"vertical-align: 0px;\" \/><\/p>\n<p id=\"fs-id3017179\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-9a2c1f3940c201a0d5bb4093949d294f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#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;&#109;&#61;&#123;&#110;&#125;&#94;&#123;&#50;&#125;&#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;&#92;&#115;&#113;&#114;&#116;&#123;&#109;&#125;&#61;&#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;&#102;&#111;&#114;&#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;&#110;&#125;&#92;&#103;&#101;&#32;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"258\" style=\"vertical-align: -4px;\" \/>.<\/p>\n<p><span id=\"fs-id1382029\" data-type=\"media\" data-alt=\"A picture of an m inside a square root sign is shown. The sign is labeled as a radical sign and the m is labeled as the radicand.\"><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_05_07_003_img.jpg\" alt=\"A picture of an m inside a square root sign is shown. The sign is labeled as a radical sign and the m is labeled as the radicand.\" data-media-type=\"image\/jpeg\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id2670991\">We can also use the radical sign for the square root of zero. Because <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-af92fb1999e59ab67ac70e3a3f8b4be1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#48;&#125;&#94;&#123;&#50;&#125;&#61;&#48;&#44;&#92;&#115;&#113;&#114;&#116;&#123;&#48;&#125;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"113\" style=\"vertical-align: -4px;\" \/>. Notice that zero has only one square root.<\/p>\n<p id=\"fs-id1396522\">The chart shows the square roots of the first <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-b59223d2c162a0fdd163af8597f0845e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"16\" style=\"vertical-align: -1px;\" \/> perfect square numbers.<\/p>\n<p><a href=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/uploads\/sites\/688\/2019\/06\/CNX_BMath_Figure_05_07_010_img.jpg\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_05_07_010_img.jpg\" alt=\"A table is shown with 2 columns. The first column contains the values: 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 column contains the values: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, and 15.\" width=\"598\" height=\"59\" data-media-type=\"image\/jpg\" \/><\/a><\/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-id2758986\" data-type=\"problem\">\n<p id=\"fs-id2203262\">Simplify: a) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-673807eb1b7a017e51ed00ed285fde50_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#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;&#50;&#53;&#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;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"31\" style=\"vertical-align: -2px;\" \/> b) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-38d602ee820db5ef10de9601cf5fc30a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#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;&#49;&#50;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"40\" style=\"vertical-align: -2px;\" \/>.<\/p>\n<\/div>\n<div id=\"fs-id2346035\" data-type=\"solution\">\n<div data-type=\"title\"><\/div>\n<div data-type=\"title\"><strong>Solution<\/strong><\/div>\n<table id=\"eip-idm859956128\" class=\"unnumbered unstyled\" summary=\".\" data-label=\"\">\n<tbody>\n<tr>\n<td>a)<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/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;\" \/><\/td>\n<\/tr>\n<tr>\n<td>Since <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/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><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-8995e2084120c1c1f4b53f490d281bc4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"8\" style=\"vertical-align: 0px;\" \/><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table id=\"eip-idm1481723040\" class=\"unnumbered unstyled\" summary=\".\" data-label=\"\">\n<tbody>\n<tr>\n<td>b)<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-f7472a15648ee154629be2bf540389b2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#49;&#50;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"41\" style=\"vertical-align: -2px;\" \/><\/td>\n<\/tr>\n<tr>\n<td>Since <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/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><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-2981688f5b26fd15fa08c897afa741ad_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"30\" style=\"vertical-align: -1px;\" \/><\/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 1.1<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div data-type=\"problem\">\n<p id=\"fs-id2940472\">Simplify: a) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-02f56383be72472e5c879882e3e100c9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#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;&#51;&#54;&#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;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"31\" style=\"vertical-align: -2px;\" \/> b) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-e8e37657b06a4f9ae6a864f6291dee00_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#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;&#49;&#54;&#57;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"40\" style=\"vertical-align: -2px;\" \/>.<\/p>\n<\/div>\n<div data-type=\"solution\">\n<details>\n<summary>Show answer<\/summary>\n<ol id=\"eip-id1168468305633\" class=\"circled\" type=\"a\">\n<li>6<\/li>\n<li>13<\/li>\n<\/ol>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id2323788\" class=\"try\" data-type=\"note\">\n<div id=\"fs-id2483689\" data-type=\"exercise\">\n<div data-type=\"problem\">\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 1.2<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div data-type=\"problem\">\n<p>Simplify: a) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-d273204578807443717e8f35d60688f1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#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;&#49;&#54;&#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;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"31\" style=\"vertical-align: -2px;\" \/> b) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-2b94992755ce45df0f14eb5b72d1b782_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#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;&#49;&#57;&#54;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"40\" style=\"vertical-align: -2px;\" \/>.<\/p>\n<\/div>\n<div data-type=\"solution\">\n<details>\n<summary>Show answer<\/summary>\n<ol id=\"eip-id1168469858541\" class=\"circled\" type=\"a\">\n<li>4<\/li>\n<li>14<\/li>\n<\/ol>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1714044\">Every positive number has two <span class=\"no-emphasis\" data-type=\"term\">square root<\/span>s and the radical sign indicates the positive one. We write <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-1448fbbd7cb69d1321c7f81d8b4ed1d6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#49;&#48;&#48;&#125;&#61;&#49;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"82\" 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\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-9d9bb8d5f1159905956148d24f6cee45_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#115;&#113;&#114;&#116;&#123;&#49;&#48;&#48;&#125;&#61;&#45;&#49;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"110\" style=\"vertical-align: -2px;\" \/>.<\/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 data-type=\"problem\">\n<p id=\"fs-id2372735\">Simplify. a) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-fae8e9fbf0e23593d382fff95f06d863_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#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;&#45;&#92;&#115;&#113;&#114;&#116;&#123;&#57;&#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;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"41\" style=\"vertical-align: -2px;\" \/> b) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-c07566911aacd30a9b109fbedd0bbe7d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#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;&#45;&#92;&#115;&#113;&#114;&#116;&#123;&#49;&#52;&#52;&#46;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"64\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\">\n<div data-type=\"title\"><\/div>\n<div data-type=\"title\"><strong>Solution<\/strong><\/div>\n<table id=\"eip-id1168469874827\" class=\"unnumbered unstyled\" summary=\".\" data-label=\"\">\n<tbody>\n<tr>\n<td>a)<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-e0fdde7500f5c112c432e935127a2f15_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#115;&#113;&#114;&#116;&#123;&#57;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"37\" style=\"vertical-align: -2px;\" \/><\/td>\n<\/tr>\n<tr>\n<td>The negative is in front of the radical sign.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/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<table id=\"eip-id1168466190277\" class=\"unnumbered unstyled\" summary=\".\" data-label=\"\">\n<tbody>\n<tr>\n<td>b)<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-7c194cb6015885b3f6f9cff89950a835_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#115;&#113;&#114;&#116;&#123;&#49;&#52;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"55\" style=\"vertical-align: -2px;\" \/><\/td>\n<\/tr>\n<tr>\n<td>The negative is in front of the radical sign.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-9eae543c53f9301fa411ef541a1f4433_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"30\" style=\"vertical-align: -1px;\" \/><\/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 2.1<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div data-type=\"problem\">\n<p>Simplify: a) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-2e9c037d10e35c2712af8ce7e18d18eb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#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;&#45;&#92;&#115;&#113;&#114;&#116;&#123;&#52;&#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;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"41\" style=\"vertical-align: -2px;\" \/> b) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-0eeb0c03dbeebd56364e736b39cd9142_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#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;&#45;&#92;&#115;&#113;&#114;&#116;&#123;&#50;&#50;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"59\" style=\"vertical-align: -2px;\" \/>.<\/p>\n<\/div>\n<div id=\"fs-id1753416\" data-type=\"solution\">\n<details>\n<summary>Show answer<\/summary>\n<ol id=\"eip-id1168469781883\" class=\"circled\" type=\"a\">\n<li>\u22122<\/li>\n<li>\u221215<\/li>\n<\/ol>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 2.2<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div data-type=\"problem\">\n<p id=\"fs-id1363622\">Simplify: a) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-919ab5675d5492994e7231c0c4f38aaa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#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;&#45;&#92;&#115;&#113;&#114;&#116;&#123;&#56;&#49;&#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;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"50\" style=\"vertical-align: -2px;\" \/> b) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-ccc0330bd1ca70e0a703835d1da2a2e4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#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;&#45;&#92;&#115;&#113;&#114;&#116;&#123;&#54;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"50\" style=\"vertical-align: -2px;\" \/>.