{"id":18389,"date":"2022-04-15T23:31:02","date_gmt":"2022-04-15T23:31:02","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/?post_type=chapter&#038;p=18389"},"modified":"2022-04-28T23:17:05","modified_gmt":"2022-04-28T23:17:05","slug":"cr-17-complex-fractions","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/chapter\/cr-17-complex-fractions\/","title":{"raw":"CR.17: Complex Fractions","rendered":"CR.17: Complex Fractions"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Simplify complex rational expressions<\/li>\r\n<\/ul>\r\n<\/div>\r\nFractions and rational expressions can be interpreted as quotients. When both the dividend (numerator) and divisor (denominator) include fractions or rational expressions, you have something more <i>complex<\/i> than usual. Do not fear\u2014you have all the tools you need to simplify these quotients!\r\n\r\nA <strong>complex fraction<\/strong> is the quotient of two fractions. These complex fractions are never considered to be in simplest form, but they can always be simplified using division of fractions. Remember, to divide fractions, you multiply by the reciprocal.\r\n\r\n<i>Before<\/i> you multiply the numbers, it is often helpful to factor the fractions. You can then cancel factors.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nSimplify.\r\n<p style=\"text-align: center;\">[latex]\\displaystyle\\Large \\dfrac{\\,\\frac{12}{35}\\,}{\\,\\frac{6}{7}\\,}[\/latex]<\/p>\r\n\r\n<div class=\"qa-wrapper\" style=\"display: block;\">\r\n\r\n<span class=\"show-answer collapsed\" style=\"cursor: pointer;\" data-target=\"q770219\">Show Solution<\/span>\r\n<div id=\"q770219\" class=\"hidden-answer\" style=\"display: none;\">\r\n\r\nRewrite the complex fraction as a division problem.\r\n<p style=\"text-align: center;\">[latex]\\displaystyle\\large \\frac{\\,\\frac{12}{35}\\,}{\\,\\frac{6}{7}\\,}=\\normalsize\\frac{12}{35}\\div \\frac{6}{7}[\/latex]<\/p>\r\nRewrite the division as multiplication and take the reciprocal of the divisor.\r\n<p style=\"text-align: center;\">[latex] =\\frac{12}{35}\\cdot \\frac{7}{6}[\/latex]<\/p>\r\nFactor the numerator and denominator looking for common factors before multiplying numbers together.\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{l}=\\frac{2\\cdot 6\\cdot 7}{5\\cdot 7\\cdot 6}\\\\\\\\=\\frac{2}{5}\\cdot \\frac{6\\cdot 7}{6\\cdot 7}\\\\\\\\=\\frac{2}{5}\\cdot 1\\end{array}[\/latex]<\/p>\r\n[latex]\\displaystyle\\Large \\frac{\\,\\frac{12}{35}\\,}{\\,\\frac{6}{7}\\,}=\\normalsize\\frac{2}{5}[\/latex]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nSimplify.\r\n<p style=\"text-align: center;\">[latex]\\displaystyle\\Large \\dfrac{\\,\\frac{6}{28}\\,}{\\,\\frac{2}{7}\\,}[\/latex]<\/p>\r\n[reveal-answer q=\"703480\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"703480\"]\r\n[latex]\\frac{\\frac{6}{28}}{\\frac{2}{7}}=\\frac{3}{4}[\/latex]\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nIf two fractions appear in the numerator or denominator (or both), first combine them. Then simplify the quotient as shown above.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nSimplify.\r\n<p style=\"text-align: center;\">[latex]\\displaystyle\\Large \\frac{\\,\\frac{3}{4}+\\frac{1}{2}\\,}{\\,\\frac{4}{5}-\\frac{1}{10}\\,}[\/latex]<\/p>\r\n\r\n<div class=\"qa-wrapper\" style=\"display: block;\">\r\n\r\n<span class=\"show-answer collapsed\" style=\"cursor: pointer;\" data-target=\"q96511\">Show Solution<\/span>\r\n<div id=\"q96511\" class=\"hidden-answer\" style=\"display: none;\">\r\n\r\nFirst combine the numerator and denominator by adding or subtracting. You may need to find a common denominator first. Note that we do not show the steps for finding a common denominator, so please review that in the previous section if you are confused.\r\n<p style=\"text-align: center;\">[latex]\\displaystyle\\Large \\frac{\\,\\frac{3}{4}+\\frac{1}{2}\\,}{\\,\\frac{4}{5}-\\frac{1}{10}\\,}=\\frac{\\,\\frac{5}{4}\\,}{\\,\\frac{7}{10}\\,}[\/latex]<\/p>\r\nRewrite the complex fraction as a division problem.\r\n<p style=\"text-align: center;\">[latex]\\displaystyle\\Large \\frac{\\,\\,\\frac{5}{4}\\,\\,}{\\,\\,\\frac{7}{10}\\,\\,}=\\normalsize\\frac{5}{4}\\div \\frac{7}{10}[\/latex]<\/p>\r\nRewrite the division as multiplication and take the reciprocal of the divisor.\r\n<p style=\"text-align: center;\">[latex] =\\dfrac{5}{4}\\cdot \\frac{10}{7}[\/latex]<\/p>\r\nMultiply and simplify as needed.\r\n<p style=\"text-align: center;\">[latex]\\dfrac{5}{4}\\cdot \\frac{10}{7}=\\frac{5\\cdot5\\cdot2}{2\\cdot2\\cdot7}=\\frac{25}{14}[\/latex]<\/p>\r\n[latex]\\displaystyle\\Large \\frac{\\,\\frac{3}{4}+\\frac{1}{2}\\,}{\\,\\frac{4}{5}-\\frac{1}{10}\\,}=\\normalsize\\frac{25}{14}[\/latex]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nSimplify.\r\n<p style=\"text-align: center;\">[latex]\\displaystyle\\Large \\frac{\\,\\frac{1}{3}-\\frac{1}{4}\\,}{\\,\\frac{7}{12}-\\frac{1}{48}\\,}[\/latex]<\/p>\r\n[reveal-answer q=\"703490\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"703490\"]\r\n[latex]\\displaystyle\\Large \\frac{\\,\\frac{1}{3}-\\frac{1}{4}\\,}{\\,\\frac{7}{12}-\\frac{1}{48}\\,}=\\normalsize\\frac{4}{27}[\/latex]\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nIn the following video, we will show a couple more examples of how to simplify complex fractions.