{"id":3276,"date":"2018-12-11T13:51:30","date_gmt":"2018-12-11T18:51:30","guid":{"rendered":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/chapter\/simplify-complex-rational-expressions\/"},"modified":"2018-12-11T13:51:30","modified_gmt":"2018-12-11T18:51:30","slug":"simplify-complex-rational-expressions","status":"publish","type":"chapter","link":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/chapter\/simplify-complex-rational-expressions\/","title":{"raw":"Simplify Complex Rational Expressions","rendered":"Simplify Complex Rational Expressions"},"content":{"raw":"\n[latexpage]<div class=\"textbox textbox--learning-objectives\"><h3 itemprop=\"educationalUse\">Learning Objectives<\/h3>By the end of this section, you will be able to: <ul><li>Simplify a complex rational expression by writing it as division<\/li><li>Simplify a complex rational expression by using the LCD<\/li><\/ul><\/div><div data-type=\"note\" id=\"fs-id1167835308956\" class=\"be-prepared\"><p>Before you get started, take this readiness quiz.<\/p><ol id=\"fs-id1167835416944\" type=\"1\"><li>Simplify: \\(\\frac{\\frac{3}{5}}{\\frac{9}{10}}.\\)<div data-type=\"newline\"><br><\/div> If you missed this problem, review <a href=\"\/contents\/425620d9-51dd-45e5-8a21-953998a4a77f#fs-id1167829590640\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li><li>Simplify: \\(\\frac{1-\\frac{1}{3}}{{4}^{2}+4\u00b75}.\\)<div data-type=\"newline\"><br><\/div> If you missed this problem, review <a href=\"\/contents\/425620d9-51dd-45e5-8a21-953998a4a77f#fs-id1167829627815\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li><li>Solve: \\(\\frac{1}{2x}+\\frac{1}{4}=\\frac{1}{8}.\\)<div data-type=\"newline\"><br><\/div> If you missed this problem, review <a href=\"\/contents\/9f100e8f-6d15-4cae-bc22-c306e9d7d55c#fs-id1167833239741\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li><\/ol><\/div><div class=\"bc-section section\" data-depth=\"1\"><h3 data-type=\"title\">Simplify a Complex Rational Expression by Writing it as Division<\/h3><p id=\"fs-id1167834227963\">Complex fractions are fractions in which the numerator or denominator contains a fraction. We previously simplified complex fractions like these:<\/p><div data-type=\"equation\" id=\"fs-id1167835419382\" class=\"unnumbered\" data-label=\"\">\\(\\begin{array}{cccccc}\\frac{\\frac{3}{4}}{\\frac{5}{8}}\\hfill &amp; &amp; &amp; &amp; &amp; \\frac{\\frac{x}{2}}{\\frac{xy}{6}}\\hfill \\end{array}\\)<\/div><p id=\"fs-id1167835301808\">In this section, we will simplify complex rational expressions, which are rational expressions with rational expressions in the numerator or denominator.<\/p><div data-type=\"note\" id=\"fs-id1167834489858\"><div data-type=\"title\">Complex Rational Expression<\/div><p id=\"fs-id1167835320604\">A <span data-type=\"term\">complex rational expression<\/span> is a rational expression in which the numerator and\/or the denominator contains a rational expression.<\/p><\/div><p>Here are a few complex rational expressions:<\/p><div data-type=\"equation\" id=\"fs-id1167831884939\" class=\"unnumbered\" data-label=\"\">\\(\\frac{\\frac{4}{y-3}}{\\frac{8}{{y}^{2}-9}}\\phantom{\\rule{3em}{0ex}}\\frac{\\frac{1}{x}+\\frac{1}{y}}{\\frac{x}{y}-\\frac{y}{x}}\\phantom{\\rule{3em}{0ex}}\\frac{\\frac{2}{x+6}}{\\frac{4}{x-6}-\\frac{4}{{x}^{2}-36}}\\)<\/div><p id=\"fs-id1167835384743\">Remember, we always exclude values that would make any denominator zero.<\/p><p id=\"fs-id1167834179741\">We will use two methods to simplify complex rational expressions.<\/p><p id=\"fs-id1167832056143\">We have already seen this complex rational expression earlier in this chapter.<\/p><div data-type=\"equation\" class=\"unnumbered\" data-label=\"\">\\(\\frac{\\frac{6{x}^{2}-7x+2}{4x-8}}{\\frac{2{x}^{2}-8x+3}{{x}^{2}-5x+6}}\\)<\/div><p id=\"fs-id1167831825117\">We noted that fraction bars tell us to divide, so rewrote it as the division problem:<\/p><div data-type=\"equation\" id=\"fs-id1167835370704\" class=\"unnumbered\" data-label=\"\">\\(\\left(\\frac{6{x}^{2}-7x+2}{4x-8}\\right)\u00f7\\left(\\frac{2{x}^{2}-8x+3}{{x}^{2}-5x+6}\\right).\\)<\/div><p>Then, we multiplied the first rational expression by the reciprocal of the second, just like we do when we divide two fractions.<\/p><p id=\"fs-id1167835319282\">This is one method to simplify complex rational expressions. We make sure the complex rational expression is of the form where one fraction is over one fraction. We then write it as if we were dividing two fractions.<\/p><div data-type=\"example\" id=\"fs-id1167826978829\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1167831836544\"><div data-type=\"problem\" id=\"fs-id1167834433716\"><p id=\"fs-id1167834346605\">Simplify the complex rational expression by writing it as division: \\(\\frac{\\frac{6}{x-4}}{\\frac{3}{{x}^{2}-16}}.\\)<\/p><\/div><div data-type=\"solution\"><p id=\"fs-id1167831910119\">\\(\\begin{array}{cccc}&amp; &amp; &amp; \\hfill \\phantom{\\rule{4em}{0ex}}\\frac{\\frac{6}{x-4}}{\\frac{3}{{x}^{2}-16}}\\hfill \\\\ \\\\ \\\\ \\text{Rewrite the complex fraction as division.}\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{4em}{0ex}}\\frac{6}{x-4}\u00f7\\frac{3}{{x}^{2}-16}\\hfill \\\\ \\\\ \\\\ \\begin{array}{c}\\text{Rewrite as the product of first times the}\\hfill \\\\ \\text{reciprocal of the second.}\\hfill \\end{array}\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{4em}{0ex}}\\frac{6}{x-4}\u00b7\\frac{{x}^{2}-16}{3}\\hfill \\\\ \\\\ \\\\ \\text{Factor.}\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{4em}{0ex}}\\frac{3\u00b72}{x-4}\u00b7\\frac{\\left(x-4\\right)\\left(x+4\\right)}{3}\\hfill \\\\ \\\\ \\\\ \\text{Multiply.}\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{4em}{0ex}}\\frac{3\u00b72\\left(x-4\\right)\\left(x+4\\right)}{3\\left(x-4\\right)}\\hfill \\\\ \\\\ \\\\ \\text{Remove common factors.}\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{4em}{0ex}}\\frac{\\overline{)3}\u00b72\\overline{)\\left(x-4\\right)}\\left(x+4\\right)}{\\overline{)3}\\overline{)\\left(x-4\\right)}}\\hfill \\\\ \\\\ \\\\ \\text{Simplify.}\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{4em}{0ex}}2\\left(x+4\\right)\\hfill \\end{array}\\)<\/p><p id=\"fs-id1167832057148\">Are there any value(s) of <em data-effect=\"italics\">x<\/em> that should not be allowed? The original complex rational expression had denominators of \\(x-4\\) and \\({x}^{2}-16.\\) This expression would be undefined if \\(x=4\\) or \\(x=-4.\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167832058388\" class=\"try\"><div data-type=\"exercise\"><div data-type=\"problem\" id=\"fs-id1167834408509\"><p id=\"fs-id1167835239962\">Simplify the complex rational expression by writing it as division: \\(\\frac{\\frac{2}{{x}^{2}-1}}{\\frac{3}{x+1}}.\\)<\/p><\/div><div data-type=\"solution\"><p id=\"fs-id1167832053233\">\\(\\frac{2}{3\\left(x-1\\right)}\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167835609958\"><div data-type=\"problem\" id=\"fs-id1167821178912\"><p id=\"fs-id1167835381150\">Simplify the complex rational expression by writing it as division: \\(\\frac{\\frac{1}{{x}^{2}-7x+12}}{\\frac{2}{x-4}}.\\)<\/p><\/div><div data-type=\"solution\"><p id=\"fs-id1167835343984\">\\(\\frac{1}{2\\left(x-3\\right)}\\)<\/p><\/div><\/div><\/div><p id=\"fs-id1167835353494\">Fraction bars act as grouping symbols. So to follow the Order of Operations, we simplify the numerator and denominator as much as possible before we can do the division.<\/p><div data-type=\"example\" id=\"fs-id1167835236104\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1167834474319\"><div data-type=\"problem\" id=\"fs-id1167831894391\"><p>Simplify the complex rational expression by writing it as division: \\(\\frac{\\frac{1}{3}+\\frac{1}{6}}{\\frac{1}{2}-\\frac{1}{3}}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835305870\"><table id=\"fs-id1167828435628\" class=\"unnumbered unstyled\" summary=\"The quantity one-third plus one-sixth divided by the quantity one-half minus one-third. Rewrite the numerator and denominator by finding the lowest common denominator. The lowest common denominator of one-third, one-sixth, one-half, and one-third is 6. In the numerator of the expression, multiply the numerator and denominator of one-third by 2. In the denominator of the expression, multiply the numerator and denominator of one-half by 3, and the numerator and denominator of one-third by 2. Simplify the numerator and the denominator. The result is the quantity two-sixths plus one-sixth all divided by the quantity three-sixths minus two-sixths. Rewrite the complex rational expression as a division problem, three-sixth divided by one-sixth. Multiply three-sixth by the reciprocal of one-sixth, which is 6 over 1. The result is three-sixth times six ones. Simplify the expression. The result is 3.\" data-label=\"\"><tbody><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" data-alt=\".\"><img src=\"CNX_IntAlg_Figure_07_03_001a_img_new.jpg#fixme#fixme\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Simplify the numerator and denominator.<div data-type=\"newline\"><br><\/div>Find the LCD and add the fractions in the numerator.<div data-type=\"newline\"><br><\/div>Find the LCD and subtract the fractions in the<div data-type=\"newline\"><br><\/div>denominator.<\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167834066374\" data-alt=\".\"><img src=\"CNX_IntAlg_Figure_07_03_001b_img_new.jpg#fixme#fixme\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Simplify the numerator and denominator.<\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167832138932\" data-alt=\".\"><img src=\"CNX_IntAlg_Figure_07_03_001c_img_new.jpg#fixme#fixme\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Rewrite the complex rational expression as a division<div data-type=\"newline\"><br><\/div>problem.<\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" data-alt=\".\"><img src=\"CNX_IntAlg_Figure_07_03_001e_img_new.jpg#fixme#fixme\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Multiply the first by the reciprocal of the second.<\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167835307222\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_03_001f_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Simplify.<\/td><td data-valign=\"top\" data-align=\"center\">3<\/td><\/tr><\/tbody><\/table><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167835254948\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167835362198\"><div data-type=\"problem\"><p>Simplify the complex rational expression by writing it as division: \\(\\frac{\\frac{1}{2}+\\frac{2}{3}}{\\frac{5}{6}+\\frac{1}{12}}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835339384\"><p id=\"fs-id1167834064255\">\\(\\frac{14}{11}\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167835320372\" class=\"try\"><div data-type=\"exercise\"><div data-type=\"problem\" id=\"fs-id1167835479676\"><p>Simplify the complex rational expression by writing it as division: \\(\\frac{\\frac{3}{4}-\\frac{1}{3}}{\\frac{1}{8}+\\frac{5}{6}}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835202220\"><p id=\"fs-id1167835254714\">\\(\\frac{10}{23}\\)<\/p><\/div><\/div><\/div><p id=\"fs-id1165926931185\">We follow the same procedure when the complex rational expression contains variables.<\/p><div data-type=\"example\" id=\"fs-id1167830705566\" class=\"textbox textbox--examples\"><div data-type=\"title\">How to Simplify a Complex Rational Expression using Division<\/div><div data-type=\"exercise\" id=\"fs-id1167834094691\"><div data-type=\"problem\" id=\"fs-id1167835283646\"><p>Simplify the complex rational expression by writing it as division: \\(\\frac{\\frac{1}{x}+\\frac{1}{y}}{\\frac{x}{y}-\\frac{y}{x}}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167831076629\"><span data-type=\"media\" data-alt=\"Step 1 is to simplify the sum in the numerator and the difference in the denominator of complex rational expression, the quantity 1 divided by x plus 1 divided by y all divided by the quantity x divided by y minus y divided by x. The common denominator of the fractions in the complex rational expression is x y. Multiply the numerator and denominator of 1 divided by x by y over y. Multiply the numerator and denominator of 1 divided by y by x over x. Multiply the numerator and denominator of x divided by y by x over x. Multiply the numerator and denominator of y over x by y over y. The result is the quantity y divided by x y plus x divided by x y all divided by the quantity x squared divided by x y minus y squared divided by x y. Add the fractions in the numerator and subtract the fractions in the denominator. The result is the sum of y and x divided by x y all divided by the difference between x squared and y squared divided by x y. We now have just one rational expression in the numerator and one in the denominator.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_03_002a_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 1 is to simplify the sum in the numerator and the difference in the denominator of complex rational expression, the quantity 1 divided by x plus 1 divided by y all divided by the quantity x divided by y minus y divided by x. The common denominator of the fractions in the complex rational expression is x y. Multiply the numerator and denominator of 1 divided by x by y over y. Multiply the numerator and denominator of 1 divided by y by x over x. Multiply the numerator and denominator of x divided by y by x over x. Multiply the numerator and denominator of y over x by y over y. The result is the quantity y divided by x y plus x divided by x y all divided by the quantity x squared divided by x y minus y squared divided by x y. Add the fractions in the numerator and subtract the fractions in the denominator. The result is the sum of y and x divided by x y all divided by the difference between x squared and y squared divided by x y. We now have just one rational expression in the numerator and one in the denominator.\"><\/span><span data-type=\"media\" id=\"fs-id1167835327260\" data-alt=\"Step 2 is to rewrite the complex rational expression as a division problem. Write the numerator divided by the denominator. The result is the quantity of the sum y and x divided by x y all divided by the quantity of the difference between x squared and y squared divided by x y.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_03_002b_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 2 is to rewrite the complex rational expression as a division problem. Write the numerator divided by the denominator. The result is the quantity of the sum y and x divided by x y all divided by the quantity of the difference between x squared and y squared divided by x y.\"><\/span><span data-type=\"media\" id=\"fs-id1167834066451\" data-alt=\"Step 3 is to divided the expressions. Multiply the first expression by the reciprocal of the second expression. The result is the quantity of the sum y and x divided by x y times the quantity x y divided by the difference between x squared and y squared. Factor any expressions if possible. The result is the product of x y and the sum of y and x all divided by the product of x y, the difference between x and y, and the sum of x and y. Remove the common factors, x y and the sum of x and y. Simplify. The result is 1 divided by the quantity x minus y.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_03_002c_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 3 is to divided the expressions. Multiply the first expression by the reciprocal of the second expression. The result is the quantity of the sum y and x divided by x y times the quantity x y divided by the difference between x squared and y squared. Factor any expressions if possible. The result is the product of x y and the sum of y and x all divided by the product of x y, the difference between x and y, and the sum of x and y. Remove the common factors, x y and the sum of x and y. Simplify. The result is 1 divided by the quantity x minus y.\"><\/span><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167835303159\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167834423424\"><div data-type=\"problem\" id=\"fs-id1167834301239\"><p id=\"fs-id1167835253848\">Simplify the complex rational expression by writing it as division: \\(\\frac{\\frac{1}{x}+\\frac{1}{y}}{\\frac{1}{x}-\\frac{1}{y}}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835356218\"><p id=\"fs-id1167835371395\">\\(\\frac{y+x}{y-x}\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167835309227\"><div data-type=\"problem\" id=\"fs-id1167834161618\"><p id=\"fs-id1167832092221\">Simplify the complex rational expression by writing it as division: \\(\\frac{\\frac{1}{a}+\\frac{1}{b}}{\\frac{1}{{a}^{2}}-\\frac{1}{{b}^{2}}}\\).<\/p><\/div><div data-type=\"solution\"><p>\\(\\frac{ab}{b-a}\\)<\/p><\/div><\/div><\/div><p id=\"fs-id1167835233515\">We summarize the steps here.<\/p><div data-type=\"note\" id=\"fs-id1167835319109\" class=\"howto\"><div data-type=\"title\">Simplify a complex rational expression by writing it as division.<\/div><ol id=\"fs-id1167835280276\" type=\"1\" class=\"stepwise\"><li>Simplify the numerator and denominator.<\/li><li>Rewrite the complex rational expression as a division problem.<\/li><li>Divide the expressions.<\/li><\/ol><\/div><div data-type=\"example\" id=\"fs-id1167835337830\" class=\"textbox textbox--examples\"><div data-type=\"exercise\"><div data-type=\"problem\" id=\"fs-id1167835498830\"><p>Simplify the complex rational expression by writing it as division: \\(\\frac{n-\\frac{4n}{n+5}}{\\frac{1}{n+5}+\\frac{1}{n-5}}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835243968\"><table id=\"fs-id1167834308269\" class=\"unnumbered unstyled can-break\" summary=\"Simplify the complex rational expression n minus the quantity 4 n divided by the quantity n plus 5 all divided by the sum of 1 divided by the quantity n plus 5 and 1 divided by the quantity n minus 5. Simplify the numerator and denominator. Find the common denominators for the numerator and denominator. The result is the n times the quantity n plus 5 all divided by 1 times the quantity n plus 5 minus the quantity 4 n divided by the quantity n plus 5 all divided by 1 times the quantity n minus 5 divided by the quantity n plus 5 times the quantity n minus 5 plus the quantity 1 times the quantity n plus 5 divided by the quantity n plus 5 times the quantity n minus 5. Simplify the numerators. The result is the quantity n squared plus 5 n divided by the quantity n plus 5 minus the quantity 4 n divided by the quantity n plus 5 all divided by 1 times the quantity n minus 5 divided by the quantity n plus 5 times the quantity n minus 5 plus 1 times the quantity n plus 5 divided by the quantity n plus 5 times the quantity n minus 5. Subtract the rational expressions in the numerator and add in the denominator. The result is the quantity n squared plus 5 n minus 4 n divided by the quantity n plus 5 all divided by the quantity n minus 5 plus n plus 5 divided by the quantity n plus 5 times the quantity n minus 5. Multiply the first expression by the reciprocal of the second. The result is the quantity n squared plus 5 n minus 4 n divided by the quantity n plus 5 times the quantity n plus 5 times the quantity n minus 5 divided by the quantity n minus 5 plus n plus 5. Simplify. We now have one rational expression over one rational expression. The result is the quantity n squared plus n divided by the quantity n plus 5 all divided by the quantity 2 n divided by the quantity n plus 5 times the quantity n minus 5. Rewrite the expression as fraction division. The result is the quotient of the quantity n squared plus n divided by the quantity n plus 5 and the quantity 2 n divided by the quantity n plus 5 times the quantity n minus 5. Multiply the first fraction times the reciprocal of the second fraction. The result is the quantity n squared plus n divided by the quantity n plus 5 times the quantity of n plus 5 times the quantity n minus 5 all divided by the quantity 2 n. Factor any expressions if possible. The result us n times the quantity n plus 1 times the quantity n plus 5 times the quantity n minus 5 all divided by the product of the quantity n plus 5 and 2 n. Remove the common factors, n and n plus 5, from the numerator and denominator. Simplify. The result is the product of the quantity n plus 1 and the quantity n minus 5 all divided by 2.\" data-label=\"\"><tbody><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167834527716\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_03_003a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Simplify the numerator and denominator.<div data-type=\"newline\"><br><\/div>Find common denominators for the numerator and<div data-type=\"newline\"><br><\/div>denominator.<\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167835370172\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_03_003b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Simplify the numerators.<\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167830699580\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_03_003c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Subtract the rational expressions in the numerator and<div data-type=\"newline\"><br><\/div>add in the denominator.<\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167835191878\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_03_003d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Simplify. (We now have one rational expression over<div data-type=\"newline\"><br><\/div>one rational expression.)