<\/p>\n<\/div>\n<div id=\"fs-id1322037\" data-type=\"solution\">\n<details>\n<summary>Show answer<\/summary>\n<ol id=\"eip-id1168468356466\" class=\"circled\" type=\"a\">\n<li>\u22129<\/li>\n<li>\u22128<\/li>\n<\/ol>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1398123\" class=\"try\" data-type=\"note\">\n<div data-type=\"exercise\"><\/div>\n<\/div>\n<h1>Square Root of a Negative Number<\/h1>\n<p id=\"fs-id1737447\">Can we simplify <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-106759f3f92dcbbd85e8e38a55b5c608_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#45;&#50;&#53;&#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\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-61e2fb1fc3a33032fc37d9990d7d438c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#50;&#53;&#63;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"39\" style=\"vertical-align: 0px;\" \/><\/p>\n<div id=\"fs-id1231669\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-b4e3e9350c688e6bde0c0b2b400dd26c_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;&#49;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#61;&#45;&#50;&#53;&#63;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"102\" style=\"vertical-align: -4px;\" \/><\/div>\n<p>None of the numbers that we have dealt with so far have a square that is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-170b493658855d7211748be1a7029d86_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#50;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"30\" style=\"vertical-align: 0px;\" \/>. Why? Any positive number squared is positive, and any negative number squared is also positive. In the next chapter we will see that all the numbers we work with are called the real numbers. So we say there is no real number equal to <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-e7cdc351b2532599665ca0e80e412978_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#45;&#50;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"46\" style=\"vertical-align: -3px;\" \/>. If we are asked to find the <span class=\"no-emphasis\" data-type=\"term\">square root<\/span> of any negative number, we say that the solution 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-id1518393\" data-type=\"problem\">\n<p id=\"fs-id1613479\">Simplify: a) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-4a3e29d87a6cf0e5940ddcfa4688ea36_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#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;&#45;&#49;&#54;&#57;&#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;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"54\" style=\"vertical-align: -3px;\" \/> b) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-b7f58b5d2684a7ae25ae6c889368a9b9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#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;&#45;&#92;&#115;&#113;&#114;&#116;&#123;&#49;&#50;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"59\" style=\"vertical-align: -2px;\" \/>.<\/p>\n<\/div>\n<div id=\"fs-id3015148\" data-type=\"solution\">\n<div data-type=\"title\"><\/div>\n<div data-type=\"title\"><strong>Solution<\/strong><\/div>\n<div data-type=\"title\"><\/div>\n<p id=\"fs-id2788439\">a) There is no real number whose square is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-f25789704087627e8b43b50632a73953_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;&#54;&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"40\" style=\"vertical-align: -1px;\" \/>. Therefore, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/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;\" \/> is not a real number.<\/p>\n<p id=\"fs-id1807537\">b) The negative is in front of the radical sign, so we find the opposite of the square root of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-c71eeef0468cd1464a1c1d6ade7e3f1e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#50;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"25\" style=\"vertical-align: -1px;\" \/>.<\/p>\n<table id=\"eip-id1168466130843\" class=\"unnumbered unstyled\" summary=\".\" data-label=\"\">\n<tbody>\n<tr>\n<td><\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-629687dd5bc70328e52ee3ab951acbe5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#115;&#113;&#114;&#116;&#123;&#49;&#50;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"55\" style=\"vertical-align: -2px;\" \/><\/td>\n<\/tr>\n<tr>\n<td>The negative is in front of the radical.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-2981688f5b26fd15fa08c897afa741ad_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"30\" style=\"vertical-align: -1px;\" \/><\/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 3.1<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id2382510\" data-type=\"problem\">\n<p id=\"fs-id1834970\">Simplify: a) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-5c2a9dde26c9b193779868709e24a297_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#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;&#45;&#49;&#57;&#54;&#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;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"54\" style=\"vertical-align: -3px;\" \/> b) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-a18436b9e1b27fcaec2f2e644cf09c89_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#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;&#45;&#92;&#115;&#113;&#114;&#116;&#123;&#56;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"50\" style=\"vertical-align: -2px;\" \/>.<\/p>\n<\/div>\n<div id=\"fs-id2470950\" data-type=\"solution\">\n<details>\n<summary>Show answer<\/summary>\n<ol id=\"eip-id1168468279641\" class=\"circled\" type=\"a\">\n<li>not a real number<\/li>\n<li>\u22129<\/li>\n<\/ol>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id2170175\" class=\"try\" data-type=\"note\">\n<div id=\"fs-id3000454\" data-type=\"exercise\">\n<div id=\"fs-id2382510\" data-type=\"problem\">\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 3.2<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id2488039\" data-type=\"problem\">\n<p id=\"fs-id2139146\">Simplify: a) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-2cd3e8b723a78af7fe384e9294c683e4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#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;&#45;&#52;&#57;&#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;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"45\" style=\"vertical-align: -3px;\" \/> b) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-b7f58b5d2684a7ae25ae6c889368a9b9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#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;&#45;&#92;&#115;&#113;&#114;&#116;&#123;&#49;&#50;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"59\" style=\"vertical-align: -2px;\" \/>.<\/p>\n<\/div>\n<div id=\"fs-id2708177\" data-type=\"solution\">\n<details>\n<summary>Show answer<\/summary>\n<ol id=\"eip-id1168469427053\" class=\"circled\" type=\"a\">\n<li>\u22127<\/li>\n<li>not a real number<\/li>\n<\/ol>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<h1>Square Roots and the Order of Operations<\/h1>\n<p id=\"fs-id2199152\">When using the order of operations to simplify an expression that has square roots, we treat the radical sign as a grouping symbol. We simplify any expressions under the radical sign before performing other operations.<\/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-id2628173\" data-type=\"problem\">\n<p id=\"fs-id2445144\">Simplify: a) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-d14e08a5499cf847f0fb326e00aedbbb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#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;&#50;&#53;&#125;&#43;&#92;&#115;&#113;&#114;&#116;&#123;&#49;&#52;&#52;&#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;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"94\" style=\"vertical-align: -2px;\" \/> b) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-c6006c136a38db33c3181aaf72361ef5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#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;&#50;&#53;&#43;&#49;&#52;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"80\" style=\"vertical-align: -3px;\" \/>.<\/p>\n<\/div>\n<div id=\"fs-id2202087\" data-type=\"solution\">\n<div data-type=\"title\"><\/div>\n<div data-type=\"title\"><strong>Solution<\/strong><\/div>\n<table id=\"eip-id1168466048150\" class=\"unnumbered unstyled\" summary=\".\" data-label=\"\">\n<tbody>\n<tr>\n<td>a) Use the order of operations.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-e880c444afedd5c7e7f54bb030b0ac26_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#50;&#53;&#125;&#43;&#92;&#115;&#113;&#114;&#116;&#123;&#49;&#52;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"95\" style=\"vertical-align: -2px;\" \/><\/td>\n<\/tr>\n<tr>\n<td>Simplify each radical.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-3bdbf543842797ae2984fb46fabeecdc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;&#43;&#49;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"48\" style=\"vertical-align: -2px;\" \/><\/td>\n<\/tr>\n<tr>\n<td>Add.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-b976c8c10558ac50c42d953f3ee38707_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#55;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"17\" style=\"vertical-align: -1px;\" \/><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table id=\"eip-id1168469785439\" class=\"unnumbered unstyled\" summary=\".\" data-label=\"\">\n<tbody>\n<tr>\n<td>b) Use the order of operations.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-d33999e6510d30f646826a90715b1614_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#50;&#53;&#43;&#49;&#52;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"80\" style=\"vertical-align: -3px;\" \/><\/td>\n<\/tr>\n<tr>\n<td>Add under the radical sign.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-dd4c05e7550847a7adda1c07adcd62f6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#49;&#54;&#57;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"41\" style=\"vertical-align: -2px;\" \/><\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-837d0d6b05c238eddf106b68ec9dcfd6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"17\" style=\"vertical-align: -1px;\" \/><\/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 4.1<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div data-type=\"problem\">\n<p id=\"fs-id1829806\">Simplify: a) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-734bfdbc6b736cbee08f4290685d0286_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#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;&#57;&#125;&#43;&#92;&#115;&#113;&#114;&#116;&#123;&#49;&#54;&#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;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"77\" style=\"vertical-align: -2px;\" \/> b) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-e1320dc05dedbf1304cda385290ab44a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#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;&#57;&#43;&#49;&#54;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"62\" style=\"vertical-align: -3px;\" \/>.<\/p>\n<\/div>\n<div id=\"fs-id2583182\" data-type=\"solution\">\n<details>\n<summary>Show answer<\/summary>\n<ol id=\"eip-id1168468305202\" class=\"circled\" type=\"a\">\n<li>7<\/li>\n<li>5<\/li>\n<\/ol>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id2195519\" class=\"try\" data-type=\"note\">\n<div id=\"fs-id2644747\" data-type=\"exercise\">\n<div id=\"fs-id2213867\" data-type=\"problem\">\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 4.2<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id2213867\" data-type=\"problem\">\n<p>Simplify: a) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-0a632fc9e7d5bb9c8158e623cfe18bdf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#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;&#54;&#52;&#43;&#50;&#50;&#53;&#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;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"79\" style=\"vertical-align: -3px;\" \/> b) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-07d16f3417871031890d5a679be0bd2c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#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;&#54;&#52;&#125;&#43;&#92;&#115;&#113;&#114;&#116;&#123;&#50;&#50;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"94\" style=\"vertical-align: -2px;\" \/>.<\/p>\n<\/div>\n<div id=\"fs-id1529759\" data-type=\"solution\">\n<details>\n<summary>Show answer<\/summary>\n<ol id=\"eip-id1168468526118\" class=\"circled\" type=\"a\">\n<li>17<\/li>\n<li>23<\/li>\n<\/ol>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<p>Notice the different answers in parts a) and b) of <a href=\"#fs-id2628173\">(Example 4)<\/a>. It is important to follow the order of operations correctly. In a), we took each square root first and then added them. In b), we added under the radical sign first and then found the square root.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<h1>Estimate Square Roots<\/h1>\n<p id=\"fs-id1731496\">So far we have only worked with square roots of perfect squares. The square roots of other numbers are not whole numbers.