{"id":1917,"date":"2021-12-02T19:39:58","date_gmt":"2021-12-03T00:39:58","guid":{"rendered":"https:\/\/opentextbc.ca\/intermediatealgebraberg\/back-matter\/midterm-2-prep-answer-key\/"},"modified":"2022-11-02T10:38:24","modified_gmt":"2022-11-02T14:38:24","slug":"midterm-2-prep-answer-key","status":"publish","type":"back-matter","link":"https:\/\/opentextbc.ca\/intermediatealgebraberg\/back-matter\/midterm-2-prep-answer-key\/","title":{"raw":"Midterm 2: Prep Answer Key","rendered":"Midterm 2: Prep Answer Key"},"content":{"raw":"<h1>Midterm Two Review<\/h1>\n<ol>\n \t<li>\n<table class=\"lines\" style=\"border-collapse: collapse; width: 50%;\" border=\"0\"><caption>[latex]x-2y=-4[\/latex]<\/caption>\n<tbody>\n<tr>\n<th style=\"width: 6.2678%; text-align: center;\" scope=\"col\">[latex]x[\/latex]<\/th>\n<th style=\"width: 5.12821%; text-align: center;\" scope=\"col\">[latex]y[\/latex]<\/th>\n<\/tr>\n<tr>\n<td style=\"width: 6.2678%; text-align: center;\">\u22124<\/td>\n<td style=\"width: 5.12821%; text-align: center;\">0<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 6.2678%; text-align: center;\">0<\/td>\n<td style=\"width: 5.12821%; text-align: center;\">2<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 6.2678%; text-align: center;\">\u22122<\/td>\n<td style=\"width: 5.12821%; text-align: center;\">1<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table class=\"lines\" style=\"border-collapse: collapse; width: 50%;\" border=\"0\"><caption>[latex]x+y=5[\/latex]<\/caption>\n<tbody>\n<tr>\n<th style=\"width: 27.4929%; text-align: center;\" scope=\"col\">[latex]x[\/latex]<\/th>\n<th style=\"width: 25.3561%; text-align: center;\" scope=\"col\">[latex]y[\/latex]<\/th>\n<\/tr>\n<tr>\n<td style=\"width: 27.4929%; text-align: center;\">0<\/td>\n<td style=\"width: 25.3561%; text-align: center;\">5<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 27.4929%; text-align: center;\">5<\/td>\n<td style=\"width: 25.3561%; text-align: center;\">0<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 27.4929%; text-align: center;\">2<\/td>\n<td style=\"width: 25.3561%; text-align: center;\">3<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<img class=\"alignnone size-medium wp-image-1915\" src=\"https:\/\/opentextbc.ca\/intermediatealgebraberg\/wp-content\/uploads\/sites\/304\/2021\/12\/Midterm2.1-300x289.jpg\" alt=\"\" width=\"300\" height=\"289\"><\/li>\n \t<li>[latex]\\phantom{a}[\/latex]\n[latex]\\begin{array}[t]{rrcrlrl}\n2x&amp;-&amp;y&amp;=&amp;0&amp;\\Rightarrow &amp;y=2x \\\\\n3x&amp;+&amp;4y&amp;=&amp;-22&amp;&amp; \\\\ \\\\\n\\therefore 3x&amp;+&amp;4(2x)&amp;=&amp;-22&amp;&amp; \\\\\n3x&amp;+&amp;8x&amp;=&amp;-22&amp;&amp; \\\\\n&amp;&amp;11x&amp;=&amp;-22&amp;&amp; \\\\\n&amp;&amp;x&amp;=&amp;-2&amp;&amp; \\\\ \\\\\n&amp;&amp;y&amp;=&amp;2x&amp;&amp; \\\\\n&amp;&amp;y&amp;=&amp;2(-2)&amp;=&amp;-4 \\\\\n\\end{array}[\/latex]\n[latex](-2,-4)[\/latex]<\/li>\n \t<li>[latex]\\phantom{a}[\/latex]\n[latex]\\begin{array}[t]{rrrrrl}\n&amp;(2x&amp;-&amp;5y&amp;=&amp;15)(2) \\\\\n&amp;(3x&amp;+&amp;2y&amp;=&amp;13)(5) \\\\\n\\hline\n&amp;4x&amp;-&amp;10y&amp;=&amp;30 \\\\\n+&amp;15x&amp;+&amp;10y&amp;=&amp;65 \\\\\n\\hline\n&amp;&amp;&amp;19x&amp;=&amp;95 \\\\\n&amp;&amp;&amp;x&amp;=&amp;5 \\\\ \\\\\n&amp;\\therefore 3(5)&amp;+&amp;2y&amp;=&amp;\\phantom{-}13 \\\\\n&amp;15&amp;+&amp;2y&amp;=&amp;\\phantom{-}13 \\\\\n&amp;-15&amp;&amp;&amp;&amp;-15 \\\\\n\\hline\n&amp;&amp;&amp;2y&amp;=&amp;-2 \\\\\n&amp;&amp;&amp;y&amp;=&amp;-1\n\\end{array}[\/latex]\n[latex](5,-1)[\/latex]<\/li>\n \t<li>[latex]\\phantom{a}[\/latex]\n[latex]\\begin{array}[t]{rr}\n\\begin{array}[t]{rrrrrrrl}\n&amp;&amp;&amp;(5x&amp;+&amp;6z&amp;=&amp;-4)(-1) \\\\ \\\\\n&amp;5x&amp;+&amp;y&amp;+&amp;6z&amp;=&amp;-2 \\\\\n+&amp;-5x&amp;&amp;&amp;-&amp;6z&amp;=&amp;\\phantom{-}4 \\\\\n\\hline\n&amp;&amp;&amp;&amp;&amp;y&amp;=&amp;2 \\\\ \\\\\n&amp;&amp;&amp;\\therefore 2y&amp;-&amp;3z&amp;=&amp;\\phantom{-}3 \\\\\n&amp;&amp;&amp;2(2)&amp;-&amp;3z&amp;=&amp;\\phantom{-}3 \\\\\n&amp;&amp;&amp;-4&amp;&amp;&amp;&amp;-4 \\\\\n\\hline\n&amp;&amp;&amp;&amp;&amp;-3z&amp;=&amp;-1 \\\\\n&amp;&amp;&amp;&amp;&amp;z&amp;=&amp;\\dfrac{1}{3} \\\\\n\\end{array}\n&amp; \\hspace{0.25in}\n\\begin{array}[t]{rrrrr}\n5x&amp;+&amp;6z&amp;=&amp;-4 \\\\\n5x&amp;+&amp;6\\left(\\dfrac{1}{3}\\right)&amp;=&amp;-4 \\\\\n5x&amp;+&amp;2&amp;=&amp;-4 \\\\\n&amp;-&amp;2&amp;&amp;-2 \\\\\n\\hline\n&amp;&amp;5x&amp;=&amp;-6 \\\\\n&amp;&amp;x&amp;=&amp;-\\dfrac{6}{5}\n\\end{array}\n\\end{array}[\/latex]\n[latex]-\\dfrac{6}{5}, 2, \\dfrac{1}{3}[\/latex]<\/li>\n \t<li>[latex]\\phantom{a}[\/latex]\n[latex]\\begin{array}[t]{rrrrrr}\n&amp;4a^2&amp;-&amp;9a&amp;+&amp;2 \\\\\n&amp;-a^2&amp;+&amp;4a&amp;+&amp;5 \\\\\n+&amp;3a^2&amp;-&amp;a&amp;+&amp;9 \\\\\n\\hline\n&amp;6a^2&amp;-&amp;6a&amp;+&amp;16\n\\end{array}[\/latex]<\/li>\n \t<li>[latex]8x^4-12x^2y^2-15x^2y^2-3x^4\\Rightarrow 5x^4-27x^2y^2[\/latex]<\/li>\n \t<li>[latex]6-2\\left[3x-20x+8-1\\right][\/latex]\n[latex]\\begin{array}[t]{l}\n6-2\\left[-17x+7\\right] \\\\\n6+34x-14 \\\\\n34x-8\n\\end{array}[\/latex]<\/li>\n \t<li>[latex]25a^{-10}b^6\\text{ or } \\dfrac{25b^6}{a^{10}}[\/latex]<\/li>\n \t<li>[latex]8a^2(a^2+10a+25) [\/latex]\n[latex]8a^4+80a^3+200a^2[\/latex]<\/li>\n \t<li>[latex]4ab^2(a^2-4)[\/latex]\n[latex]4a^3b^2-16ab^2[\/latex]<\/li>\n \t<li>[latex]\\phantom{a}[\/latex]\n[latex]\\begin{array}[t]{rrrrrrrr}\n&amp;x^2&amp;-&amp;4x&amp;+&amp;7\\phantom{x}&amp;&amp; \\\\\n\\times &amp;&amp;&amp;x&amp;-&amp;3\\phantom{x}&amp;&amp; \\\\\n\\hline\n&amp;x^3&amp;-&amp;4x^2&amp;+&amp;7x&amp;&amp; \\\\\n+&amp;&amp;-&amp;3x^2&amp;+&amp;12x&amp;-&amp;21 \\\\\n\\hline\n&amp;x^3&amp;-&amp;7x^2&amp;+&amp;19x&amp;-&amp;21 \\\\\n\\end{array}[\/latex]<\/li>\n \t<li>[latex]\\phantom{a}[\/latex]\n[latex]\\begin{array}[t]{rrrrrrrrrr}\n&amp;2x^2&amp;+&amp;x&amp;-&amp;3\\phantom{x^2}&amp;&amp;&amp;&amp; \\\\\n\\times &amp;2x^2&amp;+&amp;x&amp;-&amp;3\\phantom{x^2}&amp;&amp;&amp;&amp; \\\\\n\\hline\n&amp;4x^4&amp;+&amp;2x^3&amp;-&amp;6x^2&amp;&amp;&amp;&amp; \\\\\n&amp;&amp;&amp;2x^3&amp;+&amp;x^2&amp;-&amp;3x&amp;&amp; \\\\\n+&amp;&amp;&amp;&amp;&amp;-6x^2&amp;-&amp;3x&amp;+&amp;9 \\\\\n\\hline\n&amp;4x^4&amp;+&amp;4x^3&amp;-&amp;11x^2&amp;-&amp;6x&amp;+&amp;9\n\\end{array}[\/latex]<\/li>\n \t<li>[latex]\\phantom{a}[\/latex]\n[latex]\\begin{array}[t]{rrrrrrrrrr}\n&amp;x^2&amp;+&amp;5x&amp;-&amp;2\\phantom{x^2}&amp;&amp;&amp;&amp; \\\\\n\\times &amp;2x^2&amp;-&amp;x&amp;+&amp;3\\phantom{x^2}&amp;&amp;&amp;&amp; \\\\\n\\hline\n&amp;2x^4&amp;+&amp;10x^3&amp;-&amp;4x^2&amp;&amp;&amp;&amp; \\\\\n&amp;&amp;&amp;-x^3&amp;-&amp;5x^2&amp;+&amp;2x&amp;&amp; \\\\\n+&amp;&amp;&amp;&amp;&amp;3x^2&amp;+&amp;15x&amp;-&amp;6 \\\\\n\\hline\n&amp;2x^4&amp;+&amp;9x^3&amp;-&amp;6x^2&amp;+&amp;17x&amp;-&amp;6\n\\end{array}[\/latex]<\/li>\n \t<li>[latex]\\phantom{a}[\/latex]\n[latex]\\begin{array}[t]{rrrrrrrrrr}\n(x+4)(x+4)&amp;\\Rightarrow &amp;&amp;x^2&amp;+&amp;8x&amp;+&amp;16&amp;&amp; \\\\\n&amp;&amp;\\times&amp;&amp;&amp;x&amp;+&amp;4&amp;&amp; \\\\\n\\hline\n&amp;&amp;&amp;x^3&amp;+&amp;8x^2&amp;+&amp;16x&amp;&amp; \\\\\n&amp;&amp;+&amp;&amp;&amp;4x^2&amp;+&amp;32x&amp;+&amp;64 \\\\\n\\hline\n&amp;&amp;&amp;x^3&amp;+&amp;12x^2&amp;+&amp;48x&amp;+&amp;64\n\\end{array}[\/latex]<\/li>\n \t<li>[latex]r^{-4-3}s^{9+9}\\Rightarrow r^{-7}s^{18}\\Rightarrow \\dfrac{s^{18}}{r^7}[\/latex]\n[latex]\\dfrac{s^{18}}{r^7}[\/latex]<\/li>\n \t<li>[latex](x^{-2--2}y^{-3-4})^{-1}[\/latex]\n[latex]\\begin{array}[t]{l}\n(1\\cancel{x^0}y^{-7})^{-1} \\\\\ny^7\ny^7\n\\end{array}[\/latex]<\/li>\n \t<li>[latex]\\phantom{a}[\/latex]\n<img class=\"aligncenter wp-image-1916 size-full\" src=\"https:\/\/opentextbc.ca\/intermediatealgebraberg\/wp-content\/uploads\/sites\/304\/2022\/11\/MT2PREP-e1649114251301.png\" alt=\"\" width=\"200\" height=\"167\"><\/li>\n \t<li>[latex]2^3\\cdot 11[\/latex]<\/li>\n \t<li>[latex]2^5\\cdot 3\\cdot 7\n\\left\\{\n\\begin{array}{l}\n84=2^2\\cdot 3\\cdot 7 \\\\\n96=2^5\\cdot 3\n\\end{array}\\right.[\/latex]<\/li>\n \t<li>[latex]x(5y+6z)-3(5y+6z)[\/latex]\n[latex](5y+6z)(x-3)[\/latex]<\/li>\n \t<li>[latex]-12=4\\times -3[\/latex]\n[latex]1=4+-3 [\/latex]\n[latex]x^2+4x-3x-12[\/latex]\n[latex]x(x+4)-3(x+4)[\/latex]\n[latex](x+4)(x-3)[\/latex]<\/li>\n \t<li>[latex]x^2(x+1)-4(x+1)[\/latex]\n[latex](x+1)(x^2-4)[\/latex]\n[latex]x+1)(x-2)(x+2)[\/latex]<\/li>\n \t<li>[latex]x^3-(3y)^3[\/latex]\n[latex](x-3y)(x^2+3xy+9y^2)[\/latex]<\/li>\n \t<li>[latex](x^2-36)(x^2+1)[\/latex]\n[latex](x-6)(x+6)(x^2+1)[\/latex]<\/li>\n \t<li>[latex]\\phantom{a}[\/latex]\n[latex]\\begin{array}[t]{lll}\n\\begin{array}[t]{rrrrl}\n(A&amp;+&amp;B&amp;=&amp;70)(-4) \\\\\n4A&amp;+&amp;7B&amp;=&amp;430\n\\end{array}\n&amp; \\Rightarrow \\hspace{0.25in}\n\\begin{array}[t]{rrrrrl}\n&amp;-4A&amp;-&amp;4B&amp;=&amp;-280 \\\\\n+&amp;4A&amp;+&amp;7B&amp;=&amp;\\phantom{-}430 \\\\\n\\hline\n&amp;&amp;&amp;3B&amp;=&amp;\\phantom{-}150 \\\\ \\\\\n&amp;&amp;&amp;B&amp;=&amp;\\dfrac{150}{3}\\text{ or }50\n\\end{array}\n&amp; \\hspace{0.25in}\n\\begin{array}[t]{rrrrr}\n\\therefore A&amp;+&amp;B&amp;=&amp;70 \\\\ \\\\\nA&amp;+&amp;50&amp;=&amp;70 \\\\\n&amp;&amp;-50&amp;&amp;-50 \\\\\n\\hline\n&amp;&amp;A&amp;=&amp;20\n\\end{array}\n\\end{array}[\/latex]<\/li>\n \t<li>[latex]\\phantom{a}[\/latex]\n[latex]\\begin{array}[t]{rrcrrrl}\n5x&amp;+&amp;21(2)&amp;=&amp;11(x&amp;+&amp;2) \\\\ \\\\\n5x&amp;+&amp;42&amp;=&amp;11x&amp;+&amp;22 \\\\\n-5x&amp;-&amp;22&amp;&amp;-5x&amp;-&amp;22 \\\\\n\\hline\n&amp;&amp;20&amp;=&amp;6x&amp;&amp; \\\\ \\\\\n&amp;&amp;x&amp;=&amp;\\dfrac{20}{6}&amp;=&amp;3\\dfrac{1}{3}\\text{ litres} \\\\\n\\end{array}[\/latex]<\/li>\n \t<li>[latex]B+G=16\\Rightarrow B=16-G\\text{ or }G=16-B [\/latex]\n[latex]\\begin{array}[t]{ll}\n\\begin{array}[t]{rrrrrrr}\nG&amp;-&amp;4&amp;=&amp;3(B&amp;-&amp;4) \\\\\n16-B&amp;-&amp;4&amp;=&amp;3B&amp;-&amp;12 \\\\\n+B&amp;+&amp;12&amp;&amp;+B&amp;+&amp;12 \\\\\n\\hline\n&amp;&amp;24&amp;=&amp;4B&amp;&amp; \\\\ \\\\\n&amp;&amp;B&amp;=&amp;\\dfrac{24}{4}&amp;=&amp;6\n\\end{array}\n&amp; \\hspace{0.25in}\n\\begin{array}[t]{rrrrr}\n\\therefore G&amp;=&amp;16&amp;-&amp;B \\\\\nG&amp;=&amp;16&amp;-&amp;6 \\\\\nG&amp;=&amp;10&amp;&amp;\n\\end{array}\n\\end{array}[\/latex]<\/li>\n<\/ol>","rendered":"<h1>Midterm Two Review<\/h1>\n<ol>\n<li>\n<table class=\"lines\" style=\"border-collapse: collapse; width: 50%;\">\n<caption>[latex]x-2y=-4[\/latex]<\/caption>\n<tbody>\n<tr>\n<th style=\"width: 6.2678%; text-align: center;\" scope=\"col\">[latex]x[\/latex]<\/th>\n<th style=\"width: 5.12821%; text-align: center;\" scope=\"col\">[latex]y[\/latex]<\/th>\n<\/tr>\n<tr>\n<td style=\"width: 6.2678%; text-align: center;\">\u22124<\/td>\n<td style=\"width: 5.12821%; text-align: center;\">0<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 6.2678%; text-align: center;\">0<\/td>\n<td style=\"width: 5.12821%; text-align: center;\">2<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 6.2678%; text-align: center;\">\u22122<\/td>\n<td style=\"width: 5.12821%; text-align: center;\">1<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table class=\"lines\" style=\"border-collapse: collapse; width: 50%;\">\n<caption>[latex]x+y=5[\/latex]<\/caption>\n<tbody>\n<tr>\n<th style=\"width: 27.4929%; text-align: center;\" scope=\"col\">[latex]x[\/latex]<\/th>\n<th style=\"width: 25.3561%; text-align: center;\" scope=\"col\">[latex]y[\/latex]<\/th>\n<\/tr>\n<tr>\n<td style=\"width: 27.4929%; text-align: center;\">0<\/td>\n<td style=\"width: 25.3561%; text-align: center;\">5<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 27.4929%; text-align: center;\">5<\/td>\n<td style=\"width: 25.3561%; text-align: center;\">0<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 27.4929%; text-align: center;\">2<\/td>\n<td style=\"width: 25.3561%; text-align: center;\">3<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-medium wp-image-1915\" src=\"https:\/\/opentextbc.ca\/intermediatealgebraberg\/wp-content\/uploads\/sites\/304\/2021\/12\/Midterm2.1-300x289.jpg\" alt=\"\" width=\"300\" height=\"289\" srcset=\"https:\/\/opentextbc.ca\/intermediatealgebraberg\/wp-content\/uploads\/sites\/304\/2021\/12\/Midterm2.1-300x289.jpg 300w, https:\/\/opentextbc.ca\/intermediatealgebraberg\/wp-content\/uploads\/sites\/304\/2021\/12\/Midterm2.1-65x63.jpg 65w, https:\/\/opentextbc.ca\/intermediatealgebraberg\/wp-content\/uploads\/sites\/304\/2021\/12\/Midterm2.1-225x217.jpg 225w, https:\/\/opentextbc.ca\/intermediatealgebraberg\/wp-content\/uploads\/sites\/304\/2021\/12\/Midterm2.1-350x337.jpg 350w, https:\/\/opentextbc.ca\/intermediatealgebraberg\/wp-content\/uploads\/sites\/304\/2021\/12\/Midterm2.1.jpg 412w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/li>\n<li>[latex]\\phantom{a}[\/latex]<br \/>\n[latex]\\begin{array}[t]{rrcrlrl} 2x&-&y&=&0&\\Rightarrow &y=2x \\\\ 3x&+&4y&=&-22&& \\\\ \\\\ \\therefore 3x&+&4(2x)&=&-22&& \\\\ 3x&+&8x&=&-22&& \\\\ &&11x&=&-22&& \\\\ &&x&=&-2&& \\\\ \\\\ &&y&=&2x&& \\\\ &&y&=&2(-2)&=&-4 \\\\ \\end{array}[\/latex]<br \/>\n[latex](-2,-4)[\/latex]<\/li>\n<li>[latex]\\phantom{a}[\/latex]<br \/>\n[latex]\\begin{array}[t]{rrrrrl} &(2x&-&5y&=&15)(2) \\\\ &(3x&+&2y&=&13)(5) \\\\ \\hline &4x&-&10y&=&30 \\\\ +&15x&+&10y&=&65 \\\\ \\hline &&&19x&=&95 \\\\ &&&x&=&5 \\\\ \\\\ &\\therefore 3(5)&+&2y&=&\\phantom{-}13 \\\\ &15&+&2y&=&\\phantom{-}13 \\\\ &-15&&&&-15 \\\\ \\hline &&&2y&=&-2 \\\\ &&&y&=&-1 \\end{array}[\/latex]<br \/>\n[latex](5,-1)[\/latex]<\/li>\n<li>[latex]\\phantom{a}[\/latex]<br \/>\n[latex]\\begin{array}[t]{rr} \\begin{array}[t]{rrrrrrrl} &&&(5x&+&6z&=&-4)(-1) \\\\ \\\\ &5x&+&y&+&6z&=&-2 \\\\ +&-5x&&&-&6z&=&\\phantom{-}4 \\\\ \\hline &&&&&y&=&2 \\\\ \\\\ &&&\\therefore 2y&-&3z&=&\\phantom{-}3 \\\\ &&&2(2)&-&3z&=&\\phantom{-}3 \\\\ &&&-4&&&&-4 \\\\ \\hline &&&&&-3z&=&-1 \\\\ &&&&&z&=&\\dfrac{1}{3} \\\\ \\end{array} & \\hspace{0.25in} \\begin{array}[t]{rrrrr} 5x&+&6z&=&-4 \\\\ 5x&+&6\\left(\\dfrac{1}{3}\\right)&=&-4 \\\\ 5x&+&2&=&-4 \\\\ &-&2&&-2 \\\\ \\hline &&5x&=&-6 \\\\ &&x&=&-\\dfrac{6}{5} \\end{array} \\end{array}[\/latex]<br \/>\n[latex]-\\dfrac{6}{5}, 