\r\n\r\n<iframe src=\"https:\/\/www.youtube.com\/embed\/lQCwze2w7OU?feature=oembed&amp;rel=0\" width=\"500\" height=\"281\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe>\r\n<h2>Simplifying Complex Rational Expressions<\/h2>\r\nA complex rational expression is a rational expression that contains additional rational expressions in the numerator, the denominator, or both. We can simplify complex rational expressions by rewriting the numerator and denominator as single rational expressions and dividing. The complex rational expression [latex]\\dfrac{a}{\\dfrac{1}{b}+c}[\/latex] can be simplified by rewriting the numerator as the fraction [latex]\\dfrac{a}{1}[\/latex] and combining the expressions in the denominator as [latex]\\dfrac{1+bc}{b}[\/latex]. We can then rewrite the expression as a multiplication problem using the reciprocal of the denominator. We get [latex]\\dfrac{a}{1}\\cdot \\dfrac{b}{1+bc}[\/latex] which is equal to [latex]\\dfrac{ab}{1+bc}[\/latex].\r\n<div class=\"textbox\">\r\n<h3>How To: Given a complex rational expression, simplify it<\/h3>\r\n<ol>\r\n \t<li>Combine the expressions in the numerator into a single rational expression by adding or subtracting.<\/li>\r\n \t<li>Combine the expressions in the denominator into a single rational expression by adding or subtracting.<\/li>\r\n \t<li>Rewrite as the numerator divided by the denominator.<\/li>\r\n \t<li>Rewrite as multiplication.<\/li>\r\n \t<li>Multiply.<\/li>\r\n \t<li>Simplify.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Simplifying Complex Rational Expressions<\/h3>\r\nSimplify: [latex]\\dfrac{y+\\dfrac{1}{x}}{\\dfrac{x}{y}}[\/latex] .\r\n\r\n[reveal-answer q=\"967019\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"967019\"]\r\n\r\nBegin by combining the expressions in the numerator into one expression.\r\n<div style=\"text-align: center;\">[latex]\\begin{array}{cc}y\\cdot \\dfrac{x}{x}+\\dfrac{1}{x}\\hfill &amp; \\text{Multiply by }\\dfrac{x}{x}\\text{to get LCD as denominator}.\\hfill \\\\ \\dfrac{xy}{x}+\\dfrac{1}{x}\\hfill &amp; \\\\ \\dfrac{xy+1}{x}\\hfill &amp; \\text{Add numerators}.\\hfill \\end{array}[\/latex]<\/div>\r\nNow the numerator is a single rational expression and the denominator is a single rational expression.\r\n<div style=\"text-align: center;\">[latex]\\dfrac{\\dfrac{xy+1}{x}}{\\dfrac{x}{y}}[\/latex]<\/div>\r\nWe can rewrite this as division and then multiplication.\r\n<div style=\"text-align: center;\">[latex]\\begin{array}{cc}\\dfrac{xy+1}{x}\\div \\dfrac{x}{y}\\hfill &amp; \\\\ \\dfrac{xy+1}{x}\\cdot \\dfrac{y}{x}\\hfill &amp; \\text{Rewrite as multiplication}\\text{.}\\hfill \\\\ \\dfrac{y\\left(xy+1\\right)}{{x}^{2}}\\hfill &amp; \\text{Multiply}\\text{.}\\hfill \\end{array}[\/latex]<\/div>\r\n<div>[\/hidden-answer]<\/div>\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nSimplify: [latex]\\dfrac{\\dfrac{x}{y}-\\dfrac{y}{x}}{y}[\/latex]\r\n\r\n[reveal-answer q=\"40643\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"40643\"]\r\n\r\n[latex]\\dfrac{{x}^{2}-{y}^{2}}{x{y}^{2}}[\/latex][\/hidden-answer]\r\n\r\n[ohm_question]3078-3080-59554[\/ohm_question]\r\n\r\n<\/div>\r\n<div class=\"textbox\">\r\n<h3>Q &amp; A<\/h3>\r\n<strong>Can a complex rational expression always be simplified?<\/strong>\r\n\r\n<em>Yes. We can always rewrite a complex rational expression as a simplified rational expression.<\/em>\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nSimplify.\r\n<p style=\"text-align: center;\">[latex]\\dfrac{\\dfrac{5x^2}{9}}{\\dfrac{15x^3}{27}}[\/latex]<\/p>\r\n\r\n<div class=\"qa-wrapper\" style=\"display: block;\">\r\n\r\n<span class=\"show-answer collapsed\" style=\"cursor: pointer;\" data-target=\"q688236\">Show Solution<\/span>\r\n<div id=\"q688236\" class=\"hidden-answer\" style=\"display: none;\">\r\n\r\nState the quotient in simplest form. Rewrite division as multiplication by the reciprocal.\r\n<p style=\"text-align: center;\">[latex]\\frac{5x^{2}}{9}\\cdot\\frac{27}{15x^{3}}[\/latex]<\/p>\r\nFactor the numerators and denominators.\r\n<p style=\"text-align: center;\">[latex]\\frac{5\\cdot{x}\\cdot{x}}{3\\cdot3}\\cdot\\frac{3\\cdot3\\cdot3}{5\\cdot3\\cdot{x}\\cdot{x}\\cdot{x}}[\/latex]<\/p>\r\nCancel common factors.\r\n<p style=\"text-align: center;\">[latex]\\large\\begin{array}{c}\\frac{\\cancel{5}\\cdot{\\cancel{x}}\\cdot{\\cancel{x}}}{\\cancel{3}\\cdot\\cancel{3}}\\cdot\\frac{\\cancel{3}\\cdot\\cancel{3}\\cdot\\cancel{3}}{\\cancel{5}\\cdot\\cancel{3}\\cdot{\\cancel{x}}\\cdot{\\cancel{x}}\\cdot{x}}=\\frac{1}{x}\\end{array}[\/latex]<\/p>\r\n[latex] \\displaystyle \\frac{5{{x}^{2}}}{9}\\div \\frac{15{{x}^{3}}}{27}=\\frac{1}{x}[\/latex]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nSimplify.\r\n<p style=\"text-align: center;\">[latex]\\dfrac{\\dfrac{3x^2}{x+2}}{\\dfrac{6x^4}{x^2+5x+6}}[\/latex]<\/p>\r\n<span class=\"show-answer collapsed\" style=\"cursor: pointer;\" data-target=\"q53255\">Show Solution<\/span>\r\n<div id=\"q53255\" class=\"hidden-answer\" style=\"display: none;\">\r\n\r\nRewrite division as multiplication by the reciprocal.\r\n<p style=\"text-align: center;\">[latex]\\frac{3x^{2}}{x+2}\\cdot\\frac{\\left(x^{2}+5x+6\\right)}{6x^{4}}[\/latex]<\/p>\r\nFactor the numerators and denominators.