<\/p>\n<p><span id=\"fs-id1788761\" data-type=\"media\" data-alt=\"A table is shown with 2 columns. The first column is labeled \u201cNumber\u201d and contains the values: 4, 5, 6, 7, 8, 9. The second column is labeled \u201cSquare root\u201d and contains the values: square root of 4 equals 2, square root of 5, square root of 6, square root of 7, square root of 8, square root of 9 equals 3.\"><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_05_07_007_img.jpg\" alt=\"A table is shown with 2 columns. The first column is labeled \u201cNumber\u201d and contains the values: 4, 5, 6, 7, 8, 9. The second column is labeled \u201cSquare root\u201d and contains the values: square root of 4 equals 2, square root of 5, square root of 6, square root of 7, square root of 8, square root of 9 equals 3.\" data-media-type=\"image\/jpeg\" \/><\/span><\/p>\n<p id=\"fs-id1569078\">We might conclude that the square roots of numbers between <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-6111a899fd636b7a5238708f8679f6ec_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"9\" style=\"vertical-align: -1px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-e99b7c7ddd421682faf67e5b5f865882_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: 0px;\" \/> will be between <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-e584dd0bab4e6c8efc164939c28db757_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"8\" style=\"vertical-align: 0px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-4a1d3ea4963f568cabd97329456036b6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: 0px;\" \/>, and they will not be whole numbers. Based on the pattern in the table above, we could say that <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-7ba329d3f2f86c0fb47d848202e0ee7d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"23\" style=\"vertical-align: -2px;\" \/> is between <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-e584dd0bab4e6c8efc164939c28db757_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"8\" style=\"vertical-align: 0px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-4a1d3ea4963f568cabd97329456036b6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: 0px;\" \/>. Using inequality symbols, we write<\/p>\n<div id=\"fs-id3324454\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-e3c927d77ded2a74424e1d011dec1325_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#60;&#92;&#115;&#113;&#114;&#116;&#123;&#53;&#125;&#60;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"89\" style=\"vertical-align: -2px;\" \/><\/div>\n<div data-type=\"equation\" data-label=\"\"><\/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 5<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id2136031\" data-type=\"problem\">\n<p id=\"fs-id2327121\">Estimate <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-7a0cc8947fbe65a29070f931744fe58e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#54;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"32\" style=\"vertical-align: -2px;\" \/> between two consecutive whole numbers.<\/p>\n<\/div>\n<div id=\"fs-id3014912\" data-type=\"solution\">\n<div data-type=\"title\"><\/div>\n<div data-type=\"title\"><strong>Solution<\/strong><\/div>\n<div data-type=\"title\"><\/div>\n<p id=\"fs-id1385193\">Think of the perfect squares closest to <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-7029ea134aa43ac5ffd9f780e196307d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#54;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"18\" style=\"vertical-align: 0px;\" \/>. Make a small table of these perfect squares and their squares roots.<\/p>\n<p><span id=\"fs-id2419067\" data-type=\"media\" data-alt=\"A table is shown with 2 columns. The first column is labeled \u201cNumber\u201d and contains the values: 36, 49, 64, and 81. There is a balloon coming out of the table between 49 and 64 that says 60. The second column is labeled \u201cSquare root\u201d and contains the values: 6, 7, 8, and 9. There is a balloon coming out of the table between 7 and 8 that says square root of 60.\"><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_05_07_006_img.jpg\" alt=\"A table is shown with 2 columns. The first column is labeled \u201cNumber\u201d and contains the values: 36, 49, 64, and 81. There is a balloon coming out of the table between 49 and 64 that says 60. The second column is labeled \u201cSquare root\u201d and contains the values: 6, 7, 8, and 9. There is a balloon coming out of the table between 7 and 8 that says square root of 60.\" data-media-type=\"image\/jpeg\" \/><\/span><\/p>\n<table id=\"eip-id1168467173482\" class=\"unnumbered unstyled\" summary=\".\" data-label=\"\">\n<tbody>\n<tr>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-2dc20df19f836522f1f27e05e5898b28_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#76;&#111;&#99;&#97;&#116;&#101;&#32;&#54;&#48;&#32;&#98;&#101;&#116;&#119;&#101;&#101;&#110;&#32;&#116;&#119;&#111;&#32;&#99;&#111;&#110;&#115;&#101;&#99;&#117;&#116;&#105;&#118;&#101;&#32;&#112;&#101;&#114;&#102;&#101;&#99;&#116;&#32;&#115;&#113;&#117;&#97;&#114;&#101;&#115;&#46;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"397\" style=\"vertical-align: -3px;\" \/><\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-ac1b0c6e6771c630041f5a0043c5d62b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#57;&#60;&#54;&#48;&#60;&#54;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"101\" style=\"vertical-align: -1px;\" \/><\/td>\n<\/tr>\n<tr>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-ad64288b8e8bad02c1c5149081f463ec_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#54;&#48;&#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;&#105;&#115;&#32;&#98;&#101;&#116;&#119;&#101;&#101;&#110;&#32;&#116;&#104;&#101;&#105;&#114;&#32;&#115;&#113;&#117;&#97;&#114;&#101;&#32;&#114;&#111;&#111;&#116;&#115;&#46;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"263\" style=\"vertical-align: -3px;\" \/><\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-4ee4be99d6ec34a77ab20f0fdcc7b9b6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#55;&#60;&#92;&#115;&#113;&#114;&#116;&#123;&#54;&#48;&#125;&#60;&#56;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"98\" style=\"vertical-align: -2px;\" \/><\/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 5.1<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1883735\" data-type=\"problem\">\n<p id=\"fs-id2171736\">Estimate <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/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;\" \/> between two consecutive whole numbers.<\/p>\n<\/div>\n<div id=\"fs-id1883425\" data-type=\"solution\">\n<details>\n<summary>Show answer<\/summary>\n<p id=\"fs-id2906465\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/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>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 5.2<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id2084941\" data-type=\"problem\">\n<p id=\"fs-id1731218\">Estimate <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-8aff887a76828d2fe10cd43f498c4774_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#56;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"32\" style=\"vertical-align: -2px;\" \/> between two consecutive whole numbers.<\/p>\n<\/div>\n<div id=\"fs-id2139079\" data-type=\"solution\">\n<details>\n<summary>Show answer<\/summary>\n<p id=\"fs-id1410348\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-db063b8cf25b0ab809c889a257610716_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#57;&#60;&#92;&#115;&#113;&#114;&#116;&#123;&#56;&#52;&#125;&#60;&#49;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"107\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<h1>Approximate Square Roots with a Calculator<\/h1>\n<p>The square roots of\u00a0 numbers that are not\u00a0 perfect squares are not whole numbers, they are irrational numbers. Its decimal form does not stop and does not repeat. Are irrational numbers real numbers? Yes, they are. When we put together the irrational numbers and rational numbers, we get the set of real numbers.<\/p>\n<p>Let&#8217;s see how we can use calculator to find the approximate square roots of those irrational numbers.<\/p>\n<p>There are mathematical methods to approximate square roots, but it is much more convenient to use a calculator to find square roots. Find the <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-c0ade7e4ffb73323c9e50a615c59f501_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#48;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"23\" style=\"vertical-align: -2px;\" \/> or <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/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. You will to use this key to approximate square roots. When you use your calculator to find the square root of a number that is not a perfect square, the answer that you see is not the exact number. It is an approximation, 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\/introalgebra\/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 <em data-effect=\"italics\">approximately<\/em>.<\/p>\n<p id=\"fs-id2669486\">Suppose your calculator has a <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-1e27a623c404bcc31b0f469d6a6e8c78_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#45;&#100;&#105;&#103;&#105;&#116;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"58\" style=\"vertical-align: -3px;\" \/> display. Using it to find the square root of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-8995e2084120c1c1f4b53f490d281bc4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"8\" style=\"vertical-align: 0px;\" \/> will give <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-5519872b48e6bec9af49328334a75851_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#46;&#50;&#51;&#54;&#48;&#54;&#55;&#57;&#55;&#55;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"95\" style=\"vertical-align: 0px;\" \/>. This is the approximate square root of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-8995e2084120c1c1f4b53f490d281bc4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"8\" style=\"vertical-align: 0px;\" \/>. When we report the answer, we should use the \u201capproximately equal to\u201d sign instead of an equal sign.<\/p>\n<div id=\"fs-id2168295\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-b1aecfa8434c3aeb13c4cfb6da455ea8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#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;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"141\" style=\"vertical-align: -2px;\" \/>. The square root of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-8995e2084120c1c1f4b53f490d281bc4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"8\" style=\"vertical-align: 0px;\" \/> is the example of irrational number and its approximation displays nine digits after the decimal place.<\/div>\n<p id=\"fs-id2137210\">You will seldom use this many digits for applications in algebra. So, if you wanted to round <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-7ba329d3f2f86c0fb47d848202e0ee7d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"23\" style=\"vertical-align: -2px;\" \/> to two decimal places, you would write<\/p>\n<div id=\"fs-id1337932\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-9558256870dc437db77e401de32cb0de_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#53;&#125;&#92;&#97;&#112;&#112;&#114;&#111;&#120;&#32;&#50;&#46;&#50;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"78\" style=\"vertical-align: -2px;\" \/><\/div>\n<p id=\"fs-id2174920\">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-id1816562\" class=\"unnumbered\" data-type=\"equation\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-c4c0224e6a3ed14d5df44e949ba15768_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;&#50;&#46;&#50;&#51;&#54;&#48;&#54;&#55;&#57;&#55;&#56;&#125;&#94;&#123;&#50;&#125;&#38;&#32;&#61;&#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;&#50;&#46;&#50;&#52;&#125;&#94;&#123;&#50;&#125;&#38;&#32;&#61;&#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;\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"241\" style=\"vertical-align: -12px;\" \/><\/div>\n<p id=\"fs-id1811799\">The squares are close, but not exactly equal, to <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-8995e2084120c1c1f4b53f490d281bc4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"8\" style=\"vertical-align: 0px;\" \/>.