<\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167832006043\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_03_003e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Rewrite as fraction division.<\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167832058793\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_03_003f_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Multiply the first times the reciprocal of the second.<\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167826779013\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_03_003g_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Factor any expressions if possible.<\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167831148955\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_03_003h_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Remove common factors.<\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167834194708\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_03_003i_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Simplify.<\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167834156730\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_03_003j_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><\/tbody><\/table><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167826977973\" class=\"try\"><div data-type=\"exercise\"><div data-type=\"problem\"><p>Simplify the complex rational expression by writing it as division: \\(\\frac{b-\\frac{3b}{b+5}}{\\frac{2}{b+5}+\\frac{1}{b-5}}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834397941\"><p>\\(\\frac{b\\left(b+2\\right)\\left(b-5\\right)}{3b-5}\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167835303583\" class=\"try\"><div data-type=\"exercise\"><div data-type=\"problem\" id=\"fs-id1167835341039\"><p id=\"fs-id1167835264626\">Simplify the complex rational expression by writing it as division: \\(\\frac{1-\\frac{3}{c+4}}{\\frac{1}{c+4}+\\frac{c}{3}}.\\)<\/p><\/div><div data-type=\"solution\"><p id=\"fs-id1167834428859\">\\(\\frac{3}{c+3}\\)<\/p><\/div><\/div><\/div><\/div><div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167835609912\"><h3 data-type=\"title\">Simplify a Complex Rational Expression by Using the LCD<\/h3><p id=\"fs-id1167835304510\">We \u201ccleared\u201d the fractions by multiplying by the LCD when we solved equations with fractions. We can use that strategy here to simplify complex rational expressions. We will multiply the numerator and denominator by the LCD of all the rational expressions.<\/p><p id=\"fs-id1167832060555\">Let\u2019s look at the complex rational expression we simplified one way in <a href=\"#fs-id1167835236104\" class=\"autogenerated-content\">(Figure)<\/a>. We will simplify it here by multiplying the numerator and denominator by the LCD. When we multiply by \\(\\frac{\\text{LCD}}{\\text{LCD}}\\) we are multiplying by 1, so the value stays the same.<\/p><div data-type=\"example\" id=\"fs-id1167826829055\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1167831239367\"><div data-type=\"problem\" id=\"fs-id1167835303825\"><p>Simplify the complex rational expression by using the LCD: \\(\\frac{\\frac{1}{3}+\\frac{1}{6}}{\\frac{1}{2}-\\frac{1}{3}}.\\)<\/p><\/div><div data-type=\"solution\"><table id=\"fs-id1167835419447\" class=\"unnumbered unstyled\" summary=\"Simplify the sum of one-third and one-sixth all divided by the difference between one-half and one-third. The least common denominator of all the fractions in the entire expression is 6. Clear the fractions by multiplying the numerator and denominator by the least common denominator. The result is 6 times sum of one-third and one-sixth all divided by 6 times the difference between one-half and one-third. Distribute. The result is the quantity 6 times one-third plus 6 times one-sixth all divided by the quantity 6 times one-half minus 6 times one-third. Simplify. The result is the quantity 2 plus 1 divided by the quantity 3 minus 2, which simplifies to 3 divided by 1. 3 divided by 1 is 3.\" data-label=\"\"><tbody><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167831104030\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_03_004a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">The LCD of all the fractions in the whole expression is 6.<\/td><td data-valign=\"top\" data-align=\"left\"><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Clear the fractions by multiplying the numerator and<div data-type=\"newline\"><br><\/div>denominator by that LCD.<\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167834555290\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_03_004b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Distribute.<\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167835518419\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_03_004c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Simplify.<\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167835307312\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_03_004d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167834556304\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_03_004e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_03_004f_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><\/tbody><\/table><\/div><\/div><\/div><div data-type=\"note\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167835343125\"><div data-type=\"problem\" id=\"fs-id1167835370755\"><p id=\"fs-id1167835206029\">Simplify the complex rational expression by using the LCD: \\(\\frac{\\frac{1}{2}+\\frac{1}{5}}{\\frac{1}{10}+\\frac{1}{5}}.\\)<\/p><\/div><div data-type=\"solution\"><p id=\"fs-id1167834179817\">\\(\\frac{7}{3}\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167830757621\"><div data-type=\"problem\"><p>Simplify the complex rational expression by using the LCD: \\(\\frac{\\frac{1}{4}+\\frac{3}{8}}{\\frac{1}{2}-\\frac{5}{16}}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835415213\"><p>\\(\\frac{10}{3}\\)<\/p><\/div><\/div><\/div><p>We will use the same example as in <a href=\"#fs-id1167830705566\" class=\"autogenerated-content\">(Figure)<\/a>. Decide which method works better for you.<\/p><div data-type=\"example\" id=\"fs-id1167831891154\" class=\"textbox textbox--examples\"><div data-type=\"title\">How to Simplify a Complex Rational Expressing using the LCD<\/div><div data-type=\"exercise\"><div data-type=\"problem\" id=\"fs-id1167831908482\"><p>Simplify the complex rational expression by using the LCD: \\(\\frac{\\frac{1}{x}+\\frac{1}{y}}{\\frac{x}{y}-\\frac{y}{x}}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835498808\"><span data-type=\"media\" id=\"fs-id1167830964450\" data-alt=\"Step 1 is to find the least common denominator of the complex rational expression, the sum of the quantity 1 divided by x and the quantity 1 divided by y all divided by the difference between the quantity x divided by y and the quantity y divided by x.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_03_005a_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 1 is to find the least common denominator of the complex rational expression, the sum of the quantity 1 divided by x and the quantity 1 divided by y all divided by the difference between the quantity x divided by y and the quantity y divided by x.\"><\/span><span data-type=\"media\" id=\"fs-id1167835414657\" data-alt=\"Step 2 is to multiply the numerator and the denominator by the least common denominator, x y. The result is x y times the sum of the quantity 1 divided by x and the quantity 1 divided by y all divided by x y times the difference between the quantity x divided by y and the quantity y divided by x.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_03_005b_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 2 is to multiply the numerator and the denominator by the least common denominator, x y. The result is x y times the sum of the quantity 1 divided by x and the quantity 1 divided by y all divided by x y times the difference between the quantity x divided by y and the quantity y divided by x.\"><\/span><span data-type=\"media\" id=\"fs-id1167830693440\" data-alt=\"Step 3 is to simplify the expression. Distribute x y in the numerator and the denominator. The result is x y times 1 divided by x plus x y times 1 divided by y all divided by x y times x divided by y plus x y times y divided by x. It simplifies to the sum of y and x divided by the quantity x squared minus y squared. Write the denominator as the difference of squares, the quantity x minus y times the quantity x plus y. The result is the quantity y plus x all divided by the quantity x minus y times the quantity x plus y. Remove the common factor, y plus x, from the numerator and denominator. The result is 1 divided by the quantity x minus y.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_03_005c_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 3 is to simplify the expression. Distribute x y in the numerator and the denominator. The result is x y times 1 divided by x plus x y times 1 divided by y all divided by x y times x divided by y plus x y times y divided by x. It simplifies to the sum of y and x divided by the quantity x squared minus y squared. Write the denominator as the difference of squares, the quantity x minus y times the quantity x plus y. The result is the quantity y plus x all divided by the quantity x minus y times the quantity x plus y. Remove the common factor, y plus x, from the numerator and denominator. The result is 1 divided by the quantity x minus y.\"><\/span><\/div><\/div><\/div><div data-type=\"note\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167834191698\"><div data-type=\"problem\"><p id=\"fs-id1167834367143\">Simplify the complex rational expression by using the LCD: \\(\\frac{\\frac{1}{a}+\\frac{1}{b}}{\\frac{a}{b}+\\frac{b}{a}}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835346314\"><p id=\"fs-id1167835198527\">\\(\\frac{b+a}{{a}^{2}+{b}^{2}}\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167835318628\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167835253742\"><div data-type=\"problem\" id=\"fs-id1167834432188\"><p id=\"fs-id1167835341828\">Simplify the complex rational expression by using the LCD: \\(\\frac{\\frac{1}{{x}^{2}}-\\frac{1}{{y}^{2}}}{\\frac{1}{x}+\\frac{1}{y}}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835218116\"><p id=\"fs-id1167835615105\">\\(\\frac{y-x}{xy}\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" class=\"howto\"><div data-type=\"title\">Simplify a complex rational expression by using the LCD.<\/div><ol id=\"fs-id1167826799162\" type=\"1\" class=\"stepwise\"><li>Find the LCD of all fractions in the complex rational expression.<\/li><li>Multiply the numerator and denominator by the LCD.<\/li><li>Simplify the expression.<\/li><\/ol><\/div><p id=\"fs-id1167835218400\">Be sure to start by factoring all the denominators so you can find the LCD.<\/p><div data-type=\"example\" id=\"fs-id1167826987981\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1167835420261\"><div data-type=\"problem\" id=\"fs-id1167835350442\"><p id=\"fs-id1167834131359\">Simplify the complex rational expression by using the LCD: \\(\\frac{\\frac{2}{x+6}}{\\frac{4}{x-6}-\\frac{4}{{x}^{2}-36}}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834130742\"><table id=\"fs-id1167834063606\" class=\"unnumbered unstyled can-break\" summary=\"Simplify the complex rational expression, 2 divided by the quantity x plus 6 all divided by the difference between 4 divided by the quantity x minus 6 and 4 divided by the quantity x squared minus 36. Find the least common denominator all of the fractions. It is x squared minus 36 which equals the quantity x plus 6 times the quantity x minus 6. Multiply the numerator and denominator by the least common denominator. The result is the quantity x plus 6 times the quantity x minus 6 times the quantity 2 divided by the quantity x plus 6 all divided by the quantity x plus 6 times the quantity x minus 6 times the difference between the quantity 4 divided by the quantity x minus 6 and 4 divided by the product of the quantity x plus 6 and the quantity x minus 6. Simplify the expression. Distribute, the quantity x plus 6 and x minus 6, in the denominator. The result is the quantity x plus 6 times the quantity x minus 6 times the quantity 2 divided by the quantity x plus 6 all divided by the quantity x plus 6 times the quantity x minus 6 times the quantity 4 divided by the quantity x minus 6 minus the quantity x plus 6 times the quantity x minus 6 times 4 divided by the product of the quantity x plus 6 and the quantity x minus 6. Simplify by removing the common factors, x plus 6 from the numerator, and the quantity x minus 6 times the quantity x plus 6, from the denominator. The result is 2 times the quantity x minus 6 all divided by the product of 4 and the quantity x plus 6 minus 4. To simplify the denominator, distribute and combine like terms. The result is 2 times the quantity x minus 6 all divided by the quantity 4 x plus 20. Factor the denominator. The result is 2 times the quantity x minus 6 all divided by 4 times the quantity x plus 5. Notice that 4 in the denominator is 2 times 2. Remove the common factor, 2, from the numerator and denominator. The result is the quantity x minus 6 all divided by 2 times the quantity x plus 5. Notice that there are no more factors common to the numerator and denominator.\" data-label=\"\"><tbody><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167832119089\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_03_006b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Find the LCD of all fractions in the complex rational<div data-type=\"newline\"><br><\/div>expression. The LCD is \\({x}^{2}-36=\\left(x+6\\right)\\left(x-6\\right)\\).<\/td><td data-valign=\"top\" data-align=\"left\"><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Multiply the numerator and denominator by the LCD.<\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167830700886\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_03_006a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Simplify the expression.<\/td><td data-valign=\"top\" data-align=\"left\"><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Distribute in the denominator.<\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167834133885\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_03_006c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Simplify.<\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167834161475\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_03_006d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Simplify.<\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_03_006e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">To simplify the denominator, distribute<div data-type=\"newline\"><br><\/div>and combine like terms.<\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_03_006f_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Factor the denominator.<\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167831040626\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_03_006g_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Remove common factors.<\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167831895215\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_03_006h_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Simplify.<\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167835369386\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_03_006i_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Notice that there are no more factors<div data-type=\"newline\"><br><\/div>common to the numerator and denominator.<\/td><td data-valign=\"top\" data-align=\"left\"><\/td><\/tr><\/tbody><\/table><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167834517647\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167832152324\"><div data-type=\"problem\" id=\"fs-id1167834395939\"><p id=\"fs-id1167831824577\">Simplify the complex rational expression by using the LCD: \\(\\frac{\\frac{3}{x+2}}{\\frac{5}{x-2}-\\frac{3}{{x}^{2}-4}}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834161803\"><p id=\"fs-id1167835377966\">\\(\\frac{3\\left(x-2\\right)}{5x+7}\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167835284772\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167835257334\"><div data-type=\"problem\" id=\"fs-id1167831115599\"><p id=\"fs-id1167830838434\">Simplify the complex rational expression by using the LCD: \\(\\frac{\\frac{2}{x-7}-\\frac{1}{x+7}}{\\frac{6}{x+7}-\\frac{1}{{x}^{2}-49}}.\\)<\/p><\/div><div data-type=\"solution\"><p id=\"fs-id1167835489114\">\\(\\frac{x+21}{6x-43}\\)<\/p><\/div><\/div><\/div><p id=\"fs-id1165926630658\">Be sure to factor the denominators first. Proceed carefully as the math can get messy!<\/p><div data-type=\"example\" id=\"fs-id1167835595640\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1167835325063\"><div data-type=\"problem\" id=\"fs-id1167834501479\"><p id=\"fs-id1167834194400\">Simplify the complex rational expression by using the LCD: \\(\\frac{\\frac{4}{{m}^{2}-7m+12}}{\\frac{3}{m-3}-\\frac{2}{m-4}}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835344864\"><table id=\"fs-id1167835363257\" class=\"unnumbered unstyled\" summary=\"Simplify the complex rational expression, 4 divided by the quantity m squared minus 7 m plus 12 all divided by the difference between 3 divided by the quantity m minus 3 and 2 divided by the quantity m minus 4. Find the least common denominator of all fractions in the expression. The least common denominator is the quantity m minus 3 times the quantity m minus 4. Multiply the numerator and denominator by the least common denominator. The result is the quantity m minus 3 times the quantity m minus 4 times the quantity 4 divided by the quantity m minus 3 times the quantity m minus 4 all divided by the quantity m minus 3 times the quantity m minus 4 times the difference between 3 divided by the quantity m minus 3 and 2 divided by the quantity m minus 4. Simplify by removing the common factors, m minus 3 and m minus 4, from the numerator and denominator. The result is 4 divided by the product of 3 and m minus 4 minus the product of 2 and minus 3. Distribute the factors 3 and 2 in the denominator. The result is 4 divided by the quantity 3 m minus 12 minus 2 m plus 6. Combine like terms in the denominator. The result is 4 divided by the quantity m minus 6.\" data-label=\"\"><tbody><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167830838226\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_03_007b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Find the LCD of all fractions in the<div data-type=\"newline\"><br><\/div>complex rational expression.<\/td><td data-valign=\"top\" data-align=\"center\"><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">The LCD is \\(\\left(m-3\\right)\\left(m-4\\right).\\)<\/td><td data-valign=\"top\" data-align=\"center\"><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Multiply the numerator and<div data-type=\"newline\"><br><\/div>denominator by the LCD.<\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167834472376\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_03_007a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Simplify.<\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167835308942\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_03_007c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Simplify.<\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167835410534\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_03_007d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Distribute.<\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167826978771\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_03_007e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Combine like terms.