\r\n<p style=\"text-align: center;\">[latex]\\frac{3\\cdot{x}\\cdot{x}}{x+2}\\cdot\\frac{\\left(x+2\\right)\\left(x+3\\right)}{2\\cdot3\\cdot{x}\\cdot{x}\\cdot{x}\\cdot{x}}[\/latex]<\/p>\r\nCancel common factors.\r\n<p style=\"text-align: center;\">[latex]\\dfrac{\\cancel{3}\\cdot{\\cancel{x}}\\cdot{\\cancel{x}}}{\\cancel{x+2}}\\cdot\\frac{\\cancel{\\left(x+2\\right)}\\left(x+3\\right)}{2\\cdot\\cancel{3}\\cdot{\\cancel{x}}\\cdot{\\cancel{x}}\\cdot{x}\\cdot{x}}[\/latex]<\/p>\r\nSimplify.\r\n<p style=\"text-align: center;\">[latex]\\frac{(x+3)}{2{{x}^{2}}}[\/latex]<\/p>\r\n[latex] \\displaystyle \\frac{3{{x}^{2}}}{x+2}\\div \\frac{6{{x}^{4}}}{({{x}^{2}}+5x+6)}=\\frac{x+3}{2{{x}^{2}}}[\/latex].\r\n\r\n<\/div>\r\n<\/div>\r\nNotice that once you rewrite the division as multiplication by a reciprocal, you follow the same process you used to multiply rational expressions.\r\n\r\nIn the video that follows, we present another example of dividing rational expressions.\r\n\r\n<iframe src=\"https:\/\/www.youtube.com\/embed\/B1tigfgs268?feature=oembed&amp;rel=0\" width=\"500\" height=\"281\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nSimplify.\r\n<p style=\"text-align: center;\">[latex]\\dfrac{\\dfrac{16c^4d^3}{5c}}{\\dfrac{4cd^4}{d^2}}[\/latex]<\/p>\r\n<span class=\"show-answer collapsed\" style=\"cursor: pointer;\" data-target=\"q53260\">Show Solution<\/span>\r\n<div id=\"q53260\" class=\"hidden-answer\" style=\"display: none;\">\r\n\r\n[latex]\\dfrac{\\dfrac{16c^4d^3}{5c}}{\\dfrac{4cd^4}{d^2}}=\\dfrac{16c^4d^3}{5c} \\cdot \\dfrac{d^2}{4cd^4}=\\dfrac{16c^4d^5}{20c^2d^4}=\\dfrac{4c^2d}{5} [\/latex]\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nSimplify.\r\n<p style=\"text-align: center;\">[latex]\\displaystyle\\Large \\frac{\\,\\,\\frac{x+5}{{{x}^{2}}-16}\\,}{\\,\\,\\frac{{{x}^{2}}-\\,\\,25}{x-4}\\,}[\/latex]<\/p>\r\n\r\n<div class=\"qa-wrapper\" style=\"display: block;\">\r\n\r\n<span class=\"show-answer collapsed\" style=\"cursor: pointer;\" data-target=\"q245262\">Show Solution<\/span>\r\n<div id=\"q245262\" class=\"hidden-answer\" style=\"display: none;\">\r\n\r\nRewrite the complex rational expression as a division problem.\r\n<p style=\"text-align: center;\">[latex] =\\frac{x+5}{{{x}^{2}}-16}\\div \\frac{{{x}^{2}}-25}{x-4}[\/latex]<\/p>\r\nRewrite the division as multiplication and take the reciprocal of the divisor. Note that the excluded values for this are [latex]-4[\/latex], [latex]4[\/latex], [latex]-5[\/latex], and [latex]5[\/latex], because those values make the denominators of one of the fractions zero.\r\n<p style=\"text-align: center;\">[latex] =\\frac{x+5}{{{x}^{2}}-16}\\cdot \\frac{x-4}{{{x}^{2}}-25}[\/latex]<\/p>\r\nFactor the numerator and denominator, looking for common factors. In this case, [latex]x+5[\/latex] and [latex]x\u20134[\/latex] are common factors of the numerator and denominator.\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{l}=\\frac{(x+5)(x-4)}{(x+4)(x-4)(x+5)(x-5)}\\\\\\\\=\\frac{1}{(x+4)(x-5)}\\end{array}[\/latex]<\/p>\r\n[latex]\\displaystyle\\Large \\frac{\\,\\,\\frac{x+5}{{{x}^{2}}-16}\\,}{\\,\\,\\frac{{{x}^{2}}-25}{x-4}\\,}\\normalsize=\\frac{1}{(x+4)(x-5)},x\\ne -4,4,-5,5[\/latex]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nSimplify.\r\n<p style=\"text-align: center;\">[latex]\\displaystyle\\Large \\frac{\\,\\,\\frac{t}{t+9}\\,}{\\,\\,\\frac{4}{t^2-81}\\,}[\/latex]<\/p>\r\n\r\n<div class=\"qa-wrapper\" style=\"display: block;\">\r\n\r\n<span class=\"show-answer collapsed\" style=\"cursor: pointer;\" data-target=\"q245290\">Show Solution<\/span>\r\n<div id=\"q245290\" class=\"hidden-answer\" style=\"display: none;\">\r\n\r\n[latex]\\displaystyle\\Large \\frac{\\,\\,\\frac{t}{t+9}\\,}{\\,\\,\\frac{4}{t^2-81}\\,}\\normalsize=\\dfrac{t}{t+9} \\cdot \\dfrac{t^2-81}{4}=\\dfrac{t}{t+9} \\cdot \\dfrac{(t+9)(t-9)}{4}=\\frac{t(t-9)}{4}[\/latex]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\nIn the next video example, we will show that simplifying a complex fraction may require factoring first.\r\n\r\n<iframe src=\"https:\/\/www.youtube.com\/embed\/fAaqo8gGW9Y?feature=oembed&amp;rel=0\" width=\"500\" height=\"281\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe>\r\n\r\nThe same ideas can be used when simplifying complex rational expressions that include more than one rational expression in the numerator or denominator. However, there is a shortcut that can be used. Compare these two examples of simplifying a complex fraction.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nSimplify.\r\n<p style=\"text-align: center;\">[latex] \\displaystyle\\dfrac{\\,\\,\\normalsize1-\\dfrac{9}{{{x}^{2}}}\\,\\,}{\\,\\,\\normalsize1+\\dfrac{5}{x}\\normalsize+\\dfrac{6}{{{x}^{2}}}\\,\\,}[\/latex]<\/p>\r\n\r\n<div class=\"qa-wrapper\" style=\"display: block;\">\r\n\r\n<span class=\"show-answer collapsed\" style=\"cursor: pointer;\" data-target=\"q344101\">Show Solution<\/span>\r\n<div id=\"q344101\" class=\"hidden-answer\" style=\"display: none;\">\r\n\r\nCombine the expressions in the numerator and denominator. To do this, rewrite the expressions using a common denominator. There is an excluded value of [latex]0[\/latex] because this makes the denominators of the fractions zero.