<\/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 data-type=\"problem\">\n<p id=\"fs-id3961155\">Round <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/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;\" \/> to two decimal places using a calculator.<\/p>\n<\/div>\n<div id=\"fs-id1862325\" data-type=\"solution\">\n<div data-type=\"title\"><\/div>\n<div data-type=\"title\"><strong>Solution<\/strong><\/div>\n<table id=\"eip-id1168468614270\" class=\"unnumbered unstyled\" summary=\".\" data-label=\"\">\n<tbody>\n<tr>\n<td><\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/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>Use the calculator square root key.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/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;\" \/><\/td>\n<\/tr>\n<tr>\n<td>Round to two decimal places.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/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><\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/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<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 6.1<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id2472223\" data-type=\"problem\">\n<p>Round <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/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;\" \/> to two decimal places.<\/p>\n<\/div>\n<div id=\"fs-id1166761303640\" data-type=\"solution\">\n<details>\n<summary>Show answer<\/summary>\n<p>\u2248 3.32<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 6.2<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1613047\" data-type=\"problem\">\n<p id=\"fs-id1362232\">Round <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-76af51a98cbbe021608a650abf2703dd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#49;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"32\" style=\"vertical-align: -2px;\" \/> to two decimal places.<\/p>\n<\/div>\n<div id=\"fs-id4326999\" data-type=\"solution\">\n<details>\n<summary>Show answer<\/summary>\n<p id=\"fs-id1863006\">\u2248 3.61<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<h1>Simplify Variable Expressions with Square Roots<\/h1>\n<p id=\"fs-id1619406\">Expressions with square root that we have looked at so far have not had any variables. What happens when we have to find a square root of a variable expression?<\/p>\n<p id=\"fs-id2244406\">Consider <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-e00d4a84322af57c96a287485e46ed2e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#57;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"41\" style=\"vertical-align: -1px;\" \/>, where <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-46bb299f47d94e1927057d14ab78801f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#92;&#103;&#101;&#32;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"43\" style=\"vertical-align: -3px;\" \/>. Can you think of an expression whose square is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-c813f57f5f73fa83186c9bcd43e56e98_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#57;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#63;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"35\" style=\"vertical-align: 0px;\" \/><\/p>\n<div id=\"fs-id2281080\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-e7eae854775a4eff1e841e52742a7063_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;&#63;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#38;&#32;&#61;&#38;&#32;&#57;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#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;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#38;&#32;&#61;&#38;&#32;&#57;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#115;&#111;&#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;&#57;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#125;&#61;&#51;&#120;&#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=\"251\" style=\"vertical-align: -16px;\" \/><\/div>\n<p id=\"fs-id2279868\">When we use a variable in a square root expression, for our work, we will assume that the variable represents a non-negative number. In every example and exercise that follows, each variable in a square root expression is greater than or equal to zero.<\/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-id3016846\" data-type=\"problem\">\n<p id=\"fs-id1518785\">Simplify: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-2485b45e92893d7423874b476efc5a20_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"32\" style=\"vertical-align: -1px;\" \/>.<\/p>\n<\/div>\n<div id=\"fs-id1969210\" data-type=\"solution\">\n<div data-type=\"title\"><\/div>\n<div data-type=\"title\"><strong>Solution<\/strong><\/div>\n<div data-type=\"title\"><\/div>\n<p id=\"eip-id1170368510506\">Think about what we would have to square to get <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-b40448f90dbf1bf9cce1035e2f3b1120_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"17\" style=\"vertical-align: 0px;\" \/>. Algebraically, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-eb8eb61329846cb5113a6991ad5dcb1c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#63;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#61;&#123;&#120;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"69\" style=\"vertical-align: -4px;\" \/><\/p>\n<table id=\"eip-id1168467419284\" class=\"unnumbered unstyled\" summary=\".\" data-label=\"\">\n<tbody>\n<tr>\n<td><\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-2485b45e92893d7423874b476efc5a20_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"32\" style=\"vertical-align: -1px;\" \/><\/td>\n<\/tr>\n<tr>\n<td>Since <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-c18fb69d198808137d59cabbcaec13dc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#61;&#123;&#120;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"71\" style=\"vertical-align: -4px;\" \/><\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-ede05c264bba0eda080918aaa09c4658_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" 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 7.1<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id2260926\" data-type=\"problem\">\n<p id=\"fs-id2784470\">Simplify: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-a01ee42597ae0dc0abfac93a97f3a516_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"35\" style=\"vertical-align: -5px;\" \/>.<\/p>\n<\/div>\n<div id=\"fs-id1568264\" data-type=\"solution\">\n<details>\n<summary>Show answer<\/summary>\n<p id=\"fs-id1789141\"><em data-effect=\"italics\">y<\/em><\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 7.2<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1733267\" data-type=\"problem\">\n<p id=\"fs-id1171104390264\">Simplify: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-20b6796cd65d6ace6ec5de67299c8b6d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#123;&#109;&#125;&#94;&#123;&#50;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -1px;\" \/>.<\/p>\n<\/div>\n<div id=\"fs-id2268730\" data-type=\"solution\">\n<details>\n<summary>Show answer<\/summary>\n<p id=\"fs-id2900277\"><em data-effect=\"italics\">m<\/em><\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1171103937955\" class=\"try\" data-type=\"note\">\n<div id=\"fs-id1202617\" data-type=\"exercise\">\n<div id=\"fs-id1733267\" data-type=\"problem\">\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-id1968764\" data-type=\"problem\">\n<p id=\"fs-id2208836\">Simplify: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-f3cb721a3dc7ed82a3ffadaed2bfaea1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#49;&#54;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"50\" style=\"vertical-align: -1px;\" \/>.<\/p>\n<\/div>\n<div id=\"fs-id2376108\" data-type=\"solution\">\n<div data-type=\"title\"><\/div>\n<div data-type=\"title\"><strong>Solution<\/strong><\/div>\n<table id=\"eip-id1168466092245\" class=\"unnumbered unstyled\" summary=\".\" data-label=\"\">\n<tbody>\n<tr>\n<td><\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-f3cb721a3dc7ed82a3ffadaed2bfaea1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#49;&#54;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"50\" style=\"vertical-align: -1px;\" \/><\/td>\n<\/tr>\n<tr>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-5af164c53f4b9b28e6dfef4ec4284e8b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#83;&#105;&#110;&#99;&#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;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#61;&#49;&#54;&#123;&#120;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"143\" style=\"vertical-align: -4px;\" \/><\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-5fa4d35b5c4377934245a54e07e43e7f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"19\" style=\"vertical-align: -1px;\" \/><\/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 8.1<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id2170992\" data-type=\"problem\">\n<p id=\"fs-id2597350\">Simplify: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-bf582aa9da74ebdfebd6d4926d0f09b3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#54;&#52;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"50\" style=\"vertical-align: -1px;\" \/>.<\/p>\n<\/div>\n<div id=\"fs-id1752622\" data-type=\"solution\">\n<details>\n<summary>Show answer<\/summary>\n<p id=\"fs-id2359180\">8<em data-effect=\"italics\">x<\/em><\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 8.2<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id2660664\" data-type=\"problem\">\n<p id=\"fs-id3408056\">Simplify: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-c113322310609cc3093ec647c45725d5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#49;&#54;&#57;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"61\" style=\"vertical-align: -5px;\" \/>.<\/p>\n<\/div>\n<div id=\"fs-id2686481\" data-type=\"solution\">\n<details>\n<summary>Show answer<\/summary>\n<p id=\"fs-id2675196\">13<em data-effect=\"italics\">y<\/em><\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\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-id1714033\" data-type=\"problem\">\n<p id=\"fs-id2793037\">Simplify: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-3341aa08dd5916aec7d643c328fde596_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#115;&#113;&#114;&#116;&#123;&#56;&#49;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"66\" style=\"vertical-align: -5px;\" \/>.<\/p>\n<\/div>\n<div id=\"fs-id1528886\" data-type=\"solution\">\n<div data-type=\"title\"><\/div>\n<div data-type=\"title\"><strong>Solution<\/strong><\/div>\n<table id=\"eip-id1168468389692\" class=\"unnumbered unstyled\" summary=\".\" data-label=\"\">\n<tbody>\n<tr>\n<td><\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-3341aa08dd5916aec7d643c328fde596_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#115;&#113;&#114;&#116;&#123;&#56;&#49;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"66\" style=\"vertical-align: -5px;\" \/><\/td>\n<\/tr>\n<tr>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-9e973a8d5717805f0a6961bdf1c66e89_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#83;&#105;&#110;&#99;&#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;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#57;&#121;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#61;&#56;&#49;&#123;&#121;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"141\" style=\"vertical-align: -4px;\" \/><\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-7c6653a12baf4e7b8241293c86e05b23_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#57;&#121;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"31\" style=\"vertical-align: -4px;\" \/><\/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 9.1<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div data-type=\"problem\">\n<p id=\"fs-id2140225\">Simplify: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-52c83217941d0ccbf2619a4401bc0e68_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#115;&#113;&#114;&#116;&#123;&#49;&#50;&#49;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"75\" style=\"vertical-align: -5px;\" \/>.