<\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167835352202\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_03_007f_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><\/tbody><\/table><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167835358450\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167835376231\"><div data-type=\"problem\" id=\"fs-id1167835349843\"><p id=\"fs-id1167835335378\">Simplify the complex rational expression by using the LCD: \\(\\frac{\\frac{3}{{x}^{2}+7x+10}}{\\frac{4}{x+2}+\\frac{1}{x+5}}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834552539\"><p id=\"fs-id1167835524236\">\\(\\frac{3}{5x+22}\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167831076568\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167835231354\"><div data-type=\"problem\" id=\"fs-id1167835303310\"><p id=\"fs-id1167831116163\">Simplify the complex rational expression by using the LCD: \\(\\frac{\\frac{4y}{y+5}+\\frac{2}{y+6}}{\\frac{3y}{{y}^{2}+11y+30}}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835422942\"><p id=\"fs-id1167831217847\">\\(\\frac{2\\left(2{y}^{2}+13y+5\\right)}{3y}\\)<\/p><\/div><\/div><\/div><div data-type=\"example\" id=\"fs-id1167830698667\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1167834214131\"><div data-type=\"problem\" id=\"fs-id1167835351498\"><p id=\"fs-id1167834562517\">Simplify the complex rational expression by using the LCD: \\(\\frac{\\frac{y}{y+1}}{1+\\frac{1}{y-1}}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835419888\"><table id=\"fs-id1167835303162\" class=\"unnumbered unstyled can-break\" summary=\"Simplify the complex rational expression, y divided by quantity y plus 1 all divided by 1 plus the quantity 1 divided by the quantity y minus 1. Find the lowest common denominator of all fractions in the complex rational expression. It is the quantity y plus 1 times the quantity y minus 1. Multiply the numerator and denominator by the least common denominator. The result is the quantity y plus 1 times the quantity y minus 1 times the quantity y divided by quantity y plus 1 all divided by the quantity y plus 1 times the quantity y minus 1 times the quantity 1 plus the quantity 1 divided by the quantity y minus 1. Distribute the least common denominator, the quantity y plus 1 times the quantity y minus 1, in the denominator. The result is the quantity y plus 1 times the quantity y minus 1 times the quantity y divided by the quantity y plus 1 all divided by the quantity y plus 1 times the quantity y minus 1 times 1 plus the quantity y plus 1 times the quantity y minus 1 times the quantity 1 divided by the quantity y minus 1. Simplify by removing the common factor, y plus 1, in the denominator and the common factor, y minus 1, in the denominator. Simplify. The result is the product of the quantity y minus 1 times y all divided by the product of the quantity y plus 1 and the quantity y minus 1 plus the quantity y plus 1. Simplify the denominator and leave the numerator factored. The result is y times the quantity y minus 1 all divided by the quantity y squared minus 1 plus y plus 1, which simplifies to y times the quantity y minus 1 all divided by the quantity y squared plus y. Factor the denominator. The result is y times the quantity y minus 1 all divided by y times the quantity y plus 1. Remove the common factor, y, in the numerator and denominator. Then simplify. The result is the quantity y minus 1 divided by the quantity y plus 1.\" data-label=\"\"><tbody><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167834099610\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_03_008c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Find the LCD of all fractions in the complex rational expression.<\/td><td data-valign=\"top\" data-align=\"center\"><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">The LCD is \\(\\left(y+1\\right)\\left(y-1\\right).\\)<\/td><td data-valign=\"top\" data-align=\"center\"><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Multiply the numerator and denominator by the LCD.<\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167834408523\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_03_008a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Distribute in the denominator and simplify.<\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167834185818\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_03_008b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Simplify.<\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167834397261\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_03_008d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Simplify the denominator and leave the<div data-type=\"newline\"><br><\/div>numerator factored.<\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167834547111\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_03_008e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167831949102\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_03_008f_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Factor the denominator and remove factors<div data-type=\"newline\"><br><\/div>common with the numerator.<\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167831874793\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_03_008g_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Simplify.<\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167831239675\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_03_008h_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><\/tbody><\/table><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167835254637\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167831954222\"><div data-type=\"problem\" id=\"fs-id1167830868588\"><p id=\"fs-id1167826804019\">Simplify the complex rational expression by using the LCD: \\(\\frac{\\frac{x}{x+3}}{1+\\frac{1}{x+3}}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835378002\"><p id=\"fs-id1167831871449\">\\(\\frac{x}{x+4}\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167834130402\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167830701079\"><div data-type=\"problem\" id=\"fs-id1167834194517\"><p id=\"fs-id1167835352898\">Simplify the complex rational expression by using the LCD: \\(\\frac{1+\\frac{1}{x-1}}{\\frac{3}{x+1}}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834196182\"><p id=\"fs-id1167834533150\">\\(\\frac{x\\left(x+1\\right)}{3\\left(x-1\\right)}\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167832056804\" class=\"media-2\"><p id=\"fs-id1167832226318\">Access this online resource for additional instruction and practice with complex fractions.<\/p><ul id=\"fs-id1171791232430\" data-display=\"block\"><li><a href=\"https:\/\/openstax.org\/l\/37CompFrac\">Complex Fractions<\/a><\/li><\/ul><\/div><\/div><div class=\"textbox\" data-depth=\"1\" id=\"fs-id1167835337547\"><h3 data-type=\"title\">Key Concepts<\/h3><ul id=\"fs-id1167832060089\" data-bullet-style=\"bullet\"><li><strong data-effect=\"bold\">How to simplify a complex rational expression by writing it as division.<\/strong><ol id=\"fs-id1167835415230\" type=\"1\" class=\"stepwise\"><li>Simplify the numerator and denominator.<\/li><li>Rewrite the complex rational expression as a division problem.<\/li><li>Divide the expressions.<\/li><\/ol><\/li><li><strong data-effect=\"bold\">How to simplify a complex rational expression by using the LCD.<\/strong><ol id=\"fs-id1167832055842\" type=\"1\" class=\"stepwise\"><li>Find the LCD of all fractions in the complex rational expression.<\/li><li>Multiply the numerator and denominator by the LCD.<\/li><li>Simplify the expression.<\/li><\/ol><\/li><\/ul><\/div><div class=\"textbox\" data-depth=\"1\"><div class=\"practice-perfect\" data-depth=\"2\" id=\"fs-id1167831045942\"><h4 data-type=\"title\">Practice Makes Perfect<\/h4><p id=\"fs-id1167827987888\"><strong data-effect=\"bold\">Simplify a Complex Rational Expression by Writing it as Division<\/strong><\/p><p id=\"fs-id1167831919787\">In the following exercises, simplify each complex rational expression by writing it as division.<\/p><div data-type=\"exercise\" id=\"fs-id1167835378752\"><div data-type=\"problem\" id=\"fs-id1167826808701\"><p id=\"fs-id1167826808703\">\\(\\frac{\\frac{2a}{a+4}}{\\frac{4{a}^{2}}{{a}^{2}-16}}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167826799193\"><p id=\"fs-id1167826799195\">\\(\\frac{a-4}{2a}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834430061\"><div data-type=\"problem\" id=\"fs-id1167834430063\"><p id=\"fs-id1167835355318\">\\(\\frac{\\frac{3b}{b-5}}{\\frac{{b}^{2}}{{b}^{2}-25}}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834536678\"><div data-type=\"problem\" id=\"fs-id1167832116081\"><p id=\"fs-id1167834130111\">\\(\\frac{\\frac{5}{{c}^{2}+5c-14}}{\\frac{10}{c+7}}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167828421158\"><p id=\"fs-id1167831896612\">\\(\\frac{1}{2\\left(c-2\\right)}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835320086\"><div data-type=\"problem\" id=\"fs-id1167835325218\"><p id=\"fs-id1167835325220\">\\(\\frac{\\frac{8}{{d}^{2}+9d+18}}{\\frac{12}{d+6}}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834340141\"><div data-type=\"problem\" id=\"fs-id1167834340143\"><p id=\"fs-id1167834397422\">\\(\\frac{\\frac{1}{2}+\\frac{5}{6}}{\\frac{2}{3}+\\frac{7}{9}}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167827958452\"><p id=\"fs-id1167827958454\">\\(\\frac{12}{13}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834156636\"><div data-type=\"problem\" id=\"fs-id1167834156638\"><p id=\"fs-id1167835419844\">\\(\\frac{\\frac{1}{2}+\\frac{3}{4}}{\\frac{3}{5}+\\frac{7}{10}}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167831826361\"><div data-type=\"problem\" id=\"fs-id1167831826364\"><p id=\"fs-id1167835284035\">\\(\\frac{\\frac{2}{3}-\\frac{1}{9}}{\\frac{3}{4}+\\frac{5}{6}}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835615375\"><p id=\"fs-id1167835615377\">\\(\\frac{20}{57}\\)<\/p><\/div><\/div><div data-type=\"exercise\"><div data-type=\"problem\" id=\"fs-id1167835511370\"><p id=\"fs-id1167835511372\">\\(\\frac{\\frac{1}{2}-\\frac{1}{6}}{\\frac{2}{3}+\\frac{3}{4}}\\)<\/p><\/div><\/div><div data-type=\"exercise\"><div data-type=\"problem\" id=\"fs-id1167826781274\"><p id=\"fs-id1167835310538\">\\(\\frac{\\frac{n}{m}+\\frac{1}{n}}{\\frac{1}{n}-\\frac{n}{m}}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835595865\"><p id=\"fs-id1167835595867\">\\(\\frac{{n}^{2}+m}{m-{n}^{2}}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835370045\"><div data-type=\"problem\" id=\"fs-id1167832043728\"><p id=\"fs-id1167832043730\">\\(\\frac{\\frac{1}{p}+\\frac{p}{q}}{\\frac{q}{p}-\\frac{1}{q}}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834184589\"><div data-type=\"problem\" id=\"fs-id1167835615398\"><p id=\"fs-id1167835615400\">\\(\\frac{\\frac{1}{r}+\\frac{1}{t}}{\\frac{1}{{r}^{2}}-\\frac{1}{{t}^{2}}}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835365577\"><p id=\"fs-id1167835362982\">\\(\\frac{rt}{t-r}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834301178\"><div data-type=\"problem\"><p id=\"fs-id1167834372197\">\\(\\frac{\\frac{2}{v}+\\frac{2}{w}}{\\frac{1}{{v}^{2}}-\\frac{1}{{w}^{2}}}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167831884421\"><div data-type=\"problem\" id=\"fs-id1167835419295\"><p id=\"fs-id1167835419297\">\\(\\frac{x-\\frac{2x}{x+3}}{\\frac{1}{x+3}+\\frac{1}{x-3}}\\)<\/p><\/div><div data-type=\"solution\"><p id=\"fs-id1167835328130\">\\(\\frac{\\left(x+1\\right)\\left(x-3\\right)}{2}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167827988185\"><div data-type=\"problem\" id=\"fs-id1167831821797\"><p id=\"fs-id1167831821799\">\\(\\frac{y-\\frac{2y}{y-4}}{\\frac{2}{y-4}+\\frac{2}{y+4}}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835381633\"><div data-type=\"problem\" id=\"fs-id1167835381635\"><p id=\"fs-id1167826983786\">\\(\\frac{2-\\frac{2}{a+3}}{\\frac{1}{a+3}+\\frac{a}{2}}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835191141\"><p id=\"fs-id1167835191143\">\\(\\frac{4}{a+1}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167830693679\"><div data-type=\"problem\" id=\"fs-id1167835370000\"><p id=\"fs-id1167835370002\">\\(\\frac{4+\\frac{4}{b-5}}{\\frac{1}{b-5}+\\frac{b}{4}}\\)<\/p><\/div><\/div><p id=\"fs-id1167834191287\"><strong data-effect=\"bold\">Simplify a Complex Rational Expression by Using the LCD<\/strong><\/p><p id=\"fs-id1167835180600\">In the following exercises, simplify each complex rational expression by using the LCD.<\/p><div data-type=\"exercise\" id=\"fs-id1167835596470\"><div data-type=\"problem\" id=\"fs-id1167835596472\"><p id=\"fs-id1167835363534\">\\(\\frac{\\frac{1}{3}+\\frac{1}{8}}{\\frac{1}{4}+\\frac{1}{12}}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834309892\"><p id=\"fs-id1167834309894\">\\(\\frac{11}{8}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834472639\"><div data-type=\"problem\" id=\"fs-id1167826880090\"><p id=\"fs-id1167826880092\">\\(\\frac{\\frac{1}{4}+\\frac{1}{9}}{\\frac{1}{6}+\\frac{1}{12}}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167831920751\"><div data-type=\"problem\" id=\"fs-id1167831239067\"><p id=\"fs-id1167831239069\">\\(\\frac{\\frac{5}{6}+\\frac{2}{9}}{\\frac{7}{18}-\\frac{1}{3}}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835280282\"><p id=\"fs-id1167835217864\">\\(19\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167831040329\"><div data-type=\"problem\" id=\"fs-id1167831040331\"><p id=\"fs-id1167834556376\">\\(\\frac{\\frac{1}{6}+\\frac{4}{15}}{\\frac{3}{5}-\\frac{1}{2}}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835190900\"><div data-type=\"problem\" id=\"fs-id1167835190903\"><p id=\"fs-id1167835370378\">\\(\\frac{\\frac{c}{d}+\\frac{1}{d}}{\\frac{1}{d}-\\frac{d}{c}}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834064044\"><p id=\"fs-id1167834064046\">\\(\\frac{{c}^{2}+c}{c-{d}^{2}}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834539410\"><div data-type=\"problem\" id=\"fs-id1167834539412\"><p id=\"fs-id1167826804612\">\\(\\frac{\\frac{1}{m}+\\frac{m}{n}}{\\frac{n}{m}-\\frac{1}{n}}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835355981\"><div data-type=\"problem\" id=\"fs-id1167834238843\"><p id=\"fs-id1167834238845\">\\(\\frac{\\frac{1}{p}+\\frac{1}{q}}{\\frac{1}{{p}^{2}}-\\frac{1}{{q}^{2}}}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835233678\"><p id=\"fs-id1167834525271\">\\(\\frac{pq}{q-p}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167826804725\"><div data-type=\"problem\" id=\"fs-id1167835420186\"><p id=\"fs-id1167835420188\">\\(\\frac{\\frac{2}{r}+\\frac{2}{t}}{\\frac{1}{{r}^{2}}-\\frac{1}{{t}^{2}}}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167832053579\"><div data-type=\"problem\" id=\"fs-id1167834300263\"><p id=\"fs-id1167834300265\">\\(\\frac{\\frac{2}{x+5}}{\\frac{3}{x-5}+\\frac{1}{{x}^{2}-25}}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834432252\"><p>\\(\\frac{2x-10}{3x+16}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835304946\"><div data-type=\"problem\" id=\"fs-id1167835304948\"><p id=\"fs-id1167835342970\">\\(\\frac{\\frac{5}{y-4}}{\\frac{3}{y+4}+\\frac{2}{{y}^{2}-16}}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167831239449\"><div data-type=\"problem\" id=\"fs-id1167834432088\"><p id=\"fs-id1167834432090\">\\(\\frac{\\frac{5}{{z}^{2}-64}+\\frac{3}{z+8}}{\\frac{1}{z+8}+\\frac{2}{z-8}}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835338228\"><p id=\"fs-id1167835338230\">\\(\\frac{3z-19}{3z+8}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835328334\"><div data-type=\"problem\" id=\"fs-id1167835328337\"><p id=\"fs-id1167831887230\">\\(\\frac{\\frac{3}{s+6}+\\frac{5}{s-6}}{\\frac{1}{{s}^{2}-36}+\\frac{4}{s+6}}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834394776\"><div data-type=\"problem\" id=\"fs-id1167830836798\"><p id=\"fs-id1167830836800\">\\(\\frac{\\frac{4}{{a}^{2}-2a-15}}{\\frac{1}{a-5}+\\frac{2}{a+3}}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167832043716\"><p id=\"fs-id1167835569735\">\\(\\frac{4}{3a-7}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835320616\"><div data-type=\"problem\" id=\"fs-id1167834279886\"><p id=\"fs-id1167834279888\">\\(\\frac{\\frac{5}{{b}^{2}-6b-27}}{\\frac{3}{b-9}+\\frac{1}{b+3}}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835416735\"><div data-type=\"problem\" id=\"fs-id1167835416737\"><p id=\"fs-id1167831025160\">\\(\\frac{\\frac{5}{c+2}-\\frac{3}{c+7}}{\\frac{5c}{{c}^{2}+9c+14}}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834219571\"><p id=\"fs-id1167834219574\">\\(\\frac{2c+29}{5c}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167831837544\"><div data-type=\"problem\"><p id=\"fs-id1167830838233\">\\(\\frac{\\frac{6}{d-4}-\\frac{2}{d+7}}{\\frac{2d}{{d}^{2}+3d-28}}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834194525\"><div data-type=\"problem\" id=\"fs-id1167835363301\"><p id=\"fs-id1167835363303\">\\(\\frac{2+\\frac{1}{p-3}}{\\frac{5}{p-3}}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835510188\"><p id=\"fs-id1167831872142\">\\(\\frac{2p-5}{5}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835514456\"><div data-type=\"problem\" id=\"fs-id1167826857291\"><p id=\"fs-id1167826857294\">\\(\\frac{\\frac{n}{n-2}}{3+\\frac{5}{n-2}}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834279936\"><div data-type=\"problem\" id=\"fs-id1167834395888\"><p id=\"fs-id1167834395890\">\\(\\frac{\\frac{m}{m+5}}{4+\\frac{1}{m-5}}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835305735\"><p id=\"fs-id1167832053222\">\\(\\frac{m\\left(m-5\\right)}{\\left(4m-19\\right)\\left(m+5\\right)}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835381804\"><div data-type=\"problem\" id=\"fs-id1167831882524\"><p id=\"fs-id1167831882526\">\\(\\frac{7+\\frac{2}{q-2}}{\\frac{1}{q+2}}\\)<\/p><\/div><\/div><p id=\"fs-id1167831832518\">In the following exercises, simplify each complex rational expression using either method.<\/p><div data-type=\"exercise\" id=\"fs-id1167831832522\"><div data-type=\"problem\" id=\"fs-id1167835360939\"><p id=\"fs-id1167835360942\">\\(\\frac{\\frac{3}{4}-\\frac{2}{7}}{\\frac{1}{2}+\\frac{5}{14}}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835623266\"><p id=\"fs-id1167835623269\">\\(\\frac{13}{24}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834252374\"><div data-type=\"problem\" id=\"fs-id1167834131419\"><p id=\"fs-id1167834131421\">\\(\\frac{\\frac{v}{w}+\\frac{1}{v}}{\\frac{1}{v}-\\frac{v}{w}}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835356825\"><div data-type=\"problem\" id=\"fs-id1167835356827\"><p id=\"fs-id1167826798856\">\\(\\frac{\\frac{2}{a+4}}{\\frac{1}{{a}^{2}-16}}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835240317\"><p id=\"fs-id1167835479556\">\\(2\\left(a-4\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835356359\"><div data-type=\"problem\" id=\"fs-id1167835356361\"><p id=\"fs-id1167830700735\">\\(\\frac{\\frac{3}{{b}^{2}-3b-40}}{\\frac{5}{b+5}-\\frac{2}{b-8}}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167831115961\"><div data-type=\"problem\" id=\"fs-id1167826869962\"><p>\\(\\frac{\\frac{3}{m}+\\frac{3}{n}}{\\frac{1}{{m}^{2}}-\\frac{1}{{n}^{2}}}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835389924\"><p id=\"fs-id1167835366569\">\\(\\frac{3mn}{n-m}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167831957810\"><div data-type=\"problem\" id=\"fs-id1167831957812\"><p id=\"fs-id1167831822064\">\\(\\frac{\\frac{2}{r-9}}{\\frac{1}{r+9}+\\frac{3}{{r}^{2}-81}}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835327394\"><div data-type=\"problem\" id=\"fs-id1167835321549\"><p id=\"fs-id1167835321551\">\\(\\frac{x-\\frac{3x}{x+2}}{\\frac{3}{x+2}+\\frac{3}{x-2}}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834533453\"><p id=\"fs-id1167835318576\">\\(\\frac{\\left(x-1\\right)\\left(x-2\\right)}{6}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167826781484\"><div data-type=\"problem\" id=\"fs-id1167835363110\"><p id=\"fs-id1167835363113\">\\(\\frac{\\frac{y}{y+3}}{2+\\frac{1}{y-3}}\\)<\/p><\/div><\/div><\/div><div class=\"writing\" data-depth=\"2\" id=\"fs-id1167834473754\"><h4 data-type=\"title\">Writing Exercises<\/h4><div data-type=\"exercise\" id=\"fs-id1167830702565\"><div data-type=\"problem\" id=\"fs-id1167834556395\"><p id=\"fs-id1167834556397\">In this section, you learned to simplify the complex fraction \\(\\frac{\\frac{3}{x+2}}{\\frac{x}{{x}^{2}-4}}\\) two ways: rewriting it as a division problem or multiplying the numerator and denominator by the LCD. Which method do you prefer? Why?<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834061445\"><p id=\"fs-id1167834061447\">Answers will vary.<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167831954191\"><div data-type=\"problem\" id=\"fs-id1167832076124\"><p id=\"fs-id1167832076126\">Efraim wants to start simplifying the complex fraction \\(\\frac{\\frac{1}{a}+\\frac{1}{b}}{\\frac{1}{a}-\\frac{1}{b}}\\) by cancelling the variables from the numerator and denominator, \\(\\frac{\\frac{1}{\\overline{)a}}+\\frac{1}{\\overline{)b}}}{\\frac{1}{\\overline{)a}}-\\frac{1}{\\overline{)b}}}.\\) Explain what is wrong with Efraim\u2019s plan.<\/p><\/div><\/div><\/div><div class=\"bc-section section\" data-depth=\"2\" id=\"fs-id1167830757635\"><h4 data-type=\"title\">Self Check<\/h4><p id=\"fs-id1167831922168\"><span class=\"token\">\u24d0<\/span> After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.<\/p><span data-type=\"media\" id=\"fs-id1167835180627\" data-alt=\"This table has four columns and three rows. The first row is a header and it labels each column, \u201cI can\u2026\u201d, \u201cConfidently,\u201d \u201cWith some help,\u201d and \u201cNo-I don\u2019t get it!\u201d In row 2, the I can was simplify a complex rational expression by writing it as division. In row 3, the I can was simplify a complex rational expression by using the least common denominator.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_03_201_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This table has four columns and three rows. The first row is a header and it labels each column, \u201cI can\u2026\u201d, \u201cConfidently,\u201d \u201cWith some help,\u201d and \u201cNo-I don\u2019t get it!\u201d In row 2, the I can was simplify a complex rational expression by writing it as division. In row 3, the I can was simplify a complex rational expression by using the least common denominator.\"><\/span><p><span class=\"token\">\u24d1<\/span> After looking at the checklist, do you think you are well-prepared for the next section? Why or why not?<\/p><\/div><\/div><div data-type=\"glossary\" class=\"textbox shaded\"><h3 data-type=\"glossary-title\">Glossary<\/h3><dl id=\"fs-id1167834308149\"><dt>complex rational expression<\/dt><dd id=\"fs-id1167834099322\">A complex rational expression is a rational expression in which the numerator and\/or denominator contains a rational expression.