\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{l}=\\frac{\\frac{{{x}^{2}}}{{{x}^{2}}}-\\frac{9}{{{x}^{2}}}}{\\frac{{{x}^{2}}}{{{x}^{2}}}+\\frac{5x}{{{x}^{2}}}+\\frac{6}{{{x}^{2}}}}\\\\\\\\=\\frac{\\frac{{{x}^{2}}-9}{{{x}^{2}}}}{\\frac{{{x}^{2}}+5x+6}{{{x}^{2}}}}\\end{array}[\/latex]<\/p>\r\nRewrite the complex rational expression as a division problem. (When you are comfortable with the step of rewriting the complex rational fraction as a division problem, you might skip this step and go straight to rewriting it as multiplication.)\r\n<p style=\"text-align: center;\">[latex] =\\frac{{{x}^{2}}-9}{{{x}^{2}}}\\div \\frac{{{x}^{2}}+5x+6}{{{x}^{2}}}[\/latex]<\/p>\r\nRewrite the division as multiplication and take the reciprocal of the divisor.\r\n<p style=\"text-align: center;\">[latex] =\\frac{{{x}^{2}}-9}{{{x}^{2}}}\\cdot \\frac{{{x}^{2}}}{{{x}^{2}}+5x+6}[\/latex]<\/p>\r\nFactor the numerator and denominator looking for common factors. In this case, [latex]x+3[\/latex] and [latex]x^{2}[\/latex] are common factors. We can now see there are two additional excluded values, [latex]-2[\/latex] and [latex]-3[\/latex].\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}=\\frac{(x+3)(x-3){{x}^{2}}}{{{x}^{2}}(x+3)(x+2)}\\\\\\\\=\\frac{(x-3)}{(x+2)}\\cdot \\frac{{{x}^{2}}(x+3)}{{{x}^{2}}(x+3)}\\end{array}[\/latex]<\/p>\r\n[latex] \\frac{1-\\frac{9}{{{x}^{2}}}}{1+\\frac{5}{x}+\\frac{6}{{{x}^{2}}}}=\\frac{x-3}{x+2},x\\ne -3,-2,0[\/latex]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nSimplify.\r\n<p style=\"text-align: center;\">[latex] \\frac{1-\\frac{9}{{{x}^{2}}}}{1+\\frac{5}{x}+\\frac{6}{{{x}^{2}}}}[\/latex]<\/p>\r\n\r\n<div class=\"qa-wrapper\" style=\"display: block;\">\r\n\r\n<span class=\"show-answer collapsed\" style=\"cursor: pointer;\" data-target=\"q926024\">Show Solution<\/span>\r\n<div id=\"q926024\" class=\"hidden-answer\" style=\"display: none;\">\r\n\r\nBefore combining the expressions, find a common denominator for all of the rational expressions. In this case, [latex]x^{2}[\/latex] is a common denominator. Multiply by [latex]1[\/latex] in the form of a fraction with the common denominator in both the numerator and denominator. In this case, multiply by [latex] \\frac{{{x}^{2}}}{{{x}^{2}}}[\/latex]. There is an excluded value of [latex]0[\/latex] because this makes the denominators of the fractions zero.\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{l}=\\frac{1-\\frac{9}{{{x}^{2}}}}{1+\\frac{5}{x}+\\frac{6}{{{x}^{2}}}}\\cdot \\frac{{{x}^{2}}}{{{x}^{2}}}\\\\\\\\=\\frac{\\left( 1-\\frac{9}{{{x}^{2}}} \\right){{x}^{2}}}{\\left( 1+\\frac{5}{x}+\\frac{6}{{{x}^{2}}} \\right){{x}^{2}}}\\\\\\\\=\\frac{{{x}^{2}}-9}{{{x}^{2}}+5x+6}\\end{array}[\/latex]<\/p>\r\nNotice that the expression is no longer complex! You can simplify by factoring and identifying common factors. We can now see there are two additional excluded values, [latex]-2[\/latex] and [latex]-3[\/latex].\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{l}=\\frac{(x+3)(x-3)}{(x+3)(x+2)}\\\\\\\\=\\frac{x+3}{x+3}\\cdot \\frac{x-3}{x+2}\\\\\\\\=1\\cdot \\frac{x-3}{x+2}\\end{array}[\/latex]<\/p>\r\n[latex] \\frac{1-\\frac{9}{{{x}^{2}}}}{1+\\frac{5}{x}+\\frac{6}{{{x}^{2}}}}=\\frac{x-3}{x+2},x\\ne -3,-2,0[\/latex]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\nYou may find the second method easier to use, but do try both ways to see what you prefer.\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nSimplify.\r\n<p style=\"text-align: center;\">[latex] \\displaystyle\\dfrac{\\,\\,\\normalsize \\dfrac{7}{{{y}^2}}-\\dfrac{1}{{2y}}\\,\\,}{\\,\\,\\normalsize \\dfrac{2}{y}\\normalsize+\\dfrac{7}{{3y}}\\,\\,}[\/latex]<\/p>\r\n\r\n<div class=\"qa-wrapper\" style=\"display: block;\">\r\n\r\n<span class=\"show-answer collapsed\" style=\"cursor: pointer;\" data-target=\"q926020\">Show Solution<\/span>\r\n<div id=\"q926020\" class=\"hidden-answer\" style=\"display: none;\">\r\n\r\n[latex] \\displaystyle\\dfrac{\\,\\,\\normalsize \\dfrac{7}{{{y}^2}}-\\dfrac{1}{{2y}}\\,\\,}{\\,\\,\\normalsize \\dfrac{2}{y}\\normalsize+\\dfrac{7}{{3y}}\\,\\,}=\\frac{3(14-y)}{26y}[\/latex]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\nWatch the video example below for a similar problem.\r\n\r\n<iframe src=\"https:\/\/www.youtube.com\/embed\/P5dfmX_FNPk?feature=oembed&amp;rel=0\" width=\"500\" height=\"281\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe>\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nSimplify.\r\n<p style=\"text-align: center;\">[latex] \\displaystyle\\dfrac{\\,\\,\\normalsize \\dfrac{7}{y}+y\\,\\,}{\\,\\,\\normalsize \\dfrac{5}{y}\\normalsize-y\\,\\,}[\/latex]<\/p>\r\n\r\n<div class=\"qa-wrapper\" style=\"display: block;\">\r\n\r\n<span class=\"show-answer collapsed\" style=\"cursor: pointer;\" data-target=\"q344100\">Show Solution<\/span>\r\n<div id=\"q344100\" class=\"hidden-answer\" style=\"display: none;\">\r\n\r\nFirst, find the LCD of the denominators that occur in the numerator of the complex fraction.\r\n\r\nIf necessary, factor the denominators before finding the LCD. In this problem, there is just one denominator, [latex]y[\/latex]. The LCD of the numerator is [latex]y[\/latex].\r\n\r\nMultiply the [latex]y[\/latex] in the numerator by [latex]\\frac{y}{y}[\/latex], so that the denominators match the LCD.\r\n\r\n[latex]\\dfrac{\\dfrac{7}{y}+y}{\\dfrac{5}{y}-y}=\\dfrac{\\dfrac{7}{y}+y \\cdot \\dfrac{y}{y}}{\\dfrac{5}{y}-y}=\\dfrac{\\dfrac{7}{y}+\\dfrac{y^2}{y}}{\\dfrac{5}{y}-y}[\/latex]\r\n\r\nAdd the fractions in the numerator.