<\/p>\n<\/div>\n<div id=\"fs-id1724861\" data-type=\"solution\">\n<details>\n<summary>Show answer<\/summary>\n<p id=\"fs-id2907510\">\u221211<em data-effect=\"italics\">y<\/em><\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 9.2<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1773567\" data-type=\"problem\">\n<p id=\"fs-id1773569\">Simplify: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-d6e3cfca798f8c36a09c0eae5026d412_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#115;&#113;&#114;&#116;&#123;&#49;&#48;&#48;&#123;&#112;&#125;&#94;&#123;&#50;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"75\" style=\"vertical-align: -5px;\" \/>.<\/p>\n<\/div>\n<div id=\"fs-id3961816\" data-type=\"solution\">\n<details>\n<summary>Show answer<\/summary>\n<p>\u221210<em data-effect=\"italics\">p<\/em><\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1546320\" class=\"try\" data-type=\"note\">\n<div id=\"fs-id2345827\" data-type=\"exercise\">\n<div id=\"fs-id3961816\" data-type=\"solution\">\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-id1943547\" data-type=\"problem\">\n<p id=\"fs-id1588623\">Simplify: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-9f7d6d068dfdb43ef15ff162cba35f29_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#51;&#54;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"70\" style=\"vertical-align: -5px;\" \/>.<\/p>\n<\/div>\n<div id=\"fs-id2684921\" data-type=\"solution\">\n<div data-type=\"title\"><\/div>\n<div data-type=\"title\"><strong>Solution<\/strong><\/div>\n<table id=\"eip-id1168466004603\" class=\"unnumbered unstyled\" summary=\".\" data-label=\"\">\n<tbody>\n<tr>\n<td><\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-9f7d6d068dfdb43ef15ff162cba35f29_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#51;&#54;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"70\" style=\"vertical-align: -5px;\" \/><\/td>\n<\/tr>\n<tr>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-20a8d02c147ff85071ce3686d5577de7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#83;&#105;&#110;&#99;&#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;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#54;&#120;&#121;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#61;&#51;&#54;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#123;&#121;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"169\" style=\"vertical-align: -4px;\" \/><\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-2a21284c7f098b67212f336007f0ccfc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#54;&#120;&#121;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"28\" style=\"vertical-align: -4px;\" \/><\/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 10.1<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1883085\" data-type=\"problem\">\n<p id=\"fs-id13852320\">Simplify: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-6bac04a4d0e37e1fa943ae4b4fbd29ed_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#49;&#48;&#48;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#123;&#98;&#125;&#94;&#123;&#50;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"73\" style=\"vertical-align: -1px;\" \/>.<\/p>\n<\/div>\n<div id=\"fs-id1502705\" data-type=\"solution\">\n<details>\n<summary>Show answer<\/summary>\n<p id=\"fs-id1502707\">10<em data-effect=\"italics\">ab<\/em><\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 10.2<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id2135180\" data-type=\"problem\">\n<p id=\"fs-id1385232\">Simplify: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-41fda69c52aa7a906abe228bb5fc041b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#50;&#50;&#53;&#123;&#109;&#125;&#94;&#123;&#50;&#125;&#123;&#110;&#125;&#94;&#123;&#50;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"83\" style=\"vertical-align: -1px;\" \/>.<\/p>\n<\/div>\n<div id=\"fs-id1735566\" data-type=\"solution\">\n<details>\n<summary>Show answer<\/summary>\n<p id=\"fs-id2608595\">15<em data-effect=\"italics\">mn<\/em><\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<h1>Use Square Roots in Applications<\/h1>\n<p id=\"fs-id2451417\">As you progress through your college courses, you\u2019ll encounter several applications of square roots. Once again, if we use our strategy for applications, it will give us a plan for finding the answer!<\/p>\n<div id=\"fs-id2754333\" class=\"howto\" data-type=\"note\">\n<div data-type=\"title\">\n<div class=\"textbox textbox--key-takeaways\">\n<header class=\"textbox__header\">HOW TO: Use a strategy for applications with square roots.<\/header>\n<div class=\"textbox__content\">\n<ol id=\"eip-id1168466250274\" class=\"stepwise\" type=\"1\">\n<li>Identify what you are asked to find.<\/li>\n<li>Write a phrase that gives the information to find it.<\/li>\n<li>Translate the phrase to an expression.<\/li>\n<li>Simplify the expression.<\/li>\n<li>Write a complete sentence that answers the question.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<h1>Square Roots and Area<\/h1>\n<p id=\"fs-id1956366\">We have solved applications with <span class=\"no-emphasis\" data-type=\"term\">area<\/span> before. If we were given the length of the sides of a square, we could find its area by squaring the length of its sides. Now we can find the length of the sides of a square if we are given the area, by finding the square root of the area.<\/p>\n<p id=\"fs-id2430870\">If the area of the square is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-25b206f25506e6d6f46be832f7119ffa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#65;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"13\" style=\"vertical-align: 0px;\" \/> square units, the length of a side is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-c62fd9df4abb27be3a471524f6d544bc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#65;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"28\" style=\"vertical-align: -2px;\" \/> units. See the table below.<\/p>\n<table id=\"fs-id1973397\" class=\"aligncenter\" summary=\"A table is shown with 2 columns and 4 rows. The first column is labeled \u201cArea (square units)\u201d and contains the values: 9, 144, A. The second column is labeled \u201cLength of side (units)\u201d and contains the values: square root of 9 equals 3, square root of 144 equals 12, square root of A.\" data-frame=\"none\">\n<thead>\n<tr valign=\"top\">\n<th data-align=\"left\">Area (square units)<\/th>\n<th data-align=\"left\">Length of side (units)<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr valign=\"top\">\n<td data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-e99b7c7ddd421682faf67e5b5f865882_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: 0px;\" \/><\/td>\n<td data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-d5c9c78ea001d1be6abf9d9d58f232db_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#57;&#125;&#61;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"55\" style=\"vertical-align: -2px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-1565125d13a851bd23f335968a7b66af_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#52;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"26\" style=\"vertical-align: -1px;\" \/><\/td>\n<td data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-45ecbfeb855c1d060d58c8d20fe783b0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#49;&#52;&#52;&#125;&#61;&#49;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"81\" style=\"vertical-align: -2px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-25b206f25506e6d6f46be832f7119ffa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#65;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"13\" style=\"vertical-align: 0px;\" \/><\/td>\n<td data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-c62fd9df4abb27be3a471524f6d544bc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#65;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"28\" style=\"vertical-align: -2px;\" \/><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div id=\"fs-id1736567\" class=\"try\" data-type=\"note\">\n<div id=\"fs-id1736570\" data-type=\"exercise\">\n<div id=\"fs-id1736572\" data-type=\"problem\">\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">EXAMPLE 11<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1736581\" data-type=\"problem\">\n<p id=\"fs-id1736583\">Mike and Lychelle want to make a square patio. They have enough concrete for an area of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-150ffdc5b0bfec3eb42a9f6ea45a11e8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#48;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"27\" style=\"vertical-align: 0px;\" \/> square feet. To the nearest tenth of a foot, how long can a side of their square patio be?<\/p>\n<\/div>\n<div id=\"fs-id1979887\" data-type=\"solution\">\n<div data-type=\"title\"><\/div>\n<div data-type=\"title\"><strong>Solution<\/strong><\/div>\n<div data-type=\"title\"><\/div>\n<p id=\"fs-id2686507\">We know the area of the square is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-150ffdc5b0bfec3eb42a9f6ea45a11e8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#48;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"27\" style=\"vertical-align: 0px;\" \/> square feet and want to find the length of the side. If the area of the square is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-25b206f25506e6d6f46be832f7119ffa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#65;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"13\" style=\"vertical-align: 0px;\" \/> square units, the length of a side is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-c62fd9df4abb27be3a471524f6d544bc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#65;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"28\" style=\"vertical-align: -2px;\" \/> units.<\/p>\n<table id=\"eip-id1168468768448\" class=\"unnumbered unstyled\" summary=\".\" data-label=\"\">\n<tbody>\n<tr>\n<td>What are you asked to find?<\/td>\n<td>The length of each side of a square patio<\/td>\n<\/tr>\n<tr>\n<td>Write a phrase.<\/td>\n<td>The length of a side<\/td>\n<\/tr>\n<tr>\n<td>Translate to an expression.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-c62fd9df4abb27be3a471524f6d544bc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#65;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"28\" style=\"vertical-align: -2px;\" \/><\/td>\n<\/tr>\n<tr>\n<td>Evaluate <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-c62fd9df4abb27be3a471524f6d544bc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#65;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"28\" style=\"vertical-align: -2px;\" \/> when <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-14c0b8dc40ac91f988bc3e31927cef50_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#65;&#61;&#50;&#48;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"64\" style=\"vertical-align: 0px;\" \/>.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/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;\" \/><\/td>\n<\/tr>\n<tr>\n<td>Use your calculator.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-55b251855c06f9ca30814741cd6e121a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#52;&#46;&#49;&#52;&#50;&#49;&#51;&#53;&#46;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"85\" style=\"vertical-align: 0px;\" \/>.<\/td>\n<\/tr>\n<tr>\n<td>Round to one decimal place.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-dc69a2f6c2f001887406578714fb3f4e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#52;&#46;&#49;&#32;&#102;&#101;&#101;&#116;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"63\" style=\"vertical-align: -1px;\" \/><\/td>\n<\/tr>\n<tr>\n<td>Write a sentence.<\/td>\n<td>Each side of the patio should be <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-54be5d6078afdd4d436a3f29525d5252_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#52;&#46;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"30\" style=\"vertical-align: -1px;\" \/> feet.<\/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 11.1<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1736572\" data-type=\"problem\">\n<p id=\"fs-id1966485\">Katie wants to plant a square lawn in her front yard. She has enough sod to cover an area of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-0dd49b89fec83009d1f78b323102f98c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#55;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"27\" style=\"vertical-align: 0px;\" \/> square feet. To the nearest tenth of a foot, how long can a side of her square lawn be?