<\/dd><\/dl><\/div>\n","rendered":"<div class=\"textbox textbox--learning-objectives\">\n<h3 itemprop=\"educationalUse\">Learning Objectives<\/h3>\n<p>By the end of this section, you will be able to: <\/p>\n<ul>\n<li>Simplify a complex rational expression by writing it as division<\/li>\n<li>Simplify a complex rational expression by using the LCD<\/li>\n<\/ul>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167835308956\" class=\"be-prepared\">\n<p>Before you get started, take this readiness quiz.<\/p>\n<ol id=\"fs-id1167835416944\" type=\"1\">\n<li>Simplify: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-da436afdf10f7694cae08dbaec6ae9b0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#53;&#125;&#125;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#57;&#125;&#123;&#49;&#48;&#125;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"36\" width=\"22\" style=\"vertical-align: -14px;\" \/>\n<div data-type=\"newline\"><\/div>\n<p> If you missed this problem, review <a href=\"\/contents\/425620d9-51dd-45e5-8a21-953998a4a77f#fs-id1167829590640\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li>\n<li>Simplify: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-bd87ed4eab3bed56cd1915f89b5f7b36_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#51;&#125;&#125;&#123;&#123;&#52;&#125;&#94;&#123;&#50;&#125;&#43;&#52;&middot;&#53;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"31\" width=\"44\" style=\"vertical-align: -9px;\" \/>\n<div data-type=\"newline\"><\/div>\n<p> If you missed this problem, review <a href=\"\/contents\/425620d9-51dd-45e5-8a21-953998a4a77f#fs-id1167829627815\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li>\n<li>Solve: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4898c56a8b34f8ec529cb9ff813e6c65_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#120;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#52;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#56;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"87\" style=\"vertical-align: -6px;\" \/>\n<div data-type=\"newline\"><\/div>\n<p> If you missed this problem, review <a href=\"\/contents\/9f100e8f-6d15-4cae-bc22-c306e9d7d55c#fs-id1167833239741\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li>\n<\/ol>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\">\n<h3 data-type=\"title\">Simplify a Complex Rational Expression by Writing it as Division<\/h3>\n<p id=\"fs-id1167834227963\">Complex fractions are fractions in which the numerator or denominator contains a fraction. We previously simplified complex fractions like these:<\/p>\n<div data-type=\"equation\" id=\"fs-id1167835419382\" class=\"unnumbered\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-72dd0b47f84f09d64132e0306fff723a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#99;&#99;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#52;&#125;&#125;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#56;&#125;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#120;&#125;&#123;&#50;&#125;&#125;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#120;&#121;&#125;&#123;&#54;&#125;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"35\" width=\"113\" style=\"vertical-align: -13px;\" \/><\/div>\n<p id=\"fs-id1167835301808\">In this section, we will simplify complex rational expressions, which are rational expressions with rational expressions in the numerator or denominator.<\/p>\n<div data-type=\"note\" id=\"fs-id1167834489858\">\n<div data-type=\"title\">Complex Rational Expression<\/div>\n<p id=\"fs-id1167835320604\">A <span data-type=\"term\">complex rational expression<\/span> is a rational expression in which the numerator and\/or the denominator contains a rational expression.<\/p>\n<\/div>\n<p>Here are a few complex rational expressions:<\/p>\n<div data-type=\"equation\" id=\"fs-id1167831884939\" class=\"unnumbered\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-46457350b0887192d3eb2a6c0acd96d2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#125;&#123;&#121;&#45;&#51;&#125;&#125;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#56;&#125;&#123;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#45;&#57;&#125;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#51;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#120;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#121;&#125;&#125;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#120;&#125;&#123;&#121;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#121;&#125;&#123;&#120;&#125;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#51;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#120;&#43;&#54;&#125;&#125;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#125;&#123;&#120;&#45;&#54;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#125;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#51;&#54;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"41\" width=\"255\" style=\"vertical-align: -17px;\" \/><\/div>\n<p id=\"fs-id1167835384743\">Remember, we always exclude values that would make any denominator zero.<\/p>\n<p id=\"fs-id1167834179741\">We will use two methods to simplify complex rational expressions.<\/p>\n<p id=\"fs-id1167832056143\">We have already seen this complex rational expression earlier in this chapter.<\/p>\n<div data-type=\"equation\" class=\"unnumbered\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1a5a0e1f3d95a3445d13820672152d9c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#54;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#55;&#120;&#43;&#50;&#125;&#123;&#52;&#120;&#45;&#56;&#125;&#125;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#56;&#120;&#43;&#51;&#125;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#53;&#120;&#43;&#54;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"47\" width=\"61\" style=\"vertical-align: -20px;\" \/><\/div>\n<p id=\"fs-id1167831825117\">We noted that fraction bars tell us to divide, so rewrote it as the division problem:<\/p>\n<div data-type=\"equation\" id=\"fs-id1167835370704\" class=\"unnumbered\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e9d4dde1669f0e4273bf44dcc8c45a8f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#54;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#55;&#120;&#43;&#50;&#125;&#123;&#52;&#120;&#45;&#56;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&divide;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#56;&#120;&#43;&#51;&#125;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#53;&#120;&#43;&#54;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"33\" width=\"189\" style=\"vertical-align: -12px;\" \/><\/div>\n<p>Then, we multiplied the first rational expression by the reciprocal of the second, just like we do when we divide two fractions.<\/p>\n<p id=\"fs-id1167835319282\">This is one method to simplify complex rational expressions. We make sure the complex rational expression is of the form where one fraction is over one fraction. We then write it as if we were dividing two fractions.<\/p>\n<div data-type=\"example\" id=\"fs-id1167826978829\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167831836544\">\n<div data-type=\"problem\" id=\"fs-id1167834433716\">\n<p id=\"fs-id1167834346605\">Simplify the complex rational expression by writing it as division: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9d445ecaa502c37b463dabbbaf89936e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#54;&#125;&#123;&#120;&#45;&#52;&#125;&#125;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#54;&#125;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"39\" width=\"45\" style=\"vertical-align: -16px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\">\n<p id=\"fs-id1167831910119\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ff2f4858cfe6b13b84f9b156b8464793_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#125;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#52;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#54;&#125;&#123;&#120;&#45;&#52;&#125;&#125;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#54;&#125;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#82;&#101;&#119;&#114;&#105;&#116;&#101;&#32;&#116;&#104;&#101;&#32;&#99;&#111;&#109;&#112;&#108;&#101;&#120;&#32;&#102;&#114;&#97;&#99;&#116;&#105;&#111;&#110;&#32;&#97;&#115;&#32;&#100;&#105;&#118;&#105;&#115;&#105;&#111;&#110;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#52;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#54;&#125;&#123;&#120;&#45;&#52;&#125;&divide;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#54;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#82;&#101;&#119;&#114;&#105;&#116;&#101;&#32;&#97;&#115;&#32;&#116;&#104;&#101;&#32;&#112;&#114;&#111;&#100;&#117;&#99;&#116;&#32;&#111;&#102;&#32;&#102;&#105;&#114;&#115;&#116;&#32;&#116;&#105;&#109;&#101;&#115;&#32;&#116;&#104;&#101;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#114;&#101;&#99;&#105;&#112;&#114;&#111;&#99;&#97;&#108;&#32;&#111;&#102;&#32;&#116;&#104;&#101;&#32;&#115;&#101;&#99;&#111;&#110;&#100;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#52;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#54;&#125;&#123;&#120;&#45;&#52;&#125;&middot;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#54;&#125;&#123;&#51;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#70;&#97;&#99;&#116;&#111;&#114;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#52;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&middot;&#50;&#125;&#123;&#120;&#45;&#52;&#125;&middot;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#123;&#51;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#77;&#117;&#108;&#116;&#105;&#112;&#108;&#121;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#52;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&middot;&#50;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#123;&#51;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#82;&#101;&#109;&#111;&#118;&#101;&#32;&#99;&#111;&#109;&#109;&#111;&#110;&#32;&#102;&#97;&#99;&#116;&#111;&#114;&#115;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#52;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#111;&#118;&#101;&#114;&#108;&#105;&#110;&#101;&#123;&#41;&#51;&#125;&middot;&#50;&#92;&#111;&#118;&#101;&#114;&#108;&#105;&#110;&#101;&#123;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#123;&#92;&#111;&#118;&#101;&#114;&#108;&#105;&#110;&#101;&#123;&#41;&#51;&#125;&#92;&#111;&#118;&#101;&#114;&#108;&#105;&#110;&#101;&#123;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#83;&#105;&#109;&#112;&#108;&#105;&#102;&#121;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#52;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#50;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"479\" width=\"554\" style=\"vertical-align: -234px;\" \/><\/p>\n<p id=\"fs-id1167832057148\">Are there any value(s) of <em data-effect=\"italics\">x<\/em> that should not be allowed? The original complex rational expression had denominators of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5c34f09f62881790ee7aa7b6a26d2fd9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#45;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"41\" style=\"vertical-align: -1px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-013b79156dd2fd6b22037c7b63c9b39b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#54;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"61\" style=\"vertical-align: -1px;\" \/> This expression would be undefined if <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2145acc2878ed61214887e120f2485b7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"43\" style=\"vertical-align: -1px;\" \/> or <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-de80293e0b5a1d0b98c28c8b715d136d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#45;&#52;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"61\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167832058388\" class=\"try\">\n<div data-type=\"exercise\">\n<div data-type=\"problem\" id=\"fs-id1167834408509\">\n<p id=\"fs-id1167835239962\">Simplify the complex rational expression by writing it as division: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-114a32e8d16ff00f1b08ca01bc28e824_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#125;&#125;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#120;&#43;&#49;&#125;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"40\" width=\"39\" style=\"vertical-align: -15px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\">\n<p id=\"fs-id1167832053233\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6fe75415ad8a6a1d967a3763dffb8354_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#51;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"43\" style=\"vertical-align: -9px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167835609958\">\n<div data-type=\"problem\" id=\"fs-id1167821178912\">\n<p id=\"fs-id1167835381150\">Simplify the complex rational expression by writing it as division: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-fea86b60b16497f87c46366be0348937_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#55;&#120;&#43;&#49;&#50;&#125;&#125;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#120;&#45;&#52;&#125;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"66\" style=\"vertical-align: -13px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\">\n<p id=\"fs-id1167835343984\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-17caaffc2ed5ff2c3a9384c9559642e2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"43\" style=\"vertical-align: -9px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1167835353494\">Fraction bars act as grouping symbols. So to follow the Order of Operations, we simplify the numerator and denominator as much as possible before we can do the division.<\/p>\n<div data-type=\"example\" id=\"fs-id1167835236104\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167834474319\">\n<div data-type=\"problem\" id=\"fs-id1167831894391\">\n<p>Simplify the complex rational expression by writing it as division: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3538fca776bdcf606171cbbf26374d3e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#51;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#54;&#125;&#125;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#51;&#125;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"35\" width=\"36\" style=\"vertical-align: -13px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835305870\">\n<table id=\"fs-id1167828435628\" class=\"unnumbered unstyled\" summary=\"The quantity one-third plus one-sixth divided by the quantity one-half minus one-third. Rewrite the numerator and denominator by finding the lowest common denominator. The lowest common denominator of one-third, one-sixth, one-half, and one-third is 6. In the numerator of the expression, multiply the numerator and denominator of one-third by 2. In the denominator of the expression, multiply the numerator and denominator of one-half by 3, and the numerator and denominator of one-third by 2. Simplify the numerator and the denominator. The result is the quantity two-sixths plus one-sixth all divided by the quantity three-sixths minus two-sixths. Rewrite the complex rational expression as a division problem, three-sixth divided by one-sixth. Multiply three-sixth by the reciprocal of one-sixth, which is 6 over 1. The result is three-sixth times six ones. Simplify the expression. The result is 3.\" data-label=\"\">\n<tbody>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"CNX_IntAlg_Figure_07_03_001a_img_new.jpg#fixme#fixme\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Simplify the numerator and denominator.<\/p>\n<div data-type=\"newline\"><\/div>\n<p>Find the LCD and add the fractions in the numerator.<\/p>\n<div data-type=\"newline\"><\/div>\n<p>Find the LCD and subtract the fractions in the<\/p>\n<div data-type=\"newline\"><\/div>\n<p>denominator.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167834066374\" data-alt=\".\"><img decoding=\"async\" src=\"CNX_IntAlg_Figure_07_03_001b_img_new.jpg#fixme#fixme\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Simplify the numerator and denominator.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167832138932\" data-alt=\".\"><img decoding=\"async\" src=\"CNX_IntAlg_Figure_07_03_001c_img_new.jpg#fixme#fixme\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Rewrite the complex rational expression as a division<\/p>\n<div data-type=\"newline\"><\/div>\n<p>problem.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"CNX_IntAlg_Figure_07_03_001e_img_new.jpg#fixme#fixme\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Multiply the first by the reciprocal of the second.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167835307222\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_03_001f_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Simplify.<\/td>\n<td data-valign=\"top\" data-align=\"center\">3<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167835254948\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167835362198\">\n<div data-type=\"problem\">\n<p>Simplify the complex rational expression by writing it as division: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-aa4f595360c929aacb19a6c25a6206f4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#51;&#125;&#125;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#54;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#49;&#50;&#125;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"36\" width=\"42\" style=\"vertical-align: -14px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835339384\">\n<p id=\"fs-id1167834064255\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-524066fab9a79200ec438c667f98d937_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#52;&#125;&#123;&#49;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"14\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167835320372\" class=\"try\">\n<div data-type=\"exercise\">\n<div data-type=\"problem\" id=\"fs-id1167835479676\">\n<p>Simplify the complex rational expression by writing it as division: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c8662894fccf81d210d8eeee66c2b7dc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#52;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#51;&#125;&#125;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#56;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#54;&#125;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"35\" width=\"36\" style=\"vertical-align: -13px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835202220\">\n<p id=\"fs-id1167835254714\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-bf57e99c6ee6743160cf37cde5c4f55f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#48;&#125;&#123;&#50;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"14\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1165926931185\">We follow the same procedure when the complex rational expression contains variables.<\/p>\n<div data-type=\"example\" id=\"fs-id1167830705566\" class=\"textbox textbox--examples\">\n<div data-type=\"title\">How to Simplify a Complex Rational Expression using Division<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834094691\">\n<div data-type=\"problem\" id=\"fs-id1167835283646\">\n<p>Simplify the complex rational expression by writing it as division: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9c7059d510be5fd36687fe36f6dd5887_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#120;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#121;&#125;&#125;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#120;&#125;&#123;&#121;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#121;&#125;&#123;&#120;&#125;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"38\" style=\"vertical-align: -14px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167831076629\"><span data-type=\"media\" data-alt=\"Step 1 is to simplify the sum in the numerator and the difference in the denominator of complex rational expression, the quantity 1 divided by x plus 1 divided by y all divided by the quantity x divided by y minus y divided by x. The common denominator of the fractions in the complex rational expression is x y. Multiply the numerator and denominator of 1 divided by x by y over y. Multiply the numerator and denominator of 1 divided by y by x over x. Multiply the numerator and denominator of x divided by y by x over x. Multiply the numerator and denominator of y over x by y over y. The result is the quantity y divided by x y plus x divided by x y all divided by the quantity x squared divided by x y minus y squared divided by x y. Add the fractions in the numerator and subtract the fractions in the denominator. The result is the sum of y and x divided by x y all divided by the difference between x squared and y squared divided by x y. We now have just one rational expression in the numerator and one in the denominator.