\r\n\r\n[latex]\\dfrac{\\dfrac{7}{y}+\\dfrac{y^2}{y}}{\\dfrac{5}{y}-y}=\\dfrac{\\dfrac{7+y^2}{y}}{\\dfrac{5}{y}-y}[\/latex]\r\n\r\nAt this point, repeat the preceding steps for the denominator of the complex fraction.\r\n\r\n[latex]\\dfrac{\\dfrac{7+y^2}{y}}{\\dfrac{5}{y}-y}=\\dfrac{\\dfrac{7+y^2}{y}}{\\dfrac{5}{y}-y \\cdot \\dfrac{y}{y}}=\\dfrac{\\dfrac{7+y^2}{y}}{\\dfrac{5}{y}-\\dfrac{y^2}{y}}=\\dfrac{\\dfrac{7+y^2}{y}}{\\dfrac{5-y^2}{y}}[\/latex]\r\n\r\nYou are now ready to divide the numerator by the denominator. To accomplish this, multiply the numerator by the reciprocal of the denominator.\r\n\r\n[latex]\\dfrac{\\dfrac{7+y^2}{y}}{\\dfrac{5-y^2}{y}}=\\dfrac{7+y^2}{y} \\cdot \\dfrac{y}{5-y^2}[\/latex]\r\n\r\nThe [latex]y[\/latex] on the top and bottom cancel, therefore:\r\n[latex]\\dfrac{\\dfrac{7}{y}+y}{\\dfrac{5}{y}-y}=\\dfrac{7+y^2}{5-y^2}[\/latex]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nSimplify.\r\n<p style=\"text-align: center;\">[latex] \\displaystyle\\dfrac{\\,\\,\\normalsize \\dfrac{9}{x}-x\\,\\,}{\\,\\,\\normalsize \\dfrac{11}{x}\\normalsize+x\\,\\,}[\/latex]<\/p>\r\n\r\n<div class=\"qa-wrapper\" style=\"display: block;\">\r\n\r\n<span class=\"show-answer collapsed\" style=\"cursor: pointer;\" data-target=\"q344200\">Show Solution<\/span>\r\n<div id=\"q344200\" class=\"hidden-answer\" style=\"display: none;\">\r\n\r\n[latex]\\dfrac{\\dfrac{9}{x}-x}{\\dfrac{11}{x}+x}=\\dfrac{9-x^2}{x^2+11}[\/latex]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<h2>Summary<\/h2>\r\nComplex rational expressions are quotients with rational expressions in the divisor, dividend, or both. When written in fraction form, they appear to be fractions within a fraction. These can be simplified by first treating the quotient as a division problem. Then you can rewrite the division as multiplication and take the reciprocal of the divisor. Or you can simplify the complex rational expression by multiplying both the numerator and denominator by a denominator common to all rational expressions within the complex expression. This can help simplify the complex expression even faster.","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Simplify complex rational expressions<\/li>\n<\/ul>\n<\/div>\n<p>Fractions and rational expressions can be interpreted as quotients. When both the dividend (numerator) and divisor (denominator) include fractions or rational expressions, you have something more <i>complex<\/i> than usual. Do not fear\u2014you have all the tools you need to simplify these quotients!<\/p>\n<p>A <strong>complex fraction<\/strong> is the quotient of two fractions. These complex fractions are never considered to be in simplest form, but they can always be simplified using division of fractions. Remember, to divide fractions, you multiply by the reciprocal.<\/p>\n<p><i>Before<\/i> you multiply the numbers, it is often helpful to factor the fractions. You can then cancel factors.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Simplify.<\/p>\n<p style=\"text-align: center;\">[latex]\\displaystyle\\Large \\dfrac{\\,\\frac{12}{35}\\,}{\\,\\frac{6}{7}\\,}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block;\">\n<p><span class=\"show-answer collapsed\" style=\"cursor: pointer;\" data-target=\"q770219\">Show Solution<\/span><\/p>\n<div id=\"q770219\" class=\"hidden-answer\" style=\"display: none;\">\n<p>Rewrite the complex fraction as a division problem.<\/p>\n<p style=\"text-align: center;\">[latex]\\displaystyle\\large \\frac{\\,\\frac{12}{35}\\,}{\\,\\frac{6}{7}\\,}=\\normalsize\\frac{12}{35}\\div \\frac{6}{7}[\/latex]<\/p>\n<p>Rewrite the division as multiplication and take the reciprocal of the divisor.<\/p>\n<p style=\"text-align: center;\">[latex]=\\frac{12}{35}\\cdot \\frac{7}{6}[\/latex]<\/p>\n<p>Factor the numerator and denominator looking for common factors before multiplying numbers together.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}=\\frac{2\\cdot 6\\cdot 7}{5\\cdot 7\\cdot 6}\\\\\\\\=\\frac{2}{5}\\cdot \\frac{6\\cdot 7}{6\\cdot 7}\\\\\\\\=\\frac{2}{5}\\cdot 1\\end{array}[\/latex]<\/p>\n<p>[latex]\\displaystyle\\Large \\frac{\\,\\frac{12}{35}\\,}{\\,\\frac{6}{7}\\,}=\\normalsize\\frac{2}{5}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Simplify.<\/p>\n<p style=\"text-align: center;\">[latex]\\displaystyle\\Large \\dfrac{\\,\\frac{6}{28}\\,}{\\,\\frac{2}{7}\\,}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q703480\">Show Solution<\/span><\/p>\n<div id=\"q703480\" class=\"hidden-answer\" style=\"display: none\">\n[latex]\\frac{\\frac{6}{28}}{\\frac{2}{7}}=\\frac{3}{4}[\/latex]\n<\/div>\n<\/div>\n<\/div>\n<p>If two fractions appear in the numerator or denominator (or both), first combine them. Then simplify the quotient as shown above.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Simplify.<\/p>\n<p style=\"text-align: center;\">[latex]\\displaystyle\\Large \\frac{\\,\\frac{3}{4}+\\frac{1}{2}\\,}{\\,\\frac{4}{5}-\\frac{1}{10}\\,}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block;\">\n<p><span class=\"show-answer collapsed\" style=\"cursor: pointer;\" data-target=\"q96511\">Show Solution<\/span><\/p>\n<div id=\"q96511\" class=\"hidden-answer\" style=\"display: none;\">\n<p>First combine the numerator and denominator by adding or subtracting. You may need to find a common denominator first. Note that we do not show the steps for finding a common denominator, so please review that in the previous section if you are confused.