<\/p>\n<\/div>\n<div id=\"fs-id1588636\" data-type=\"solution\">\n<details open=\"open\">\n<summary>Show answer<\/summary>\n<p id=\"fs-id2708152\">19.2 feet<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 11.2<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id2283966\" data-type=\"problem\">\n<p id=\"fs-id2707825\">Sergio wants to make a square mosaic as an inlay for a table he is building. He has enough tile to cover an area of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-a499d99ca415f2919974a94d3e26526c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#55;&#48;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"36\" style=\"vertical-align: -1px;\" \/> square centimetres. How long can a side of his mosaic be?<\/p>\n<\/div>\n<div id=\"fs-id1377070\" data-type=\"solution\">\n<details open=\"open\">\n<summary>Show answer<\/summary>\n<p id=\"fs-id1460987\">52 centimetres<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<h1>Square Roots and Gravity<\/h1>\n<p id=\"fs-id1530631\">Another application of <span class=\"no-emphasis\" data-type=\"term\">square root<\/span>s involves <span class=\"no-emphasis\" data-type=\"term\">gravity<\/span>. On Earth, if an object is dropped from a height of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-14b463d0ecd5b350ced6cf1d6a12eef3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#104;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"10\" style=\"vertical-align: 0px;\" \/> feet, the time in seconds it will take to reach the ground is found by evaluating the expression <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-25f0194b82335423571d9e7117fbad6b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#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=\"20\" style=\"vertical-align: -6px;\" \/>. For example, if an object is dropped from a height of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-de25eb06760758d2c59eb63fca270413_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#54;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"18\" style=\"vertical-align: -1px;\" \/> feet, we can find the time it takes to reach the ground by evaluating <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-7d2a1960e89b3b24f5efbad31808ebdf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#115;&#113;&#114;&#116;&#123;&#54;&#52;&#125;&#125;&#123;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"26\" style=\"vertical-align: -6px;\" \/>.<\/p>\n<table id=\"eip-id1168466722027\" class=\"unnumbered unstyled\" summary=\".\" data-label=\"\">\n<tbody>\n<tr>\n<td><\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-7d2a1960e89b3b24f5efbad31808ebdf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#115;&#113;&#114;&#116;&#123;&#54;&#52;&#125;&#125;&#123;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"26\" style=\"vertical-align: -6px;\" \/><\/td>\n<\/tr>\n<tr>\n<td>Take the square root of 64.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-2e61c14568e7048877026c57ab0b19c5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#56;&#125;&#123;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"7\" style=\"vertical-align: -6px;\" \/><\/td>\n<\/tr>\n<tr>\n<td>Simplify the fraction.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-e584dd0bab4e6c8efc164939c28db757_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"8\" style=\"vertical-align: 0px;\" \/><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id2946358\">It would take <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-e584dd0bab4e6c8efc164939c28db757_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"8\" style=\"vertical-align: 0px;\" \/> seconds for an object dropped from a height of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-de25eb06760758d2c59eb63fca270413_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#54;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"18\" style=\"vertical-align: -1px;\" \/> feet to reach the ground.<\/p>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">EXAMPLE 12<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id3415154\" data-type=\"problem\">\n<p id=\"fs-id3415156\">Christy dropped her sunglasses from a bridge <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-94eb615ea4179120cbb0e2cf72077709_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#48;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"27\" style=\"vertical-align: -1px;\" \/> feet above a river. How many seconds does it take for the sunglasses to reach the river?<\/p>\n<\/div>\n<div id=\"fs-id4210836\" data-type=\"solution\">\n<div data-type=\"title\"><\/div>\n<div data-type=\"title\"><strong>Solution<\/strong><\/div>\n<table id=\"eip-id1168468767850\" class=\"unnumbered unstyled\" summary=\".\" data-label=\"\">\n<tbody>\n<tr>\n<td>What are you asked to find?<\/td>\n<td>The number of seconds it takes for the sunglasses to reach the river<\/td>\n<\/tr>\n<tr>\n<td>Write a phrase.<\/td>\n<td>The time it will take to reach the river<\/td>\n<\/tr>\n<tr>\n<td>Translate to an expression.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-25f0194b82335423571d9e7117fbad6b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#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=\"20\" style=\"vertical-align: -6px;\" \/><\/td>\n<\/tr>\n<tr>\n<td>Evaluate <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-25f0194b82335423571d9e7117fbad6b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#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=\"20\" style=\"vertical-align: -6px;\" \/> when <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-6567131f17a47928187a68ab9993e5ba_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#104;&#61;&#52;&#48;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"61\" style=\"vertical-align: -1px;\" \/>.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-6572298e7f74cf28fcb8e66461eecd15_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#115;&#113;&#114;&#116;&#123;&#52;&#48;&#48;&#125;&#125;&#123;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"33\" style=\"vertical-align: -6px;\" \/><\/td>\n<\/tr>\n<tr>\n<td>Find the square root of 400.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-ac54bfb341fd34f04e9224f29df35122_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#48;&#125;&#123;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"14\" style=\"vertical-align: -6px;\" \/><\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-8995e2084120c1c1f4b53f490d281bc4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"8\" style=\"vertical-align: 0px;\" \/><\/td>\n<\/tr>\n<tr>\n<td>Write a sentence.<\/td>\n<td>It will take 5 seconds for the 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 12.1<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1960973\" data-type=\"problem\">\n<p id=\"fs-id1960975\">A helicopter drops a rescue package from a height of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-7dd042f3fd5627723ddd61c30a02499d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#50;&#57;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"35\" style=\"vertical-align: -1px;\" \/> feet. How many seconds does it take for the package to reach the ground?<\/p>\n<\/div>\n<div id=\"fs-id3326094\" data-type=\"solution\">\n<details open=\"open\">\n<summary>Show answer<\/summary>\n<p id=\"fs-id3326096\">9 seconds<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 12.2<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id2742055\" data-type=\"problem\">\n<p id=\"fs-id2742057\">A window washer drops a squeegee from a platform <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-b213e548fae96ae5a5680d5875a42659_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#57;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"26\" style=\"vertical-align: -1px;\" \/> feet above the sidewalk. How many seconds does it take for the squeegee to reach the sidewalk?<\/p>\n<\/div>\n<div id=\"fs-id1884553\" data-type=\"solution\">\n<details open=\"open\">\n<summary>Show answer<\/summary>\n<p id=\"fs-id1884555\">3.5 seconds<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<h1>Square Roots and Accident Investigations<\/h1>\n<p id=\"fs-id1933926\">Police officers investigating car accidents measure the length of the skid marks on the pavement. Then they use square roots to determine the speed, in miles per hour, a car was going before applying the brakes. According to some formulas, if the length of the skid marks is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-4e8716946f6a868f015e0d62f28bc540_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#100;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"10\" style=\"vertical-align: 0px;\" \/> feet, then the speed of the car can be found by evaluating <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-a138d9bd0807e960045f2deddf52bfb9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#50;&#52;&#100;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"42\" style=\"vertical-align: -2px;\" \/>.<\/p>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">EXAMPLE 13<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id3327796\" data-type=\"problem\">\n<p id=\"fs-id3327798\">After a car accident, the skid marks for one car measured <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-7835f96172f1d141890010f111a9b574_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#57;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"26\" style=\"vertical-align: -1px;\" \/> feet. To the nearest tenth, what was the speed of the car (in mph) before the brakes were applied?<\/p>\n<\/div>\n<div id=\"fs-id2134026\" data-type=\"solution\">\n<div data-type=\"title\"><\/div>\n<div data-type=\"title\"><strong>Solution<\/strong><\/div>\n<table id=\"eip-id1168468505344\" class=\"unnumbered unstyled\" summary=\".\" data-label=\"\">\n<tbody>\n<tr>\n<td>What are you asked to find?<\/td>\n<td>The speed of the car before the brakes were applied<\/td>\n<\/tr>\n<tr>\n<td>Write a phrase.<\/td>\n<td>The speed of the car<\/td>\n<\/tr>\n<tr>\n<td>Translate to an expression.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-a138d9bd0807e960045f2deddf52bfb9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#50;&#52;&#100;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"42\" style=\"vertical-align: -2px;\" \/><\/td>\n<\/tr>\n<tr>\n<td>Evaluate<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-0a64bd4fdf387e1252806dcc108842ad_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#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;&#50;&#52;&#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;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"41\" style=\"vertical-align: -2px;\" \/>when<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-b9693e2c6a41db7140c8c1d4c3d5d581_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#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;&#100;&#61;&#49;&#57;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"59\" style=\"vertical-align: 0px;\" \/>.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-c5cec596369b28654f3b5606d441428f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#50;&#52;&#92;&#99;&#100;&#111;&#116;&#49;&#57;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"72\" style=\"vertical-align: -2px;\" \/><\/td>\n<\/tr>\n<tr>\n<td>Multiply.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-f383df7534a05659e2880874dc522dce_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#52;&#44;&#53;&#54;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"58\" style=\"vertical-align: -4px;\" \/><\/td>\n<\/tr>\n<tr>\n<td>Use your calculator.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-6ebdf5bc86f9c8f6b9413ae42a112602_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#54;&#55;&#46;&#53;&#50;&#55;&#55;&#55;&#50;&#46;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"86\" style=\"vertical-align: 0px;\" \/>.<\/td>\n<\/tr>\n<tr>\n<td>Round to tenths.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-e22ed61a7065f006b6a992ba8876c9dc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#54;&#55;&#46;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"31\" style=\"vertical-align: 0px;\" \/><\/td>\n<\/tr>\n<tr>\n<td>Write a sentence.<\/td>\n<td>The speed of the car was approximately 67.5 miles per hour.