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_03_002a_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 1 is to simplify the sum in the numerator and the difference in the denominator of complex rational expression, the quantity 1 divided by x plus 1 divided by y all divided by the quantity x divided by y minus y divided by x. The common denominator of the fractions in the complex rational expression is x y. Multiply the numerator and denominator of 1 divided by x by y over y. Multiply the numerator and denominator of 1 divided by y by x over x. Multiply the numerator and denominator of x divided by y by x over x. Multiply the numerator and denominator of y over x by y over y. The result is the quantity y divided by x y plus x divided by x y all divided by the quantity x squared divided by x y minus y squared divided by x y. Add the fractions in the numerator and subtract the fractions in the denominator. The result is the sum of y and x divided by x y all divided by the difference between x squared and y squared divided by x y. We now have just one rational expression in the numerator and one in the denominator.\" \/><\/span><span data-type=\"media\" id=\"fs-id1167835327260\" data-alt=\"Step 2 is to rewrite the complex rational expression as a division problem. Write the numerator divided by the denominator. The result is the quantity of the sum y and x divided by x y all divided by the quantity of the difference between x squared and y squared divided by x y.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_03_002b_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 2 is to rewrite the complex rational expression as a division problem. Write the numerator divided by the denominator. The result is the quantity of the sum y and x divided by x y all divided by the quantity of the difference between x squared and y squared divided by x y.\" \/><\/span><span data-type=\"media\" id=\"fs-id1167834066451\" data-alt=\"Step 3 is to divided the expressions. Multiply the first expression by the reciprocal of the second expression. The result is the quantity of the sum y and x divided by x y times the quantity x y divided by the difference between x squared and y squared. Factor any expressions if possible. The result is the product of x y and the sum of y and x all divided by the product of x y, the difference between x and y, and the sum of x and y. Remove the common factors, x y and the sum of x and y. Simplify. The result is 1 divided by the quantity x minus y.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_03_002c_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 3 is to divided the expressions. Multiply the first expression by the reciprocal of the second expression. The result is the quantity of the sum y and x divided by x y times the quantity x y divided by the difference between x squared and y squared. Factor any expressions if possible. The result is the product of x y and the sum of y and x all divided by the product of x y, the difference between x and y, and the sum of x and y. Remove the common factors, x y and the sum of x and y. Simplify. The result is 1 divided by the quantity x minus y.\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167835303159\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167834423424\">\n<div data-type=\"problem\" id=\"fs-id1167834301239\">\n<p id=\"fs-id1167835253848\">Simplify the complex rational expression by writing it as division: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3b2559451d760dd9654d60063d6f7b10_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#120;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#121;&#125;&#125;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#120;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#121;&#125;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"39\" width=\"38\" style=\"vertical-align: -15px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835356218\">\n<p id=\"fs-id1167835371395\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-68719056ca7e2746c857dbef6d62e3a8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#121;&#43;&#120;&#125;&#123;&#121;&#45;&#120;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"26\" style=\"vertical-align: -9px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167835309227\">\n<div data-type=\"problem\" id=\"fs-id1167834161618\">\n<p id=\"fs-id1167832092221\">Simplify the complex rational expression by writing it as division: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-44d7a4272ce6639bfc9f531155ef4edf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#97;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#98;&#125;&#125;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#123;&#98;&#125;&#94;&#123;&#50;&#125;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"37\" width=\"44\" style=\"vertical-align: -15px;\" \/>.<\/p>\n<\/div>\n<div data-type=\"solution\">\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7f9acbbdbea591736718748b89f4aa9c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#97;&#98;&#125;&#123;&#98;&#45;&#97;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"24\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1167835233515\">We summarize the steps here.<\/p>\n<div data-type=\"note\" id=\"fs-id1167835319109\" class=\"howto\">\n<div data-type=\"title\">Simplify a complex rational expression by writing it as division.<\/div>\n<ol id=\"fs-id1167835280276\" type=\"1\" class=\"stepwise\">\n<li>Simplify the numerator and denominator.<\/li>\n<li>Rewrite the complex rational expression as a division problem.<\/li>\n<li>Divide the expressions.<\/li>\n<\/ol>\n<\/div>\n<div data-type=\"example\" id=\"fs-id1167835337830\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\">\n<div data-type=\"problem\" id=\"fs-id1167835498830\">\n<p>Simplify the complex rational expression by writing it as division: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-91ba655d75fa212d42470f0b44c48b89_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#110;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#110;&#125;&#123;&#110;&#43;&#53;&#125;&#125;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#110;&#43;&#53;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#110;&#45;&#53;&#125;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"70\" style=\"vertical-align: -15px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835243968\">\n<table id=\"fs-id1167834308269\" class=\"unnumbered unstyled can-break\" summary=\"Simplify the complex rational expression n minus the quantity 4 n divided by the quantity n plus 5 all divided by the sum of 1 divided by the quantity n plus 5 and 1 divided by the quantity n minus 5. Simplify the numerator and denominator. Find the common denominators for the numerator and denominator. The result is the n times the quantity n plus 5 all divided by 1 times the quantity n plus 5 minus the quantity 4 n divided by the quantity n plus 5 all divided by 1 times the quantity n minus 5 divided by the quantity n plus 5 times the quantity n minus 5 plus the quantity 1 times the quantity n plus 5 divided by the quantity n plus 5 times the quantity n minus 5. Simplify the numerators. The result is the quantity n squared plus 5 n divided by the quantity n plus 5 minus the quantity 4 n divided by the quantity n plus 5 all divided by 1 times the quantity n minus 5 divided by the quantity n plus 5 times the quantity n minus 5 plus 1 times the quantity n plus 5 divided by the quantity n plus 5 times the quantity n minus 5. Subtract the rational expressions in the numerator and add in the denominator. The result is the quantity n squared plus 5 n minus 4 n divided by the quantity n plus 5 all divided by the quantity n minus 5 plus n plus 5 divided by the quantity n plus 5 times the quantity n minus 5. Multiply the first expression by the reciprocal of the second. The result is the quantity n squared plus 5 n minus 4 n divided by the quantity n plus 5 times the quantity n plus 5 times the quantity n minus 5 divided by the quantity n minus 5 plus n plus 5. Simplify. We now have one rational expression over one rational expression. The result is the quantity n squared plus n divided by the quantity n plus 5 all divided by the quantity 2 n divided by the quantity n plus 5 times the quantity n minus 5. Rewrite the expression as fraction division. The result is the quotient of the quantity n squared plus n divided by the quantity n plus 5 and the quantity 2 n divided by the quantity n plus 5 times the quantity n minus 5. Multiply the first fraction times the reciprocal of the second fraction. The result is the quantity n squared plus n divided by the quantity n plus 5 times the quantity of n plus 5 times the quantity n minus 5 all divided by the quantity 2 n. Factor any expressions if possible. The result us n times the quantity n plus 1 times the quantity n plus 5 times the quantity n minus 5 all divided by the product of the quantity n plus 5 and 2 n. Remove the common factors, n and n plus 5, from the numerator and denominator. Simplify. The result is the product of the quantity n plus 1 and the quantity n minus 5 all divided by 2.\" data-label=\"\">\n<tbody>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167834527716\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_03_003a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Simplify the numerator and denominator.<\/p>\n<div data-type=\"newline\"><\/div>\n<p>Find common denominators for the numerator and<\/p>\n<div data-type=\"newline\"><\/div>\n<p>denominator.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167835370172\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_03_003b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Simplify the numerators.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167830699580\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_03_003c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Subtract the rational expressions in the numerator and<\/p>\n<div data-type=\"newline\"><\/div>\n<p>add in the denominator.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167835191878\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_03_003d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Simplify. (We now have one rational expression over<\/p>\n<div data-type=\"newline\"><\/div>\n<p>one rational expression.)<\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167832006043\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_03_003e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Rewrite as fraction division.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167832058793\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_03_003f_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Multiply the first times the reciprocal of the second.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167826779013\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_03_003g_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Factor any expressions if possible.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167831148955\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_03_003h_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Remove common factors.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167834194708\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_03_003i_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Simplify.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167834156730\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_03_003j_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167826977973\" class=\"try\">\n<div data-type=\"exercise\">\n<div data-type=\"problem\">\n<p>Simplify the complex rational expression by writing it as division: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e16bb96a17f6f3d7fb01e7c6f8dc1a97_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#98;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#98;&#125;&#123;&#98;&#43;&#53;&#125;&#125;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#98;&#43;&#53;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#98;&#45;&#53;&#125;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"65\" style=\"vertical-align: -15px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834397941\">\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a075212fba343a26c8892bc86678ce40_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#98;&#92;&#108;&#101;&#102;&#116;&#40;&#98;&#43;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#98;&#45;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#123;&#51;&#98;&#45;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"75\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167835303583\" class=\"try\">\n<div data-type=\"exercise\">\n<div data-type=\"problem\" id=\"fs-id1167835341039\">\n<p id=\"fs-id1167835264626\">Simplify the complex rational expression by writing it as division: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f609b644bc46d9848dc55518a38b6de2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#99;&#43;&#52;&#125;&#125;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#99;&#43;&#52;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#99;&#125;&#123;&#51;&#125;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"51\" style=\"vertical-align: -15px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\">\n<p id=\"fs-id1167834428859\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-79771ebb6848fa0538fb9d5126233814_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#99;&#43;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"24\" style=\"vertical-align: -8px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167835609912\">\n<h3 data-type=\"title\">Simplify a Complex Rational Expression by Using the LCD<\/h3>\n<p id=\"fs-id1167835304510\">We \u201ccleared\u201d the fractions by multiplying by the LCD when we solved equations with fractions. We can use that strategy here to simplify complex rational expressions. We will multiply the numerator and denominator by the LCD of all the rational expressions.<\/p>\n<p id=\"fs-id1167832060555\">Let\u2019s look at the complex rational expression we simplified one way in <a href=\"#fs-id1167835236104\" class=\"autogenerated-content\">(Figure)<\/a>. We will simplify it here by multiplying the numerator and denominator by the LCD. When we multiply by <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-cfe6af780373ea0b6b283f07af7c930c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#76;&#67;&#68;&#125;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#76;&#67;&#68;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"30\" style=\"vertical-align: -6px;\" \/> we are multiplying by 1, so the value stays the same.<\/p>\n<div data-type=\"example\" id=\"fs-id1167826829055\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167831239367\">\n<div data-type=\"problem\" id=\"fs-id1167835303825\">\n<p>Simplify the complex rational expression by using the LCD: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3538fca776bdcf606171cbbf26374d3e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#51;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#54;&#125;&#125;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#51;&#125;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"35\" width=\"36\" style=\"vertical-align: -13px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\">\n<table id=\"fs-id1167835419447\" class=\"unnumbered unstyled\" summary=\"Simplify the sum of one-third and one-sixth all divided by the difference between one-half and one-third. The least common denominator of all the fractions in the entire expression is 6. Clear the fractions by multiplying the numerator and denominator by the least common denominator. The result is 6 times sum of one-third and one-sixth all divided by 6 times the difference between one-half and one-third. Distribute. The result is the quantity 6 times one-third plus 6 times one-sixth all divided by the quantity 6 times one-half minus 6 times one-third. Simplify. The result is the quantity 2 plus 1 divided by the quantity 3 minus 2, which simplifies to 3 divided by 1. 3 divided by 1 is 3.\" data-label=\"\">\n<tbody>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167831104030\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_03_004a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">The LCD of all the fractions in the whole expression is 6.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Clear the fractions by multiplying the numerator and<\/p>\n<div data-type=\"newline\"><\/div>\n<p>denominator by that LCD.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167834555290\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_03_004b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Distribute.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167835518419\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_03_004c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Simplify.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167835307312\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_03_004d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167834556304\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_03_004e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_03_004f_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167835343125\">\n<div data-type=\"problem\" id=\"fs-id1167835370755\">\n<p id=\"fs-id1167835206029\">Simplify the complex rational expression by using the LCD: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-59edfd932ce842188265b32123f07383_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#53;&#125;&#125;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#49;&#48;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#53;&#125;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"36\" width=\"42\" style=\"vertical-align: -14px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\">\n<p id=\"fs-id1167834179817\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2330587fe43276ff3e47ca1fc1bf0a57_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#55;&#125;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"7\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167830757621\">\n<div data-type=\"problem\">\n<p>Simplify the complex rational expression by using the LCD: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-75765148ab8da1925345c4449884801b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#52;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#56;&#125;&#125;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#49;&#54;&#125;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"36\" width=\"42\" style=\"vertical-align: -14px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835415213\">\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a00d32b021df4bf81cba9b2ed120b5a0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#48;&#125;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"14\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>We will use the same example as in <a href=\"#fs-id1167830705566\" class=\"autogenerated-content\">(Figure)<\/a>. Decide which method works better for you.<\/p>\n<div data-type=\"example\" id=\"fs-id1167831891154\" class=\"textbox textbox--examples\">\n<div data-type=\"title\">How to Simplify a Complex Rational Expressing using the LCD<\/div>\n<div data-type=\"exercise\">\n<div data-type=\"problem\" id=\"fs-id1167831908482\">\n<p>Simplify the complex rational expression by using the LCD: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9c7059d510be5fd36687fe36f6dd5887_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#120;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#121;&#125;&#125;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#120;&#125;&#123;&#121;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#121;&#125;&#123;&#120;&#125;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"38\" style=\"vertical-align: -14px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835498808\"><span data-type=\"media\" id=\"fs-id1167830964450\" data-alt=\"Step 1 is to find the least common denominator of the complex rational expression, the sum of the quantity 1 divided by x and the quantity 1 divided by y all divided by the difference between the quantity x divided by y and the quantity y divided by x.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_03_005a_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 1 is to find the least common denominator of the complex rational expression, the sum of the quantity 1 divided by x and the quantity 1 divided by y all divided by the difference between the quantity x divided by y and the quantity y divided by x.\" \/><\/span><span data-type=\"media\" id=\"fs-id1167835414657\" data-alt=\"Step 2 is to multiply the numerator and the denominator by the least common denominator, x y. The result is x y times the sum of the quantity 1 divided by x and the quantity 1 divided by y all divided by x y times the difference between the quantity x divided by y and the quantity y divided by x.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_03_005b_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 2 is to multiply the numerator and the denominator by the least common denominator, x y. The result is x y times the sum of the quantity 1 divided by x and the quantity 1 divided by y all divided by x y times the difference between the quantity x divided by y and the quantity y divided by x.