2, \\dfrac{1}{3}[\/latex]<\/li>\n<li>[latex]\\phantom{a}[\/latex]<br \/>\n[latex]\\begin{array}[t]{rrrrrr} &4a^2&-&9a&+&2 \\\\ &-a^2&+&4a&+&5 \\\\ +&3a^2&-&a&+&9 \\\\ \\hline &6a^2&-&6a&+&16 \\end{array}[\/latex]<\/li>\n<li>[latex]8x^4-12x^2y^2-15x^2y^2-3x^4\\Rightarrow 5x^4-27x^2y^2[\/latex]<\/li>\n<li>[latex]6-2\\left[3x-20x+8-1\\right][\/latex]<br \/>\n[latex]\\begin{array}[t]{l} 6-2\\left[-17x+7\\right] \\\\ 6+34x-14 \\\\ 34x-8 \\end{array}[\/latex]<\/li>\n<li>[latex]25a^{-10}b^6\\text{ or } \\dfrac{25b^6}{a^{10}}[\/latex]<\/li>\n<li>[latex]8a^2(a^2+10a+25)[\/latex]<br \/>\n[latex]8a^4+80a^3+200a^2[\/latex]<\/li>\n<li>[latex]4ab^2(a^2-4)[\/latex]<br \/>\n[latex]4a^3b^2-16ab^2[\/latex]<\/li>\n<li>[latex]\\phantom{a}[\/latex]<br \/>\n[latex]\\begin{array}[t]{rrrrrrrr} &x^2&-&4x&+&7\\phantom{x}&& \\\\ \\times &&&x&-&3\\phantom{x}&& \\\\ \\hline &x^3&-&4x^2&+&7x&& \\\\ +&&-&3x^2&+&12x&-&21 \\\\ \\hline &x^3&-&7x^2&+&19x&-&21 \\\\ \\end{array}[\/latex]<\/li>\n<li>[latex]\\phantom{a}[\/latex]<br \/>\n[latex]\\begin{array}[t]{rrrrrrrrrr} &2x^2&+&x&-&3\\phantom{x^2}&&&& \\\\ \\times &2x^2&+&x&-&3\\phantom{x^2}&&&& \\\\ \\hline &4x^4&+&2x^3&-&6x^2&&&& \\\\ &&&2x^3&+&x^2&-&3x&& \\\\ +&&&&&-6x^2&-&3x&+&9 \\\\ \\hline &4x^4&+&4x^3&-&11x^2&-&6x&+&9 \\end{array}[\/latex]<\/li>\n<li>[latex]\\phantom{a}[\/latex]<br \/>\n[latex]\\begin{array}[t]{rrrrrrrrrr} &x^2&+&5x&-&2\\phantom{x^2}&&&& \\\\ \\times &2x^2&-&x&+&3\\phantom{x^2}&&&& \\\\ \\hline &2x^4&+&10x^3&-&4x^2&&&& \\\\ &&&-x^3&-&5x^2&+&2x&& \\\\ +&&&&&3x^2&+&15x&-&6 \\\\ \\hline &2x^4&+&9x^3&-&6x^2&+&17x&-&6 \\end{array}[\/latex]<\/li>\n<li>[latex]\\phantom{a}[\/latex]<br \/>\n[latex]\\begin{array}[t]{rrrrrrrrrr} (x+4)(x+4)&\\Rightarrow &&x^2&+&8x&+&16&& \\\\ &&\\times&&&x&+&4&& \\\\ \\hline &&&x^3&+&8x^2&+&16x&& \\\\ &&+&&&4x^2&+&32x&+&64 \\\\ \\hline &&&x^3&+&12x^2&+&48x&+&64 \\end{array}[\/latex]<\/li>\n<li>[latex]r^{-4-3}s^{9+9}\\Rightarrow r^{-7}s^{18}\\Rightarrow \\dfrac{s^{18}}{r^7}[\/latex]<br \/>\n[latex]\\dfrac{s^{18}}{r^7}[\/latex]<\/li>\n<li>[latex](x^{-2--2}y^{-3-4})^{-1}[\/latex]<br \/>\n[latex]\\begin{array}[t]{l} (1\\cancel{x^0}y^{-7})^{-1} \\\\ y^7 y^7 \\end{array}[\/latex]<\/li>\n<li>[latex]\\phantom{a}[\/latex]<br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-1916 