<\/p>\n<p style=\"text-align: center;\">[latex]\\displaystyle\\Large \\frac{\\,\\frac{3}{4}+\\frac{1}{2}\\,}{\\,\\frac{4}{5}-\\frac{1}{10}\\,}=\\frac{\\,\\frac{5}{4}\\,}{\\,\\frac{7}{10}\\,}[\/latex]<\/p>\n<p>Rewrite the complex fraction as a division problem.<\/p>\n<p style=\"text-align: center;\">[latex]\\displaystyle\\Large \\frac{\\,\\,\\frac{5}{4}\\,\\,}{\\,\\,\\frac{7}{10}\\,\\,}=\\normalsize\\frac{5}{4}\\div \\frac{7}{10}[\/latex]<\/p>\n<p>Rewrite the division as multiplication and take the reciprocal of the divisor.<\/p>\n<p style=\"text-align: center;\">[latex]=\\dfrac{5}{4}\\cdot \\frac{10}{7}[\/latex]<\/p>\n<p>Multiply and simplify as needed.<\/p>\n<p style=\"text-align: center;\">[latex]\\dfrac{5}{4}\\cdot \\frac{10}{7}=\\frac{5\\cdot5\\cdot2}{2\\cdot2\\cdot7}=\\frac{25}{14}[\/latex]<\/p>\n<p>[latex]\\displaystyle\\Large \\frac{\\,\\frac{3}{4}+\\frac{1}{2}\\,}{\\,\\frac{4}{5}-\\frac{1}{10}\\,}=\\normalsize\\frac{25}{14}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Simplify.<\/p>\n<p style=\"text-align: center;\">[latex]\\displaystyle\\Large \\frac{\\,\\frac{1}{3}-\\frac{1}{4}\\,}{\\,\\frac{7}{12}-\\frac{1}{48}\\,}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q703490\">Show Solution<\/span><\/p>\n<div id=\"q703490\" class=\"hidden-answer\" style=\"display: none\">\n[latex]\\displaystyle\\Large \\frac{\\,\\frac{1}{3}-\\frac{1}{4}\\,}{\\,\\frac{7}{12}-\\frac{1}{48}\\,}=\\normalsize\\frac{4}{27}[\/latex]\n<\/div>\n<\/div>\n<\/div>\n<p>In the following video, we will show a couple more examples of how to simplify complex fractions.<\/p>\n<p><iframe loading=\"lazy\" src=\"https:\/\/www.youtube.com\/embed\/lQCwze2w7OU?feature=oembed&amp;rel=0\" width=\"500\" height=\"281\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h2>Simplifying Complex Rational Expressions<\/h2>\n<p>A complex rational expression is a rational expression that contains additional rational expressions in the numerator, the denominator, or both. We can simplify complex rational expressions by rewriting the numerator and denominator as single rational expressions and dividing. The complex rational expression [latex]\\dfrac{a}{\\dfrac{1}{b}+c}[\/latex] can be simplified by rewriting the numerator as the fraction [latex]\\dfrac{a}{1}[\/latex] and combining the expressions in the denominator as [latex]\\dfrac{1+bc}{b}[\/latex]. We can then rewrite the expression as a multiplication problem using the reciprocal of the denominator. We get [latex]\\dfrac{a}{1}\\cdot \\dfrac{b}{1+bc}[\/latex] which is equal to [latex]\\dfrac{ab}{1+bc}[\/latex].<\/p>\n<div class=\"textbox\">\n<h3>How To: Given a complex rational expression, simplify it<\/h3>\n<ol>\n<li>Combine the expressions in the numerator into a single rational expression by adding or subtracting.<\/li>\n<li>Combine the expressions in the denominator into a single rational expression by adding or subtracting.<\/li>\n<li>Rewrite as the numerator divided by the denominator.<\/li>\n<li>Rewrite as multiplication.<\/li>\n<li>Multiply.<\/li>\n<li>Simplify.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Simplifying Complex Rational Expressions<\/h3>\n<p>Simplify: [latex]\\dfrac{y+\\dfrac{1}{x}}{\\dfrac{x}{y}}[\/latex] .<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q967019\">Show Solution<\/span><\/p>\n<div id=\"q967019\" class=\"hidden-answer\" style=\"display: none\">\n<p>Begin by combining the expressions in the numerator into one expression.<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{array}{cc}y\\cdot \\dfrac{x}{x}+\\dfrac{1}{x}\\hfill & \\text{Multiply by }\\dfrac{x}{x}\\text{to get LCD as denominator}.\\hfill \\\\ \\dfrac{xy}{x}+\\dfrac{1}{x}\\hfill & \\\\ \\dfrac{xy+1}{x}\\hfill & \\text{Add numerators}.\\hfill \\end{array}[\/latex]<\/div>\n<p>Now the numerator is a single rational expression and the denominator is a single rational expression.<\/p>\n<div style=\"text-align: center;\">[latex]\\dfrac{\\dfrac{xy+1}{x}}{\\dfrac{x}{y}}[\/latex]<\/div>\n<p>We can rewrite this as division and then multiplication.<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{array}{cc}\\dfrac{xy+1}{x}\\div \\dfrac{x}{y}\\hfill & \\\\ \\dfrac{xy+1}{x}\\cdot \\dfrac{y}{x}\\hfill & \\text{Rewrite as multiplication}\\text{.}\\hfill \\\\ \\dfrac{y\\left(xy+1\\right)}{{x}^{2}}\\hfill & \\text{Multiply}\\text{.}\\hfill \\end{array}[\/latex]<\/div>\n<div><\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Simplify: [latex]\\dfrac{\\dfrac{x}{y}-\\dfrac{y}{x}}{y}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q40643\">Show Solution<\/span><\/p>\n<div id=\"q40643\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]\\dfrac{{x}^{2}-{y}^{2}}{x{y}^{2}}[\/latex]<\/p><\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"ohm3078\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=3078-3080-59554&theme=oea&iframe_resize_id=ohm3078&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<div class=\"textbox\">\n<h3>Q &amp; A<\/h3>\n<p><strong>Can a complex rational expression always be simplified?<\/strong><\/p>\n<p><em>Yes. We can always rewrite a complex rational expression as a simplified rational expression.