\" \/><\/span><span data-type=\"media\" id=\"fs-id1167830693440\" data-alt=\"Step 3 is to simplify the expression. Distribute x y in the numerator and the denominator. The result is x y times 1 divided by x plus x y times 1 divided by y all divided by x y times x divided by y plus x y times y divided by x. It simplifies to the sum of y and x divided by the quantity x squared minus y squared. Write the denominator as the difference of squares, the quantity x minus y times the quantity x plus y. The result is the quantity y plus x all divided by the quantity x minus y times the quantity x plus y. Remove the common factor, y plus x, from the numerator and denominator. The result is 1 divided by the quantity x minus y.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_03_005c_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 3 is to simplify the expression. Distribute x y in the numerator and the denominator. The result is x y times 1 divided by x plus x y times 1 divided by y all divided by x y times x divided by y plus x y times y divided by x. It simplifies to the sum of y and x divided by the quantity x squared minus y squared. Write the denominator as the difference of squares, the quantity x minus y times the quantity x plus y. The result is the quantity y plus x all divided by the quantity x minus y times the quantity x plus y. Remove the common factor, y plus x, from the numerator and denominator. The result is 1 divided by the quantity x minus y.\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167834191698\">\n<div data-type=\"problem\">\n<p id=\"fs-id1167834367143\">Simplify the complex rational expression by using the LCD: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-240eeb05326b8a4a0334d5b5b5d84c85_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#97;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#98;&#125;&#125;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#97;&#125;&#123;&#98;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#98;&#125;&#123;&#97;&#125;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"36\" width=\"37\" style=\"vertical-align: -14px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835346314\">\n<p id=\"fs-id1167835198527\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c61533986b5ab218ec3c5184880ad4e0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#98;&#43;&#97;&#125;&#123;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#43;&#123;&#98;&#125;&#94;&#123;&#50;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"38\" style=\"vertical-align: -9px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167835318628\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167835253742\">\n<div data-type=\"problem\" id=\"fs-id1167834432188\">\n<p id=\"fs-id1167835341828\">Simplify the complex rational expression by using the LCD: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b7198202430713bdd8cb18113bc85cef_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#125;&#125;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#120;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#121;&#125;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"40\" width=\"51\" style=\"vertical-align: -15px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835218116\">\n<p id=\"fs-id1167835615105\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-fcb0517563db91236c1ff4f0a5050697_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#121;&#45;&#120;&#125;&#123;&#120;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"26\" style=\"vertical-align: -9px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" class=\"howto\">\n<div data-type=\"title\">Simplify a complex rational expression by using the LCD.<\/div>\n<ol id=\"fs-id1167826799162\" type=\"1\" class=\"stepwise\">\n<li>Find the LCD of all fractions in the complex rational expression.<\/li>\n<li>Multiply the numerator and denominator by the LCD.<\/li>\n<li>Simplify the expression.<\/li>\n<\/ol>\n<\/div>\n<p id=\"fs-id1167835218400\">Be sure to start by factoring all the denominators so you can find the LCD.<\/p>\n<div data-type=\"example\" id=\"fs-id1167826987981\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167835420261\">\n<div data-type=\"problem\" id=\"fs-id1167835350442\">\n<p id=\"fs-id1167834131359\">Simplify the complex rational expression by using the LCD: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3ba3d3e36d0d510c631edc6adcf8e134_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#120;&#43;&#54;&#125;&#125;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#125;&#123;&#120;&#45;&#54;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#125;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#51;&#54;&#125;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"81\" style=\"vertical-align: -15px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834130742\">\n<table id=\"fs-id1167834063606\" class=\"unnumbered unstyled can-break\" summary=\"Simplify the complex rational expression, 2 divided by the quantity x plus 6 all divided by the difference between 4 divided by the quantity x minus 6 and 4 divided by the quantity x squared minus 36. Find the least common denominator all of the fractions. It is x squared minus 36 which equals the quantity x plus 6 times the quantity x minus 6. Multiply the numerator and denominator by the least common denominator. The result is the quantity x plus 6 times the quantity x minus 6 times the quantity 2 divided by the quantity x plus 6 all divided by the quantity x plus 6 times the quantity x minus 6 times the difference between the quantity 4 divided by the quantity x minus 6 and 4 divided by the product of the quantity x plus 6 and the quantity x minus 6. Simplify the expression. Distribute, the quantity x plus 6 and x minus 6, in the denominator. The result is the quantity x plus 6 times the quantity x minus 6 times the quantity 2 divided by the quantity x plus 6 all divided by the quantity x plus 6 times the quantity x minus 6 times the quantity 4 divided by the quantity x minus 6 minus the quantity x plus 6 times the quantity x minus 6 times 4 divided by the product of the quantity x plus 6 and the quantity x minus 6. Simplify by removing the common factors, x plus 6 from the numerator, and the quantity x minus 6 times the quantity x plus 6, from the denominator. The result is 2 times the quantity x minus 6 all divided by the product of 4 and the quantity x plus 6 minus 4. To simplify the denominator, distribute and combine like terms. The result is 2 times the quantity x minus 6 all divided by the quantity 4 x plus 20. Factor the denominator. The result is 2 times the quantity x minus 6 all divided by 4 times the quantity x plus 5. Notice that 4 in the denominator is 2 times 2. Remove the common factor, 2, from the numerator and denominator. The result is the quantity x minus 6 all divided by 2 times the quantity x plus 5. Notice that there are no more factors common to the numerator and denominator.\" data-label=\"\">\n<tbody>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167832119089\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_03_006b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Find the LCD of all fractions in the complex rational<\/p>\n<div data-type=\"newline\"><\/div>\n<p>expression. The LCD is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2ed1a858a42392c62807e8f936d34129_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#51;&#54;&#61;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#54;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#54;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"192\" style=\"vertical-align: -4px;\" \/>.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Multiply the numerator and denominator by the LCD.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167830700886\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_03_006a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Simplify the expression.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Distribute in the denominator.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167834133885\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_03_006c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Simplify.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167834161475\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_03_006d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Simplify.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_03_006e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">To simplify the denominator, distribute<\/p>\n<div data-type=\"newline\"><\/div>\n<p>and combine like terms.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_03_006f_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Factor the denominator.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167831040626\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_03_006g_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Remove common factors.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167831895215\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_03_006h_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Simplify.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167835369386\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_03_006i_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Notice that there are no more factors<\/p>\n<div data-type=\"newline\"><\/div>\n<p>common to the numerator and denominator.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167834517647\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167832152324\">\n<div data-type=\"problem\" id=\"fs-id1167834395939\">\n<p id=\"fs-id1167831824577\">Simplify the complex rational expression by using the LCD: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-60cf51a447d69da93fe518276644c976_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#120;&#43;&#50;&#125;&#125;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#120;&#45;&#50;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#52;&#125;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"75\" style=\"vertical-align: -15px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834161803\">\n<p id=\"fs-id1167835377966\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-cc7cf46d94b28c897fcfda44f57f59bc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#123;&#53;&#120;&#43;&#55;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"28\" width=\"43\" style=\"vertical-align: -8px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167835284772\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167835257334\">\n<div data-type=\"problem\" id=\"fs-id1167831115599\">\n<p id=\"fs-id1167830838434\">Simplify the complex rational expression by using the LCD: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1595452241e400aec39a1be328f3b8d7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#120;&#45;&#55;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#120;&#43;&#55;&#125;&#125;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#54;&#125;&#123;&#120;&#43;&#55;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#52;&#57;&#125;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"81\" style=\"vertical-align: -15px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\">\n<p id=\"fs-id1167835489114\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1ca5c629583aeec8f5ba236befbe5b26_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#120;&#43;&#50;&#49;&#125;&#123;&#54;&#120;&#45;&#52;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"40\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1165926630658\">Be sure to factor the denominators first. Proceed carefully as the math can get messy!<\/p>\n<div data-type=\"example\" id=\"fs-id1167835595640\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167835325063\">\n<div data-type=\"problem\" id=\"fs-id1167834501479\">\n<p id=\"fs-id1167834194400\">Simplify the complex rational expression by using the LCD: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-fe36d01adcb1c24589dfb8b4015a7933_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#125;&#123;&#123;&#109;&#125;&#94;&#123;&#50;&#125;&#45;&#55;&#109;&#43;&#49;&#50;&#125;&#125;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#109;&#45;&#51;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#109;&#45;&#52;&#125;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"76\" style=\"vertical-align: -13px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835344864\">\n<table id=\"fs-id1167835363257\" class=\"unnumbered unstyled\" summary=\"Simplify the complex rational expression, 4 divided by the quantity m squared minus 7 m plus 12 all divided by the difference between 3 divided by the quantity m minus 3 and 2 divided by the quantity m minus 4. Find the least common denominator of all fractions in the expression. The least common denominator is the quantity m minus 3 times the quantity m minus 4. Multiply the numerator and denominator by the least common denominator. The result is the quantity m minus 3 times the quantity m minus 4 times the quantity 4 divided by the quantity m minus 3 times the quantity m minus 4 all divided by the quantity m minus 3 times the quantity m minus 4 times the difference between 3 divided by the quantity m minus 3 and 2 divided by the quantity m minus 4. Simplify by removing the common factors, m minus 3 and m minus 4, from the numerator and denominator. The result is 4 divided by the product of 3 and m minus 4 minus the product of 2 and minus 3. Distribute the factors 3 and 2 in the denominator. The result is 4 divided by the quantity 3 m minus 12 minus 2 m plus 6. Combine like terms in the denominator. The result is 4 divided by the quantity m minus 6.\" data-label=\"\">\n<tbody>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167830838226\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_03_007b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Find the LCD of all fractions in the<\/p>\n<div data-type=\"newline\"><\/div>\n<p>complex rational expression.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">The LCD is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a6e067e4c1ac54ead144efe4c286e077_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#109;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#109;&#45;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"129\" style=\"vertical-align: -4px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"center\"><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Multiply the numerator and<\/p>\n<div data-type=\"newline\"><\/div>\n<p>denominator by the LCD.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167834472376\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_03_007a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Simplify.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167835308942\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_03_007c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Simplify.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167835410534\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_03_007d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Distribute.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167826978771\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_03_007e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Combine like terms.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167835352202\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_03_007f_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167835358450\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167835376231\">\n<div data-type=\"problem\" id=\"fs-id1167835349843\">\n<p id=\"fs-id1167835335378\">Simplify the complex rational expression by using the LCD: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0f59339437a6ed4baf41da91a02812f1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#55;&#120;&#43;&#49;&#48;&#125;&#125;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#125;&#123;&#120;&#43;&#50;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#120;&#43;&#53;&#125;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"40\" width=\"68\" style=\"vertical-align: -15px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834552539\">\n<p id=\"fs-id1167835524236\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-58159dc3afc84264f28f4e5826012a7f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#53;&#120;&#43;&#50;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"40\" style=\"vertical-align: -8px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167831076568\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167835231354\">\n<div data-type=\"problem\" id=\"fs-id1167835303310\">\n<p id=\"fs-id1167831116163\">Simplify the complex rational expression by using the LCD: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-47dde1b224af8216e56e708ea4d5723f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#121;&#125;&#123;&#121;&#43;&#53;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#121;&#43;&#54;&#125;&#125;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#121;&#125;&#123;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#43;&#49;&#49;&#121;&#43;&#51;&#48;&#125;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"42\" width=\"71\" style=\"vertical-align: -18px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835422942\">\n<p id=\"fs-id1167831217847\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f83538be24edaf7a1d8aa8ac2e5da63f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#43;&#49;&#51;&#121;&#43;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#123;&#51;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"33\" width=\"91\" style=\"vertical-align: -9px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"example\" id=\"fs-id1167830698667\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167834214131\">\n<div data-type=\"problem\" id=\"fs-id1167835351498\">\n<p id=\"fs-id1167834562517\">Simplify the complex rational expression by using the LCD: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-62d63662aa0f3f0ec2d9a6ed37b40cd3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#121;&#125;&#123;&#121;&#43;&#49;&#125;&#125;&#123;&#49;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#121;&#45;&#49;&#125;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"37\" width=\"49\" style=\"vertical-align: -15px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835419888\">\n<table id=\"fs-id1167835303162\" class=\"unnumbered unstyled can-break\" summary=\"Simplify the complex rational expression, y divided by quantity y plus 1 all divided by 1 plus the quantity 1 divided by the quantity y minus 1. Find the lowest common denominator of all fractions in the complex rational expression. It is the quantity y plus 1 times the quantity y minus 1. Multiply the numerator and denominator by the least common denominator. The result is the quantity y plus 1 times the quantity y minus 1 times the quantity y divided by quantity y plus 1 all divided by the quantity y plus 1 times the quantity y minus 1 times the quantity 1 plus the quantity 1 divided by the quantity y minus 1. Distribute the least common denominator, the quantity y plus 1 times the quantity y minus 1, in the denominator. The result is the quantity y plus 1 times the quantity y minus 1 times the quantity y divided by the quantity y plus 1 all divided by the quantity y plus 1 times the quantity y minus 1 times 1 plus the quantity y plus 1 times the quantity y minus 1 times the quantity 1 divided by the quantity y minus 1. Simplify by removing the common factor, y plus 1, in the denominator and the common factor, y minus 1, in the denominator. Simplify. The result is the product of the quantity y minus 1 times y all divided by the product of the quantity y plus 1 and the quantity y minus 1 plus the quantity y plus 1. Simplify the denominator and leave the numerator factored. The result is y times the quantity y minus 1 all divided by the quantity y squared minus 1 plus y plus 1, which simplifies to y times the quantity y minus 1 all divided by the quantity y squared plus y. Factor the denominator. The result is y times the quantity y minus 1 all divided by y times the quantity y plus 1. Remove the common factor, y, in the numerator and denominator. Then simplify. The result is the quantity y minus 1 divided by the quantity y plus 1.\" data-label=\"\">\n<tbody>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167834099610\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_03_008c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Find the LCD of all fractions in the complex rational expression.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">The LCD is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0bbcb88c1f8f96d714189f8626f49277_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#43;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"116\" style=\"vertical-align: -4px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"center\"><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Multiply the numerator and denominator by the LCD.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167834408523\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_03_008a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Distribute in the denominator and simplify.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167834185818\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_03_008b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Simplify.