size-full\" src=\"https:\/\/opentextbc.ca\/intermediatealgebraberg\/wp-content\/uploads\/sites\/304\/2022\/11\/MT2PREP-e1649114251301.png\" alt=\"\" width=\"200\" height=\"167\" srcset=\"https:\/\/opentextbc.ca\/intermediatealgebraberg\/wp-content\/uploads\/sites\/304\/2022\/11\/MT2PREP-e1649114251301.png 200w, https:\/\/opentextbc.ca\/intermediatealgebraberg\/wp-content\/uploads\/sites\/304\/2022\/11\/MT2PREP-e1649114251301-65x54.png 65w\" sizes=\"auto, (max-width: 200px) 100vw, 200px\" \/><\/li>\n<li>[latex]2^3\\cdot 11[\/latex]<\/li>\n<li>[latex]2^5\\cdot 3\\cdot 7 \\left\\{ \\begin{array}{l} 84=2^2\\cdot 3\\cdot 7 \\\\ 96=2^5\\cdot 3 \\end{array}\\right.[\/latex]<\/li>\n<li>[latex]x(5y+6z)-3(5y+6z)[\/latex]<br \/>\n[latex](5y+6z)(x-3)[\/latex]<\/li>\n<li>[latex]-12=4\\times -3[\/latex]<br \/>\n[latex]1=4+-3[\/latex]<br \/>\n[latex]x^2+4x-3x-12[\/latex]<br \/>\n[latex]x(x+4)-3(x+4)[\/latex]<br \/>\n[latex](x+4)(x-3)[\/latex]<\/li>\n<li>[latex]x^2(x+1)-4(x+1)[\/latex]<br \/>\n[latex](x+1)(x^2-4)[\/latex]<br \/>\n[latex]x+1)(x-2)(x+2)[\/latex]<\/li>\n<li>[latex]x^3-(3y)^3[\/latex]<br \/>\n[latex](x-3y)(x^2+3xy+9y^2)[\/latex]<\/li>\n<li>[latex](x^2-36)(x^2+1)[\/latex]<br \/>\n[latex](x-6)(x+6)(x^2+1)[\/latex]<\/li>\n<li>[latex]\\phantom{a}[\/latex]<br \/>\n[latex]\\begin{array}[t]{lll} \\begin{array}[t]{rrrrl} (A&+&B&=&70)(-4) \\\\ 4A&+&7B&=&430 \\end{array} & \\Rightarrow \\hspace{0.25in} \\begin{array}[t]{rrrrrl} &-4A&-&4B&=&-280 \\\\ +&4A&+&7B&=&\\phantom{-}430 \\\\ \\hline &&&3B&=&\\phantom{-}150 \\\\ \\\\ &&&B&=&\\dfrac{150}{3}\\text{ or }50 \\end{array} & \\hspace{0.25in} \\begin{array}[t]{rrrrr} \\therefore A&+&B&=&70 \\\\ \\\\ A&+&50&=&70 \\\\ &&-50&&-50 \\\\ \\hline &&A&=&20 \\end{array} \\end{array}[\/latex]<\/li>\n<li>[latex]\\phantom{a}[\/latex]<br \/>\n[latex]\\begin{array}[t]{rrcrrrl} 5x&+&21(2)&=&11(x&+&2) \\\\ \\\\ 5x&+&42&=&11x&+&22 \\\\ -5x&-&22&&-5x&-&22 \\\\ \\hline &&20&=&6x&& \\\\ \\\\ &&x&=&\\dfrac{20}{6}&=&3\\dfrac{1}{3}\\text{ litres} \\\\ \\end{array}[\/latex]<\/li>\n<li>[latex]B+G=16\\Rightarrow B=16-G\\text{ or }G=16-B[\/latex]<br \/>\n[latex]\\begin{array}[t]{ll} \\begin{array}[t]{rrrrrrr} G&-&4&=&3(B&-&4) \\\\ 16-B&-&4&=&3B&-&12 \\\\ +B&+&12&&+B&+&12 \\\\ \\hline &&24&=&4B&& \\\\ \\\\ &&B&=&\\dfrac{24}{4}&=&6 \\end{array} & \\hspace{0.25in} \\begin{array}[t]{rrrrr} \\therefore G&=&16&-&B \\\\ G&=&16&-&6 \\\\ G&=&10&& \\end{array} 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