<\/em><\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Simplify.<\/p>\n<p style=\"text-align: center;\">[latex]\\dfrac{\\dfrac{5x^2}{9}}{\\dfrac{15x^3}{27}}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block;\">\n<p><span class=\"show-answer collapsed\" style=\"cursor: pointer;\" data-target=\"q688236\">Show Solution<\/span><\/p>\n<div id=\"q688236\" class=\"hidden-answer\" style=\"display: none;\">\n<p>State the quotient in simplest form. Rewrite division as multiplication by the reciprocal.<\/p>\n<p style=\"text-align: center;\">[latex]\\frac{5x^{2}}{9}\\cdot\\frac{27}{15x^{3}}[\/latex]<\/p>\n<p>Factor the numerators and denominators.<\/p>\n<p style=\"text-align: center;\">[latex]\\frac{5\\cdot{x}\\cdot{x}}{3\\cdot3}\\cdot\\frac{3\\cdot3\\cdot3}{5\\cdot3\\cdot{x}\\cdot{x}\\cdot{x}}[\/latex]<\/p>\n<p>Cancel common factors.<\/p>\n<p style=\"text-align: center;\">[latex]\\large\\begin{array}{c}\\frac{\\cancel{5}\\cdot{\\cancel{x}}\\cdot{\\cancel{x}}}{\\cancel{3}\\cdot\\cancel{3}}\\cdot\\frac{\\cancel{3}\\cdot\\cancel{3}\\cdot\\cancel{3}}{\\cancel{5}\\cdot\\cancel{3}\\cdot{\\cancel{x}}\\cdot{\\cancel{x}}\\cdot{x}}=\\frac{1}{x}\\end{array}[\/latex]<\/p>\n<p>[latex]\\displaystyle \\frac{5{{x}^{2}}}{9}\\div \\frac{15{{x}^{3}}}{27}=\\frac{1}{x}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Simplify.<\/p>\n<p style=\"text-align: center;\">[latex]\\dfrac{\\dfrac{3x^2}{x+2}}{\\dfrac{6x^4}{x^2+5x+6}}[\/latex]<\/p>\n<p><span class=\"show-answer collapsed\" style=\"cursor: pointer;\" data-target=\"q53255\">Show Solution<\/span><\/p>\n<div id=\"q53255\" class=\"hidden-answer\" style=\"display: none;\">\n<p>Rewrite division as multiplication by the reciprocal.<\/p>\n<p style=\"text-align: center;\">[latex]\\frac{3x^{2}}{x+2}\\cdot\\frac{\\left(x^{2}+5x+6\\right)}{6x^{4}}[\/latex]<\/p>\n<p>Factor the numerators and denominators.<\/p>\n<p style=\"text-align: center;\">[latex]\\frac{3\\cdot{x}\\cdot{x}}{x+2}\\cdot\\frac{\\left(x+2\\right)\\left(x+3\\right)}{2\\cdot3\\cdot{x}\\cdot{x}\\cdot{x}\\cdot{x}}[\/latex]<\/p>\n<p>Cancel common factors.<\/p>\n<p style=\"text-align: center;\">[latex]\\dfrac{\\cancel{3}\\cdot{\\cancel{x}}\\cdot{\\cancel{x}}}{\\cancel{x+2}}\\cdot\\frac{\\cancel{\\left(x+2\\right)}\\left(x+3\\right)}{2\\cdot\\cancel{3}\\cdot{\\cancel{x}}\\cdot{\\cancel{x}}\\cdot{x}\\cdot{x}}[\/latex]<\/p>\n<p>Simplify.<\/p>\n<p style=\"text-align: center;\">[latex]\\frac{(x+3)}{2{{x}^{2}}}[\/latex]<\/p>\n<p>[latex]\\displaystyle \\frac{3{{x}^{2}}}{x+2}\\div \\frac{6{{x}^{4}}}{({{x}^{2}}+5x+6)}=\\frac{x+3}{2{{x}^{2}}}[\/latex].<\/p>\n<\/div>\n<\/div>\n<p>Notice that once you rewrite the division as multiplication by a reciprocal, you follow the same process you used to multiply rational expressions.<\/p>\n<p>In the video that follows, we present another example of dividing rational expressions.<\/p>\n<p><iframe loading=\"lazy\" src=\"https:\/\/www.youtube.com\/embed\/B1tigfgs268?feature=oembed&amp;rel=0\" width=\"500\" height=\"281\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Simplify.<\/p>\n<p style=\"text-align: center;\">[latex]\\dfrac{\\dfrac{16c^4d^3}{5c}}{\\dfrac{4cd^4}{d^2}}[\/latex]<\/p>\n<p><span class=\"show-answer collapsed\" style=\"cursor: pointer;\" data-target=\"q53260\">Show Solution<\/span><\/p>\n<div id=\"q53260\" class=\"hidden-answer\" style=\"display: none;\">\n<p>[latex]\\dfrac{\\dfrac{16c^4d^3}{5c}}{\\dfrac{4cd^4}{d^2}}=\\dfrac{16c^4d^3}{5c} \\cdot \\dfrac{d^2}{4cd^4}=\\dfrac{16c^4d^5}{20c^2d^4}=\\dfrac{4c^2d}{5}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Simplify.<\/p>\n<p style=\"text-align: center;\">[latex]\\displaystyle\\Large \\frac{\\,\\,\\frac{x+5}{{{x}^{2}}-16}\\,}{\\,\\,\\frac{{{x}^{2}}-\\,\\,25}{x-4}\\,}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block;\">\n<p><span class=\"show-answer collapsed\" style=\"cursor: pointer;\" data-target=\"q245262\">Show Solution<\/span><\/p>\n<div id=\"q245262\" class=\"hidden-answer\" style=\"display: none;\">\n<p>Rewrite the complex rational expression as a division problem.<\/p>\n<p style=\"text-align: center;\">[latex]=\\frac{x+5}{{{x}^{2}}-16}\\div \\frac{{{x}^{2}}-25}{x-4}[\/latex]<\/p>\n<p>Rewrite the division as multiplication and take the reciprocal of the divisor. Note that the excluded values for this are [latex]-4[\/latex], [latex]4[\/latex], [latex]-5[\/latex], and [latex]5[\/latex], because those values make the denominators of one of the fractions zero.<\/p>\n<p style=\"text-align: center;\">[latex]=\\frac{x+5}{{{x}^{2}}-16}\\cdot \\frac{x-4}{{{x}^{2}}-25}[\/latex]<\/p>\n<p>Factor the numerator and denominator, looking for common factors. In this case, [latex]x+5[\/latex] and [latex]x\u20134[\/latex] are common factors of the numerator and denominator.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}=\\frac{(x+5)(x-4)}{(x+4)(x-4)(x+5)(x-5)}\\\\\\\\=\\frac{1}{(x+4)(x-5)}\\end{array}[\/latex]<\/p>\n<p>[latex]\\displaystyle\\Large \\frac{\\,\\,\\frac{x+5}{{{x}^{2}}-16}\\,}{\\,\\,\\frac{{{x}^{2}}-25}{x-4}\\,}\\normalsize=\\frac{1}{(x+4)(x-5)},x\\ne -4,4,-5,5[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Simplify.