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167834397261\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_03_008d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Simplify the denominator and leave the<\/p>\n<div data-type=\"newline\"><\/div>\n<p>numerator factored.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167834547111\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_03_008e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167831949102\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_03_008f_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Factor the denominator and remove factors<\/p>\n<div data-type=\"newline\"><\/div>\n<p>common with the numerator.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167831874793\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_03_008g_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Simplify.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167831239675\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_03_008h_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167835254637\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167831954222\">\n<div data-type=\"problem\" id=\"fs-id1167830868588\">\n<p id=\"fs-id1167826804019\">Simplify the complex rational expression by using the LCD: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-76dbf6c1ea657194939b0c325f937de4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#120;&#125;&#123;&#120;&#43;&#51;&#125;&#125;&#123;&#49;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#120;&#43;&#51;&#125;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"36\" width=\"49\" style=\"vertical-align: -15px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835378002\">\n<p id=\"fs-id1167831871449\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-986cce1eda7895ffa424b660e112372a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#120;&#125;&#123;&#120;&#43;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"26\" style=\"vertical-align: -8px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167834130402\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167830701079\">\n<div data-type=\"problem\" id=\"fs-id1167834194517\">\n<p id=\"fs-id1167835352898\">Simplify the complex rational expression by using the LCD: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-56d624e6d7365719fdafc928a7ff9f8d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#120;&#45;&#49;&#125;&#125;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#120;&#43;&#49;&#125;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"49\" style=\"vertical-align: -15px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834196182\">\n<p id=\"fs-id1167834533150\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6b72c1ce2e84ea219ca735d65a580195_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#120;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#123;&#51;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"29\" width=\"44\" style=\"vertical-align: -9px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167832056804\" class=\"media-2\">\n<p id=\"fs-id1167832226318\">Access this online resource for additional instruction and practice with complex fractions.<\/p>\n<ul id=\"fs-id1171791232430\" data-display=\"block\">\n<li><a href=\"https:\/\/openstax.org\/l\/37CompFrac\">Complex Fractions<\/a><\/li>\n<\/ul>\n<\/div>\n<\/div>\n<div class=\"textbox\" data-depth=\"1\" id=\"fs-id1167835337547\">\n<h3 data-type=\"title\">Key Concepts<\/h3>\n<ul id=\"fs-id1167832060089\" data-bullet-style=\"bullet\">\n<li><strong data-effect=\"bold\">How to simplify a complex rational expression by writing it as division.<\/strong>\n<ol id=\"fs-id1167835415230\" type=\"1\" class=\"stepwise\">\n<li>Simplify the numerator and denominator.<\/li>\n<li>Rewrite the complex rational expression as a division problem.<\/li>\n<li>Divide the expressions.<\/li>\n<\/ol>\n<\/li>\n<li><strong data-effect=\"bold\">How to simplify a complex rational expression by using the LCD.<\/strong>\n<ol id=\"fs-id1167832055842\" type=\"1\" class=\"stepwise\">\n<li>Find the LCD of all fractions in the complex rational expression.<\/li>\n<li>Multiply the numerator and denominator by the LCD.<\/li>\n<li>Simplify the expression.<\/li>\n<\/ol>\n<\/li>\n<\/ul>\n<\/div>\n<div class=\"textbox\" data-depth=\"1\">\n<div class=\"practice-perfect\" data-depth=\"2\" id=\"fs-id1167831045942\">\n<h4 data-type=\"title\">Practice Makes Perfect<\/h4>\n<p id=\"fs-id1167827987888\"><strong data-effect=\"bold\">Simplify a Complex Rational Expression by Writing it as Division<\/strong><\/p>\n<p id=\"fs-id1167831919787\">In the following exercises, simplify each complex rational expression by writing it as division.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167835378752\">\n<div data-type=\"problem\" id=\"fs-id1167826808701\">\n<p id=\"fs-id1167826808703\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c7d215a350d4a8c72e7505d4e90ce6d1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#97;&#125;&#123;&#97;&#43;&#52;&#125;&#125;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#54;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"42\" width=\"39\" style=\"vertical-align: -19px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167826799193\">\n<p id=\"fs-id1167826799195\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2e80e71d8310dadfaca6116aba8ed9b8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#97;&#45;&#52;&#125;&#123;&#50;&#97;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"25\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834430061\">\n<div data-type=\"problem\" id=\"fs-id1167834430063\">\n<p id=\"fs-id1167835355318\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-93beacd5cdd980a1c547ae65ac84224e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#98;&#125;&#123;&#98;&#45;&#53;&#125;&#125;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#98;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#123;&#98;&#125;&#94;&#123;&#50;&#125;&#45;&#50;&#53;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"41\" width=\"38\" style=\"vertical-align: -18px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834536678\">\n<div data-type=\"problem\" id=\"fs-id1167832116081\">\n<p id=\"fs-id1167834130111\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-95087222c0d1b7c6cc2b3879d34dcd70_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#123;&#99;&#125;&#94;&#123;&#50;&#125;&#43;&#53;&#99;&#45;&#49;&#52;&#125;&#125;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#48;&#125;&#123;&#99;&#43;&#55;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"41\" width=\"58\" style=\"vertical-align: -15px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167828421158\">\n<p id=\"fs-id1167831896612\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-cc05c625cbdfe000d36c74f1c1443db9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#92;&#108;&#101;&#102;&#116;&#40;&#99;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"42\" style=\"vertical-align: -9px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835320086\">\n<div data-type=\"problem\" id=\"fs-id1167835325218\">\n<p id=\"fs-id1167835325220\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e0f8bfd3d85bbf72150a53249b5dd439_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#56;&#125;&#123;&#123;&#100;&#125;&#94;&#123;&#50;&#125;&#43;&#57;&#100;&#43;&#49;&#56;&#125;&#125;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#50;&#125;&#123;&#100;&#43;&#54;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"40\" width=\"60\" style=\"vertical-align: -15px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834340141\">\n<div data-type=\"problem\" id=\"fs-id1167834340143\">\n<p id=\"fs-id1167834397422\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c0807a27d77ba8e76e7980500a836769_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#54;&#125;&#125;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#51;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#55;&#125;&#123;&#57;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"36\" width=\"31\" style=\"vertical-align: -13px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167827958452\">\n<p id=\"fs-id1167827958454\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-20d7691e507a7abd73c19a9b96ecb464_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#50;&#125;&#123;&#49;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"14\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834156636\">\n<div data-type=\"problem\" id=\"fs-id1167834156638\">\n<p id=\"fs-id1167835419844\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6c1f6599708e9cc18ab4c653a11256e9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#52;&#125;&#125;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#53;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#55;&#125;&#123;&#49;&#48;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"36\" width=\"37\" style=\"vertical-align: -14px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167831826361\">\n<div data-type=\"problem\" id=\"fs-id1167831826364\">\n<p id=\"fs-id1167835284035\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7182543d9f4dd506d816f4d1ceaa76d4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#51;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#57;&#125;&#125;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#52;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#54;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"35\" width=\"31\" style=\"vertical-align: -13px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835615375\">\n<p id=\"fs-id1167835615377\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3dcce61190cd95d69e6a7b584b8d19b3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#48;&#125;&#123;&#53;&#55;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"14\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\">\n<div data-type=\"problem\" id=\"fs-id1167835511370\">\n<p id=\"fs-id1167835511372\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5b4e1af1911fd3e9985b20cddc3c9878_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#54;&#125;&#125;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#51;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#52;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"35\" width=\"31\" style=\"vertical-align: -13px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\">\n<div data-type=\"problem\" id=\"fs-id1167826781274\">\n<p id=\"fs-id1167835310538\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c77b2861ead809043f0fcc40435d012d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#110;&#125;&#123;&#109;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#110;&#125;&#125;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#110;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#110;&#125;&#123;&#109;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"35\" width=\"37\" style=\"vertical-align: -13px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835595865\">\n<p id=\"fs-id1167835595867\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f6cc974363b25ad6c360fd8a8bcc4339_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#110;&#125;&#94;&#123;&#50;&#125;&#43;&#109;&#125;&#123;&#109;&#45;&#123;&#110;&#125;&#94;&#123;&#50;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"38\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835370045\">\n<div data-type=\"problem\" id=\"fs-id1167832043728\">\n<p id=\"fs-id1167832043730\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a4cdf73a4446f8b9b417765784f20e05_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#112;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#112;&#125;&#123;&#113;&#125;&#125;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#113;&#125;&#123;&#112;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#113;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"39\" width=\"31\" style=\"vertical-align: -15px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834184589\">\n<div data-type=\"problem\" id=\"fs-id1167835615398\">\n<p id=\"fs-id1167835615400\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2f45e1bfb6956c7c799564e713c0d008_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#114;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#116;&#125;&#125;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#123;&#114;&#125;&#94;&#123;&#50;&#125;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#123;&#116;&#125;&#94;&#123;&#50;&#125;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"37\" width=\"43\" style=\"vertical-align: -15px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835365577\">\n<p id=\"fs-id1167835362982\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c64218c360818e20861f1c3e548d48d7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#114;&#116;&#125;&#123;&#116;&#45;&#114;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"23\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834301178\">\n<div data-type=\"problem\">\n<p id=\"fs-id1167834372197\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-906f5538d939722463eec7cfec3662f6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#118;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#119;&#125;&#125;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#123;&#118;&#125;&#94;&#123;&#50;&#125;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#123;&#119;&#125;&#94;&#123;&#50;&#125;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"37\" width=\"48\" style=\"vertical-align: -15px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167831884421\">\n<div data-type=\"problem\" id=\"fs-id1167835419295\">\n<p id=\"fs-id1167835419297\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6bbc5aa26f32cc223003aa40411dbc1f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#120;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#120;&#125;&#123;&#120;&#43;&#51;&#125;&#125;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#120;&#43;&#51;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#120;&#45;&#51;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"63\" style=\"vertical-align: -15px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\">\n<p id=\"fs-id1167835328130\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6e3fb063c5069baeb7f549603645a280_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"73\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167827988185\">\n<div data-type=\"problem\" id=\"fs-id1167831821797\">\n<p id=\"fs-id1167831821799\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-51ecff9048bfe0a822dbd2544a9dde33_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#121;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#121;&#125;&#123;&#121;&#45;&#52;&#125;&#125;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#121;&#45;&#52;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#121;&#43;&#52;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"39\" width=\"62\" style=\"vertical-align: -15px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835381633\">\n<div data-type=\"problem\" id=\"fs-id1167835381635\">\n<p id=\"fs-id1167826983786\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-19ad28a3f9f7b53890486d05b9e116f4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#97;&#43;&#51;&#125;&#125;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#97;&#43;&#51;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#97;&#125;&#123;&#50;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"47\" style=\"vertical-align: -15px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835191141\">\n<p id=\"fs-id1167835191143\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-28ecf3150121ba44b96b97fea9bcb061_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#125;&#123;&#97;&#43;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"25\" style=\"vertical-align: -8px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167830693679\">\n<div data-type=\"problem\" id=\"fs-id1167835370000\">\n<p id=\"fs-id1167835370002\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0838a6bbf114becc5cf5525ac9a4e962_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#125;&#123;&#98;&#45;&#53;&#125;&#125;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#98;&#45;&#53;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#98;&#125;&#123;&#52;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"37\" width=\"46\" style=\"vertical-align: -14px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167834191287\"><strong data-effect=\"bold\">Simplify a Complex Rational Expression by Using the LCD<\/strong><\/p>\n<p id=\"fs-id1167835180600\">In the following exercises, simplify each complex rational expression by using the LCD.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167835596470\">\n<div data-type=\"problem\" id=\"fs-id1167835596472\">\n<p id=\"fs-id1167835363534\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8f1396f664b16d1589fb891397858fa7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#51;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#56;&#125;&#125;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#52;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#49;&#50;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"36\" width=\"37\" style=\"vertical-align: -14px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834309892\">\n<p id=\"fs-id1167834309894\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9deb028bb2822163ee3aa88f47c67bfe_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#49;&#125;&#123;&#56;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"14\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834472639\">\n<div data-type=\"problem\" id=\"fs-id1167826880090\">\n<p id=\"fs-id1167826880092\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-500e03a3d3518b2af2f2dac8606bbf9d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#52;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#57;&#125;&#125;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#54;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#49;&#50;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"36\" width=\"37\" style=\"vertical-align: -14px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167831920751\">\n<div data-type=\"problem\" id=\"fs-id1167831239067\">\n<p id=\"fs-id1167831239069\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d2f6f6bb2d40b4be5dfcd210f7ebaffb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#54;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#57;&#125;&#125;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#55;&#125;&#123;&#49;&#56;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#51;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"37\" width=\"37\" style=\"vertical-align: -14px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835280282\">\n<p id=\"fs-id1167835217864\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-23ae1c0b89342725f5c66b11294f8bcc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"17\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167831040329\">\n<div data-type=\"problem\" id=\"fs-id1167831040331\">\n<p id=\"fs-id1167834556376\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-49d97d46260caaffb493163c54ef17cf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#54;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#125;&#123;&#49;&#53;&#125;&#125;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#53;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"35\" width=\"37\" style=\"vertical-align: -13px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835190900\">\n<div data-type=\"problem\" id=\"fs-id1167835190903\">\n<p id=\"fs-id1167835370378\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-dcba8bf3d513c03978d0d0a8a1f3259e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#99;&#125;&#123;&#100;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#100;&#125;&#125;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#100;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#100;&#125;&#123;&#99;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"36\" width=\"31\" style=\"vertical-align: -14px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834064044\">\n<p id=\"fs-id1167834064046\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-64578beeb5115758befd2951b3dfa66f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#99;&#125;&#94;&#123;&#50;&#125;&#43;&#99;&#125;&#123;&#99;&#45;&#123;&#100;&#125;&#94;&#123;&#50;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"31\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834539410\">\n<div data-type=\"problem\" id=\"fs-id1167834539412\">\n<p id=\"fs-id1167826804612\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-529750e9e8bfa85cc908ed866164236e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#109;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#109;&#125;&#123;&#110;&#125;&#125;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#110;&#125;&#123;&#109;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#110;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"35\" width=\"40\" style=\"vertical-align: -13px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835355981\">\n<div data-type=\"problem\" id=\"fs-id1167834238843\">\n<p