<\/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 13.1<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1828365\" data-type=\"problem\">\n<p id=\"fs-id1828367\">An accident investigator measured the skid marks of a car and found their length was <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-12b818a990fee7373006af5f0935f243_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#55;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"18\" style=\"vertical-align: 0px;\" \/> feet. To the nearest tenth, what was the speed of the car before the brakes were applied?<\/p>\n<\/div>\n<div id=\"fs-id1321756\" data-type=\"solution\">\n<details open=\"open\">\n<summary>Show answer<\/summary>\n<p id=\"fs-id1321759\">42.7 mph<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 13.2<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1804208\" data-type=\"problem\">\n<p id=\"fs-id2431428\">The skid marks of a vehicle involved in an accident were <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-32aeb570d3f9974c50b5f1a7e6843215_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#50;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"25\" style=\"vertical-align: -1px;\" \/> feet long. To the nearest tenth, how fast had the vehicle been going before the brakes were applied?<\/p>\n<\/div>\n<div id=\"fs-id3414964\" data-type=\"solution\">\n<details open=\"open\">\n<summary>Show answer<\/summary>\n<p id=\"fs-id1985261\">54.1 mph<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1168468515639\" class=\"links-to-literacy\" data-type=\"note\">The <em data-effect=\"italics\">Links to Literacy<\/em> activity &#8220;Sea Squares&#8221; will provide you with another view of the topics covered in this section.<\/div>\n<div id=\"fs-id2150426\" class=\"media-2\" data-type=\"note\">\n<div data-type=\"title\">ACCESS ADDITIONAL ONLINE RESOURCES<\/div>\n<ul id=\"fs-id1728151\" data-display=\"block\">\n<li><a href=\"http:\/\/www.openstax.org\/l\/24introsqroots\">Introduction to Square Roots<\/a><\/li>\n<li><a href=\"http:\/\/www.openstax.org\/l\/24estsqrtcalc\">Estimating Square Roots with a Calculator<\/a><\/li>\n<\/ul>\n<\/div>\n<p>Key Concepts<\/p>\n<ul id=\"eip-88\">\n<li><strong data-effect=\"bold\">Square Root Notation<\/strong><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/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 <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/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;\" \/>\u2019<br \/>\nIf <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-65f697b202683465e0b299437740955f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#109;&#61;&#123;&#110;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"57\" style=\"vertical-align: 0px;\" \/>, then <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-37801e81ac40053d649f9a0c06e37488_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#109;&#125;&#61;&#110;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"64\" style=\"vertical-align: -4px;\" \/>, for <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-f4e3ee6b99f2be2d578b8e2f5126199b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#110;&#92;&#103;&#101;&#32;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"43\" style=\"vertical-align: -3px;\" \/>. <span id=\"eip-id1170320621963\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_BMath_Figure_05_07_011_img.png\" alt=\".\" data-media-type=\"image\/jpg\" \/><\/span><\/li>\n<li><strong data-effect=\"bold\">Use a strategy for applications with square roots.<\/strong>\n<ul id=\"eip-id1170325410660\">\n<li>Identify what you are asked to find.<\/li>\n<li>Write a phrase that gives the information to find it.<\/li>\n<li>Translate the phrase to an expression.<\/li>\n<li>Simplify the expression.<\/li>\n<li>Write a complete sentence that answers the question.<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<h1>Practice Makes Perfect<\/h1>\n<h2 id=\"fs-id2629399\">Simplify Expressions with Square Roots<\/h2>\n<p id=\"eip-673\">In the following exercises, simplify.<\/p>\n<table style=\"border-collapse: collapse; width: 100%; height: 126px;\">\n<tbody>\n<tr style=\"height: 14px;\">\n<td style=\"width: 50%; height: 14px;\">1. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/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;\" \/><\/td>\n<td style=\"width: 50%; height: 14px;\">2. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-65ddaa07508d3929b6969a5e4e6baddf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"23\" style=\"vertical-align: -2px;\" \/><\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"width: 50%; height: 14px;\">3. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/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;\" \/><\/td>\n<td style=\"width: 50%; height: 14px;\">4. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-bb4b7e7ac9715b068342f7e4ccdd555c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#49;&#52;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"41\" style=\"vertical-align: -2px;\" \/><\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"width: 50%; height: 14px;\">5. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-b6c27859826b76f13b4d8fc41f93b0c3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#115;&#113;&#114;&#116;&#123;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"37\" style=\"vertical-align: -2px;\" \/><\/td>\n<td style=\"width: 50%; height: 14px;\">6. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-490c638692d9b2b581deea4a0e260357_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#115;&#113;&#114;&#116;&#123;&#49;&#48;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"55\" style=\"vertical-align: -2px;\" \/><\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"width: 50%; height: 14px;\">7. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/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;\" \/><\/td>\n<td style=\"width: 50%; height: 14px;\">8. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-629687dd5bc70328e52ee3ab951acbe5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#115;&#113;&#114;&#116;&#123;&#49;&#50;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"55\" style=\"vertical-align: -2px;\" \/><\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"width: 50%; height: 14px;\">9. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/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;\" \/><\/td>\n<td style=\"width: 50%; height: 14px;\">10. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-e147c65350129481b966194b9306d610_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#45;&#51;&#54;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"46\" style=\"vertical-align: -3px;\" \/><\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"width: 50%; height: 14px;\">11. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/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;\" \/><\/td>\n<td style=\"width: 50%; height: 14px;\">12. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-7dca7c0770f94a075c5454a5e22355c1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#45;&#52;&#57;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"46\" style=\"vertical-align: -3px;\" \/><\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"width: 50%; height: 14px;\">13. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-d73a31f97d2635e7af7e981311a82edd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#57;&#43;&#49;&#54;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"63\" style=\"vertical-align: -3px;\" \/><\/td>\n<td style=\"width: 50%; height: 14px;\">14. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-d33999e6510d30f646826a90715b1614_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#50;&#53;&#43;&#49;&#52;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"80\" style=\"vertical-align: -3px;\" \/><\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"width: 50%; height: 14px;\">15. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-be367e7e54b70b86ba3b19cbc2cdf844_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#57;&#125;&#43;&#92;&#115;&#113;&#114;&#116;&#123;&#49;&#54;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"77\" style=\"vertical-align: -2px;\" \/><\/td>\n<td style=\"width: 50%; height: 14px;\">16. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-e880c444afedd5c7e7f54bb030b0ac26_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#50;&#53;&#125;&#43;&#92;&#115;&#113;&#114;&#116;&#123;&#49;&#52;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"95\" style=\"vertical-align: -2px;\" \/><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h2 id=\"fs-id1618239\">Estimate Square Roots<\/h2>\n<p id=\"eip-681\">In the following exercises, estimate each square root between two consecutive whole numbers.<\/p>\n<table style=\"border-collapse: collapse; width: 100%; height: 28px;\">\n<tbody>\n<tr style=\"height: 14px;\">\n<td style=\"width: 50%; height: 14px;\">17. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/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;\" \/><\/td>\n<td style=\"width: 50%; height: 14px;\">18. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-b6d4996d03f20b82926c9d6cbf6885fe_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#53;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"32\" style=\"vertical-align: -2px;\" \/><\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"width: 50%; height: 14px;\">19. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/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;\" \/><\/td>\n<td style=\"width: 50%; height: 14px;\">20. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-39b1ee445b48aed7d76291f5be38079c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#49;&#55;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"41\" style=\"vertical-align: -2px;\" \/><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h2 id=\"fs-id1390408\"><strong data-effect=\"bold\"><br \/>\n<\/strong>Approximate Square Roots with a Calculator<\/h2>\n<p id=\"eip-83\">In the following exercises, use a calculator to approximate each square root and round to two decimal places.<\/p>\n<div id=\"fs-id1724408\" data-type=\"exercise\">\n<div id=\"fs-id1724410\" data-type=\"problem\">\n<table style=\"border-collapse: collapse; width: 100%;\">\n<tbody>\n<tr>\n<td style=\"width: 50%;\">21. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/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;\" \/><\/td>\n<td style=\"width: 50%;\">22. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-c968b67d283ebc5ed1e208adea6a5d15_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#50;&#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%;\">23. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/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;\" \/><\/td>\n<td style=\"width: 50%;\">24. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-f85c3a8abdcd1bf3006d91016951bc5c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#52;&#55;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"32\" style=\"vertical-align: -2px;\" \/><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h2 id=\"fs-id1724412\">Simplify Variable Expressions with Square Roots<\/h2>\n<\/div>\n<\/div>\n<p id=\"eip-591\">In the following exercises, simplify. (Assume all variables are greater than or equal to zero.)<\/p>\n<table style=\"border-collapse: collapse; width: 100%;\">\n<tbody>\n<tr>\n<td style=\"width: 50%;\">25. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-a01ee42597ae0dc0abfac93a97f3a516_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"35\" style=\"vertical-align: -5px;\" \/><\/td>\n<td style=\"width: 50%;\">26. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-62b84e678a40572621b6b7e78ccfdd97_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#123;&#98;&#125;&#94;&#123;&#50;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"30\" style=\"vertical-align: -1px;\" \/><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 50%;\">27. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-2bffd9c9501a3d34590954a8a1b66ef8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#52;&#57;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"50\" style=\"vertical-align: -1px;\" \/><\/td>\n<td style=\"width: 50%;\">28. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-d63ada3debd717ea558869a5488d34f1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#49;&#48;&#48;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"61\" style=\"vertical-align: -5px;\" \/><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 50%;\">29. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-4f097f04bb76f00dc5bfe2c5c1f526ae_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#115;&#113;&#114;&#116;&#123;&#54;&#52;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"63\" style=\"vertical-align: -1px;\" \/><\/td>\n<td style=\"width: 50%;\">30. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-a4ed84eaa3619e3ba10262fa522d353f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#115;&#113;&#114;&#116;&#123;&#50;&#53;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"64\" style=\"vertical-align: -1px;\" \/><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 50%;\">31. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-eea3fbf22b60d96e6f6af9ce17a2d9a2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#49;&#52;&#52;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"79\" style=\"vertical-align: -5px;\" \/><\/td>\n<td style=\"width: 50%;\">32. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-d4c13d3c9b5a2cec964412a8ad829276_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#49;&#57;&#54;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#123;&#98;&#125;&#94;&#123;&#50;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"73\" style=\"vertical-align: -1px;\" \/><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h2 id=\"fs-id1362921\"><strong data-effect=\"bold\"><br \/>\n<\/strong>Use Square Roots in Applications<\/h2>\n<p id=\"eip-412\">In the following exercises, solve. Round to one decimal place.<\/p>\n<table style=\"border-collapse: collapse; width: 100%;\">\n<tbody>\n<tr>\n<td style=\"width: 50%;\">33. <strong data-effect=\"bold\">Landscaping<\/strong> Reid wants to have a square garden plot in his backyard. He has enough compost to cover an area of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-da7843edb510e285d79daaeecac66b11_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#55;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"17\" style=\"vertical-align: 0px;\" \/> square feet. How long can a side of his garden be?<\/td>\n<td style=\"width: 50%;\">34.<strong data-effect=\"bold\"> Landscaping<\/strong> Tasha wants to make a square patio in her yard. She has enough concrete to pave an area of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-32a0d45ae691a68fec3cd6977bbe8af1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#51;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"26\" style=\"vertical-align: -1px;\" \/> square feet. How long can a side of her patio be?<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 50%;\">35.<strong data-effect=\"bold\"> Gravity<\/strong> An airplane dropped a flare from a height of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-6856b7ff4e8ae16a90f4e4233ce87495_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#44;&#48;&#50;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"43\" style=\"vertical-align: -4px;\" \/> feet above a lake. How many seconds did it take for the flare to reach the water?<\/td>\n<td style=\"width: 50%;\">36. <strong data-effect=\"bold\">Gravity<\/strong> A hang glider dropped his cell phone from a height of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-cbce5da9ad97b6a857b01a263e084e5e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#53;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"27\" style=\"vertical-align: 0px;\" \/> feet. How many seconds did it take for the cell phone to reach the ground?<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 50%;\">37. <strong data-effect=\"bold\">Gravity<\/strong> A construction worker dropped a hammer while building the Grand Canyon skywalk, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-6e74d43f675adbc7669ec3b945f1d12f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#44;&#48;&#48;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"44\" style=\"vertical-align: -4px;\" \/> feet above the Colorado River. How many seconds did it take for the hammer to reach the river?<\/td>\n<td style=\"width: 50%;\">38. <strong data-effect=\"bold\">Accident investigation<\/strong> The skid marks from a car involved in an accident measured <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-66ceeedc341151a1adecf7dd22c17553_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"18\" style=\"vertical-align: -1px;\" \/> feet. What was the speed of the car before the brakes were applied?<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 50%;\">39.<strong data-effect=\"bold\"> Accident investigation<\/strong> The skid marks from a car involved in an accident measured <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-726484184fdf7357e39932cf034052c9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#49;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"27\" style=\"vertical-align: -1px;\" \/> feet. What was the speed of the car before the brakes were applied?<\/td>\n<td style=\"width: 50%;\">40. <strong data-effect=\"bold\">Accident investigation<\/strong> An accident investigator measured the skid marks of one of the vehicles involved in an accident. The length of the skid marks was <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-855fd7154024e59965cf2d02c401d62d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#55;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"25\" style=\"vertical-align: -1px;\" \/> feet. What was the speed of the vehicle before the brakes were applied?<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 50%;\">41. <strong data-effect=\"bold\">Accident investigation<\/strong> An accident investigator measured the skid marks of one of the vehicles involved in an accident. The length of the skid marks was <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-72004f572ee9243b1cc651f30e6f5584_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#49;&#55;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"26\" style=\"vertical-align: -1px;\" \/> feet. What was the speed of the vehicle before the brakes were applied?<\/td>\n<td style=\"width: 50%;\"><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h2 style=\"text-align: left;\" data-type=\"title\">Everyday Math<\/h2>\n<table style=\"border-collapse: collapse; width: 100%;\">\n<tbody>\n<tr>\n<td style=\"width: 50%;\">42. <strong data-effect=\"bold\">Decorating<\/strong> Denise wants to install a square accent of designer tiles in her new shower. She can afford to buy <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-401bd31f396b71bb39e304b3f50555d7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#54;&#50;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"26\" style=\"vertical-align: 0px;\" \/> square centimetres of the designer tiles. How long can a side of the accent be?<\/td>\n<td style=\"width: 50%;\">43. <strong data-effect=\"bold\">Decorating<\/strong> Morris wants to have a square mosaic inlaid in his new patio. His budget allows for <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-d699610441a5c3555ba50fa8fb960844_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#44;&#48;&#50;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"43\" style=\"vertical-align: -4px;\" \/> tiles. Each tile is square with an area of one square inch. How long can a side of the mosaic be?<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h2 style=\"text-align: left;\" data-type=\"title\">Writing Exercises<\/h2>\n<table style=\"border-collapse: collapse; width: 100%;\">\n<tbody>\n<tr>\n<td style=\"width: 50%;\">44. Why is there no real number equal to <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-aafe2412d1eaab262173c75d7a4b9d35_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#45;&#54;&#52;&#125;&#63;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"53\" style=\"vertical-align: -3px;\" \/><\/td>\n<td style=\"width: 50%;\">45. What is the difference between <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-74deaa134f7b1f68fcc620bbda92abe9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#57;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"16\" style=\"vertical-align: 0px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-78cdedc689fa6afd26cf727e7d94d6f9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#57;&#125;&#63;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"31\" style=\"vertical-align: -2px;\" \/><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h1>Answers<\/h1>\n<table style=\"border-collapse: collapse; width: 100%;\">\n<tbody>\n<tr>\n<td style=\"width: 33.3333%;\">1. 6<\/td>\n<td style=\"width: 33.3333%;\">3. 8<\/td>\n<td style=\"width: 33.3333%;\">5. -2<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 33.3333%;\">7. -1<\/td>\n<td style=\"width: 33.3333%;\">9. not a real number<\/td>\n<td style=\"width: 33.3333%;\">11. not a real number<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 33.3333%;\">13. 5<\/td>\n<td style=\"width: 33.3333%;\">15. 7<\/td>\n<td style=\"width: 33.3333%;\">17. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/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;\" \/><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 33.3333%;\">19. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/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;\" \/><\/td>\n<td style=\"width: 33.3333%;\">21. 4.36<\/td>\n<td style=\"width: 33.3333%;\">23. 7.28<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 33.3333%;\">25. <em data-effect=\"italics\">y<\/em><\/td>\n<td style=\"width: 33.3333%;\">27. 7<em data-effect=\"italics\">x<\/em><\/td>\n<td style=\"width: 33.3333%;\">29. \u22128<em data-effect=\"italics\">a<\/em><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 33.3333%;\">31. 12<em data-effect=\"italics\">xy<\/em><\/td>\n<td style=\"width: 33.3333%;\">33. 8.7 feet<\/td>\n<td style=\"width: 33.3333%;\">35. 8 seconds<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 33.3333%;\">37. 15.8 seconds<\/td>\n<td style=\"width: 33.3333%;\">39. 72 mph<\/td>\n<td style=\"width: 33.3333%;\">41. 53.0 mph<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 33.3333%;\">43. 45 inches<\/td>\n<td style=\"width: 33.3333%;\">45.\u00a0Answers will vary. 9<sup>2<\/sup> reads: \u201cnine squared\u201d and means nine times itself. The expression <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-63803f6e103855103d8e3b1eb053bf6e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#57;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"23\" style=\"vertical-align: -2px;\" \/> reads: \u201cthe square root of nine\u201d which gives us the number such that if it were multiplied by itself would give you the number inside of the square root.<\/td>\n<td style=\"width: 33.3333%;\"><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h1>Attributions<\/h1>\n<p>This chapter has been adapted from \u201cSimplify and Use Square Roots\u201d in <a href=\"https:\/\/openstax.org\/details\/books\/elementary-algebra\"><em>Elementary Algebra<\/em> (OpenStax)<\/a> 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":90,"menu_order":4,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-2003","chapter","type-chapter","status-publish","hentry"],"part":1897,"_links":{"self":[{"href":"https:\/\/opentextbc.ca\/introalgebra\/wp-json\/pressbooks\/v2\/chapters\/2003","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/opentextbc.ca\/introalgebra\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/opentextbc.ca\/introalgebra\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/opentextbc.ca\/introalgebra\/wp-json\/wp\/v2\/users\/90"}],"version-history":[{"count":2,"href":"https:\/\/opentextbc.ca\/introalgebra\/wp-json\/pressbooks\/v2\/chapters\/2003\/revisions"}],"predecessor-version":[{"id":2527,"href":"https:\/\/opentextbc.ca\/introalgebra\/wp-json\/pressbooks\/v2\/chapters\/2003\/revisions\/2527"}],"part":[{"href":"https:\/\/opentextbc.ca\/introalgebra\/wp-json\/pressbooks\/v2\/parts\/1897"}],"metadata":[{"href":"https:\/\/opentextbc.ca\/introalgebra\/wp-json\/pressbooks\/v2\/chapters\/2003\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/opentextbc.ca\/introalgebra\/wp-json\/wp\/v2\/media?parent=2003"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/opentextbc.ca\/introalgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=2003"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/opentextbc.ca\/introalgebra\/wp-json\/wp\/v2\/contributor?post=2003"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/opentextbc.ca\/introalgebra\/wp-json\/wp\/v2\/license?post=2003"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}