id=\"fs-id1167834238845\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-81fd6d31a4740bcae4c0214a12e651e2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#112;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#113;&#125;&#125;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#123;&#112;&#125;&#94;&#123;&#50;&#125;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#123;&#113;&#125;&#94;&#123;&#50;&#125;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"41\" width=\"45\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835233678\">\n<p id=\"fs-id1167834525271\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-bad9cf81ae1776d6b8aff9013770adf5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#112;&#113;&#125;&#123;&#113;&#45;&#112;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"25\" style=\"vertical-align: -9px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167826804725\">\n<div data-type=\"problem\" id=\"fs-id1167835420186\">\n<p id=\"fs-id1167835420188\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0dcde1385c3183babd9a69bb27d9c89a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#114;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#116;&#125;&#125;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#123;&#114;&#125;&#94;&#123;&#50;&#125;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#123;&#116;&#125;&#94;&#123;&#50;&#125;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"37\" width=\"43\" style=\"vertical-align: -15px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167832053579\">\n<div data-type=\"problem\" id=\"fs-id1167834300263\">\n<p id=\"fs-id1167834300265\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-900589d1e94ace906a825fda62c49201_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#120;&#43;&#53;&#125;&#125;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#120;&#45;&#53;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#50;&#53;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"76\" style=\"vertical-align: -15px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834432252\">\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8399998705a9ef3e648ded480fefdc7d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#120;&#45;&#49;&#48;&#125;&#123;&#51;&#120;&#43;&#49;&#54;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"40\" style=\"vertical-align: -8px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835304946\">\n<div data-type=\"problem\" id=\"fs-id1167835304948\">\n<p id=\"fs-id1167835342970\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-38b1dc2a66f2369386701f3627df5c4f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#121;&#45;&#52;&#125;&#125;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#121;&#43;&#52;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#54;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"42\" width=\"75\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167831239449\">\n<div data-type=\"problem\" id=\"fs-id1167834432088\">\n<p id=\"fs-id1167834432090\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-bc5efbd07f8dbfa3a944ee2431de3614_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#123;&#122;&#125;&#94;&#123;&#50;&#125;&#45;&#54;&#52;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#122;&#43;&#56;&#125;&#125;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#122;&#43;&#56;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#122;&#45;&#56;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"41\" width=\"74\" style=\"vertical-align: -15px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835338228\">\n<p id=\"fs-id1167835338230\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-29feb15c7712f67a7927b559eaa9da61_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#122;&#45;&#49;&#57;&#125;&#123;&#51;&#122;&#43;&#56;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"39\" style=\"vertical-align: -8px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835328334\">\n<div data-type=\"problem\" id=\"fs-id1167835328337\">\n<p id=\"fs-id1167831887230\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-31d2c8f40ee5a6cdcb9bcd8ed46e931a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#115;&#43;&#54;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#115;&#45;&#54;&#125;&#125;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#123;&#115;&#125;&#94;&#123;&#50;&#125;&#45;&#51;&#54;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#125;&#123;&#115;&#43;&#54;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"39\" width=\"73\" style=\"vertical-align: -15px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834394776\">\n<div data-type=\"problem\" id=\"fs-id1167830836798\">\n<p id=\"fs-id1167830836800\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-cf3e5c5b3394855941a5b5201c74f27f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#125;&#123;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#45;&#50;&#97;&#45;&#49;&#53;&#125;&#125;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#97;&#45;&#53;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#97;&#43;&#51;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"40\" width=\"62\" style=\"vertical-align: -15px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167832043716\">\n<p id=\"fs-id1167835569735\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6a4b99a6bc85c289427e3bc1bfbfaab0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#125;&#123;&#51;&#97;&#45;&#55;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"32\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835320616\">\n<div data-type=\"problem\" id=\"fs-id1167834279886\">\n<p id=\"fs-id1167834279888\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6b576c0c326f925a35fade28e347ae49_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#123;&#98;&#125;&#94;&#123;&#50;&#125;&#45;&#54;&#98;&#45;&#50;&#55;&#125;&#125;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#98;&#45;&#57;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#98;&#43;&#51;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"41\" width=\"60\" style=\"vertical-align: -15px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835416735\">\n<div data-type=\"problem\" id=\"fs-id1167835416737\">\n<p id=\"fs-id1167831025160\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-dcccc542264a92098de0e0ce6aac47ed_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#99;&#43;&#50;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#99;&#43;&#55;&#125;&#125;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#99;&#125;&#123;&#123;&#99;&#125;&#94;&#123;&#50;&#125;&#43;&#57;&#99;&#43;&#49;&#52;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"41\" width=\"60\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834219571\">\n<p id=\"fs-id1167834219574\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9211205d1ba832b52a51199928e5a90e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#99;&#43;&#50;&#57;&#125;&#123;&#53;&#99;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"38\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167831837544\">\n<div data-type=\"problem\">\n<p id=\"fs-id1167830838233\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f8542e95114a1afdbb00056a4418e1cc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#54;&#125;&#123;&#100;&#45;&#52;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#100;&#43;&#55;&#125;&#125;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#100;&#125;&#123;&#123;&#100;&#125;&#94;&#123;&#50;&#125;&#43;&#51;&#100;&#45;&#50;&#56;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"40\" width=\"62\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834194525\">\n<div data-type=\"problem\" id=\"fs-id1167835363301\">\n<p id=\"fs-id1167835363303\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2cb7557b2fe2fd639344fd25c34c88ed_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#112;&#45;&#51;&#125;&#125;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#112;&#45;&#51;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"39\" width=\"44\" style=\"vertical-align: -15px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835510188\">\n<p id=\"fs-id1167831872142\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a54e6bf70bc8ee85f0c8038538ace0e4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#112;&#45;&#53;&#125;&#123;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"32\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835514456\">\n<div data-type=\"problem\" id=\"fs-id1167826857291\">\n<p id=\"fs-id1167826857294\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a9060e562683f8cd62d941e574ede7e1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#110;&#125;&#123;&#110;&#45;&#50;&#125;&#125;&#123;&#51;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#110;&#45;&#50;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"34\" width=\"45\" style=\"vertical-align: -13px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834279936\">\n<div data-type=\"problem\" id=\"fs-id1167834395888\">\n<p id=\"fs-id1167834395890\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-eda0df34ba0261079f910c01597feb15_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#109;&#125;&#123;&#109;&#43;&#53;&#125;&#125;&#123;&#52;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#109;&#45;&#53;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"34\" width=\"48\" style=\"vertical-align: -13px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835305735\">\n<p id=\"fs-id1167832053222\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e50de122ba778325ed0b2cb8b9b83649_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#109;&#92;&#108;&#101;&#102;&#116;&#40;&#109;&#45;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#109;&#45;&#49;&#57;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#109;&#43;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"29\" width=\"95\" style=\"vertical-align: -9px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835381804\">\n<div data-type=\"problem\" id=\"fs-id1167831882524\">\n<p id=\"fs-id1167831882526\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f14fcc9aa64220e05ce2f316e434c4b5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#55;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#113;&#45;&#50;&#125;&#125;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#113;&#43;&#50;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"39\" width=\"43\" style=\"vertical-align: -15px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167831832518\">In the following exercises, simplify each complex rational expression using either method.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167831832522\">\n<div data-type=\"problem\" id=\"fs-id1167835360939\">\n<p id=\"fs-id1167835360942\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ea606b0e1462a9c751b135abf801cfcc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#52;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#55;&#125;&#125;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#49;&#52;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"36\" width=\"37\" style=\"vertical-align: -14px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835623266\">\n<p id=\"fs-id1167835623269\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b506afea731584a25429b32f6b72d365_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#51;&#125;&#123;&#50;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"14\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834252374\">\n<div data-type=\"problem\" id=\"fs-id1167834131419\">\n<p id=\"fs-id1167834131421\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-03ab0f43715324e75f85ca1d69561300_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#118;&#125;&#123;&#119;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#118;&#125;&#125;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#118;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#118;&#125;&#123;&#119;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"35\" width=\"34\" style=\"vertical-align: -13px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835356825\">\n<div data-type=\"problem\" id=\"fs-id1167835356827\">\n<p id=\"fs-id1167826798856\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b6d8e5c7601c3764fbe910161c4a9ee4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#97;&#43;&#52;&#125;&#125;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#54;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"39\" width=\"39\" style=\"vertical-align: -16px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835240317\">\n<p id=\"fs-id1167835479556\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7fdaa5ab35eeb1f920ea161e15686ed6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#92;&#108;&#101;&#102;&#116;&#40;&#97;&#45;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"65\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835356359\">\n<div data-type=\"problem\" id=\"fs-id1167835356361\">\n<p id=\"fs-id1167830700735\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-82d895890ed505217b0456d3165f9a0f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#123;&#98;&#125;&#94;&#123;&#50;&#125;&#45;&#51;&#98;&#45;&#52;&#48;&#125;&#125;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#98;&#43;&#53;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#98;&#45;&#56;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"40\" width=\"60\" style=\"vertical-align: -15px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167831115961\">\n<div data-type=\"problem\" id=\"fs-id1167826869962\">\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7aebe7fdad05e96903daf124a03bf631_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#109;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#110;&#125;&#125;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#123;&#109;&#125;&#94;&#123;&#50;&#125;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#123;&#110;&#125;&#94;&#123;&#50;&#125;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"37\" width=\"51\" style=\"vertical-align: -15px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835389924\">\n<p id=\"fs-id1167835366569\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f4af63c09af0a1202050d608ff125391_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#109;&#110;&#125;&#123;&#110;&#45;&#109;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"32\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167831957810\">\n<div data-type=\"problem\" id=\"fs-id1167831957812\">\n<p id=\"fs-id1167831822064\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-df857940f9b43eeaa45dfd214811830a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#114;&#45;&#57;&#125;&#125;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#114;&#43;&#57;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#123;&#114;&#125;&#94;&#123;&#50;&#125;&#45;&#56;&#49;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"39\" width=\"74\" style=\"vertical-align: -16px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835327394\">\n<div data-type=\"problem\" id=\"fs-id1167835321549\">\n<p id=\"fs-id1167835321551\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-96ad2cf9b04e8421dafd96568bffe2a5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#120;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#120;&#125;&#123;&#120;&#43;&#50;&#125;&#125;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#120;&#43;&#50;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#120;&#45;&#50;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"63\" style=\"vertical-align: -15px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834533453\">\n<p id=\"fs-id1167835318576\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5da39d4c10169e4736f43f900efdc0d4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#123;&#54;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"73\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167826781484\">\n<div data-type=\"problem\" id=\"fs-id1167835363110\">\n<p id=\"fs-id1167835363113\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-58b74b90cc5b38a88e91dca7944643b9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#121;&#125;&#123;&#121;&#43;&#51;&#125;&#125;&#123;&#50;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#121;&#45;&#51;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"37\" width=\"44\" style=\"vertical-align: -15px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"writing\" data-depth=\"2\" id=\"fs-id1167834473754\">\n<h4 data-type=\"title\">Writing Exercises<\/h4>\n<div data-type=\"exercise\" id=\"fs-id1167830702565\">\n<div data-type=\"problem\" id=\"fs-id1167834556395\">\n<p id=\"fs-id1167834556397\">In this section, you learned to simplify the complex fraction <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6091310aa7cad038d7630a08e8174440_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#120;&#43;&#50;&#125;&#125;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#120;&#125;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#52;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"36\" width=\"33\" style=\"vertical-align: -13px;\" \/> two ways: rewriting it as a division problem or multiplying the numerator and denominator by the LCD. Which method do you prefer? Why?<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834061445\">\n<p id=\"fs-id1167834061447\">Answers will vary.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167831954191\">\n<div data-type=\"problem\" id=\"fs-id1167832076124\">\n<p id=\"fs-id1167832076126\">Efraim wants to start simplifying the complex fraction <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8afc2ef4d29d663e865974e541ad8b34_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#97;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#98;&#125;&#125;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#97;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#98;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"35\" width=\"31\" style=\"vertical-align: -13px;\" \/> by cancelling the variables from the numerator and denominator, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-526e82f299087e0d711531a51fb5993d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#92;&#111;&#118;&#101;&#114;&#108;&#105;&#110;&#101;&#123;&#41;&#97;&#125;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#92;&#111;&#118;&#101;&#114;&#108;&#105;&#110;&#101;&#123;&#41;&#98;&#125;&#125;&#125;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#92;&#111;&#118;&#101;&#114;&#108;&#105;&#110;&#101;&#123;&#41;&#97;&#125;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#92;&#111;&#118;&#101;&#114;&#108;&#105;&#110;&#101;&#123;&#41;&#98;&#125;&#125;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"46\" style=\"vertical-align: -17px;\" \/> Explain what is wrong with Efraim\u2019s plan.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"2\" id=\"fs-id1167830757635\">\n<h4 data-type=\"title\">Self Check<\/h4>\n<p id=\"fs-id1167831922168\"><span class=\"token\">\u24d0<\/span> After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.<\/p>\n<p><span data-type=\"media\" id=\"fs-id1167835180627\" data-alt=\"This table has four columns and three rows. The first row is a header and it labels each column, \u201cI can\u2026\u201d, \u201cConfidently,\u201d \u201cWith some help,\u201d and \u201cNo-I don\u2019t get it!\u201d In row 2, the I can was simplify a complex rational expression by writing it as division. In row 3, the I can was simplify a complex rational expression by using the least common denominator.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_03_201_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This table has four columns and three rows. The first row is a header and it labels each column, \u201cI can\u2026\u201d, \u201cConfidently,\u201d \u201cWith some help,\u201d and \u201cNo-I don\u2019t get it!\u201d In row 2, the I can was simplify a complex rational expression by writing it as division. In row 3, the I can was simplify a complex rational expression by using the least common denominator.\" \/><\/span><\/p>\n<p><span class=\"token\">\u24d1<\/span> After looking at the checklist, do you think you are well-prepared for the next section? Why or why not?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"glossary\" class=\"textbox shaded\">\n<h3 data-type=\"glossary-title\">Glossary<\/h3>\n<dl id=\"fs-id1167834308149\">\n<dt>complex rational expression<\/dt>\n<dd id=\"fs-id1167834099322\">A complex rational expression is a rational expression in which the numerator and\/or denominator contains a rational expression.<\/dd>\n<\/dl>\n<\/div>\n","protected":false},"author":103,"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-3276","chapter","type-chapter","status-publish","hentry"],"part":3130,"_links":{"self":[{"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/chapters\/3276","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/wp\/v2\/users\/103"}],"version-history":[{"count":0,"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/chapters\/3276\/revisions"}],"part":[{"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/parts\/3130"}],"metadata":[{"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/chapters\/3276\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/wp\/v2\/media?parent=3276"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/chapter-type?post=3276"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/wp\/v2\/contributor?post=3276"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/wp\/v2\/license?post=3276"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}