<\/p>\n<p style=\"text-align: center;\">[latex]\\displaystyle\\Large \\frac{\\,\\,\\frac{t}{t+9}\\,}{\\,\\,\\frac{4}{t^2-81}\\,}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block;\">\n<p><span class=\"show-answer collapsed\" style=\"cursor: pointer;\" data-target=\"q245290\">Show Solution<\/span><\/p>\n<div id=\"q245290\" class=\"hidden-answer\" style=\"display: none;\">\n<p>[latex]\\displaystyle\\Large \\frac{\\,\\,\\frac{t}{t+9}\\,}{\\,\\,\\frac{4}{t^2-81}\\,}\\normalsize=\\dfrac{t}{t+9} \\cdot \\dfrac{t^2-81}{4}=\\dfrac{t}{t+9} \\cdot \\dfrac{(t+9)(t-9)}{4}=\\frac{t(t-9)}{4}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>In the next video example, we will show that simplifying a complex fraction may require factoring first.<\/p>\n<p><iframe loading=\"lazy\" src=\"https:\/\/www.youtube.com\/embed\/fAaqo8gGW9Y?feature=oembed&amp;rel=0\" width=\"500\" height=\"281\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>The same ideas can be used when simplifying complex rational expressions that include more than one rational expression in the numerator or denominator. However, there is a shortcut that can be used. Compare these two examples of simplifying a complex fraction.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Simplify.<\/p>\n<p style=\"text-align: center;\">[latex]\\displaystyle\\dfrac{\\,\\,\\normalsize1-\\dfrac{9}{{{x}^{2}}}\\,\\,}{\\,\\,\\normalsize1+\\dfrac{5}{x}\\normalsize+\\dfrac{6}{{{x}^{2}}}\\,\\,}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block;\">\n<p><span class=\"show-answer collapsed\" style=\"cursor: pointer;\" data-target=\"q344101\">Show Solution<\/span><\/p>\n<div id=\"q344101\" class=\"hidden-answer\" style=\"display: none;\">\n<p>Combine the expressions in the numerator and denominator. To do this, rewrite the expressions using a common denominator. There is an excluded value of [latex]0[\/latex] because this makes the denominators of the fractions zero.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}=\\frac{\\frac{{{x}^{2}}}{{{x}^{2}}}-\\frac{9}{{{x}^{2}}}}{\\frac{{{x}^{2}}}{{{x}^{2}}}+\\frac{5x}{{{x}^{2}}}+\\frac{6}{{{x}^{2}}}}\\\\\\\\=\\frac{\\frac{{{x}^{2}}-9}{{{x}^{2}}}}{\\frac{{{x}^{2}}+5x+6}{{{x}^{2}}}}\\end{array}[\/latex]<\/p>\n<p>Rewrite the complex rational expression as a division problem. (When you are comfortable with the step of rewriting the complex rational fraction as a division problem, you might skip this step and go straight to rewriting it as multiplication.)<\/p>\n<p style=\"text-align: center;\">[latex]=\\frac{{{x}^{2}}-9}{{{x}^{2}}}\\div \\frac{{{x}^{2}}+5x+6}{{{x}^{2}}}[\/latex]<\/p>\n<p>Rewrite the division as multiplication and take the reciprocal of the divisor.<\/p>\n<p style=\"text-align: center;\">[latex]=\\frac{{{x}^{2}}-9}{{{x}^{2}}}\\cdot \\frac{{{x}^{2}}}{{{x}^{2}}+5x+6}[\/latex]<\/p>\n<p>Factor the numerator and denominator looking for common factors. In this case, [latex]x+3[\/latex] and [latex]x^{2}[\/latex] are common factors. We can now see there are two additional excluded values, [latex]-2[\/latex] and [latex]-3[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}=\\frac{(x+3)(x-3){{x}^{2}}}{{{x}^{2}}(x+3)(x+2)}\\\\\\\\=\\frac{(x-3)}{(x+2)}\\cdot \\frac{{{x}^{2}}(x+3)}{{{x}^{2}}(x+3)}\\end{array}[\/latex]<\/p>\n<p>[latex]\\frac{1-\\frac{9}{{{x}^{2}}}}{1+\\frac{5}{x}+\\frac{6}{{{x}^{2}}}}=\\frac{x-3}{x+2},x\\ne -3,-2,0[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Simplify.<\/p>\n<p style=\"text-align: center;\">[latex]\\frac{1-\\frac{9}{{{x}^{2}}}}{1+\\frac{5}{x}+\\frac{6}{{{x}^{2}}}}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block;\">\n<p><span class=\"show-answer collapsed\" style=\"cursor: pointer;\" data-target=\"q926024\">Show Solution<\/span><\/p>\n<div id=\"q926024\" class=\"hidden-answer\" style=\"display: none;\">\n<p>Before combining the expressions, find a common denominator for all of the rational expressions. In this case, [latex]x^{2}[\/latex] is a common denominator. Multiply by [latex]1[\/latex] in the form of a fraction with the common denominator in both the numerator and denominator. In this case, multiply by [latex]\\frac{{{x}^{2}}}{{{x}^{2}}}[\/latex]. There is an excluded value of [latex]0[\/latex] because this makes the denominators of the fractions zero.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}=\\frac{1-\\frac{9}{{{x}^{2}}}}{1+\\frac{5}{x}+\\frac{6}{{{x}^{2}}}}\\cdot \\frac{{{x}^{2}}}{{{x}^{2}}}\\\\\\\\=\\frac{\\left( 1-\\frac{9}{{{x}^{2}}} \\right){{x}^{2}}}{\\left( 1+\\frac{5}{x}+\\frac{6}{{{x}^{2}}} \\right){{x}^{2}}}\\\\\\\\=\\frac{{{x}^{2}}-9}{{{x}^{2}}+5x+6}\\end{array}[\/latex]<\/p>\n<p>Notice that the expression is no longer complex! You can simplify by factoring and identifying common factors. We can now see there are two additional excluded values, [latex]-2[\/latex] and [latex]-3[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}=\\frac{(x+3)(x-3)}{(x+3)(x+2)}\\\\\\\\=\\frac{x+3}{x+3}\\cdot \\frac{x-3}{x+2}\\\\\\\\=1\\cdot \\frac{x-3}{x+2}\\end{array}[\/latex]<\/p>\n<p>[latex]\\frac{1-\\frac{9}{{{x}^{2}}}}{1+\\frac{5}{x}+\\frac{6}{{{x}^{2}}}}=\\frac{x-3}{x+2},x\\ne -3,-2,0[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>You may find the second method easier to use, but do try both ways to see what you prefer.<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Simplify.<\/p>\n<p style=\"text-align: center;\">[latex]\\displaystyle\\dfrac{\\,\\,\\normalsize \\dfrac{7}{{{y}^2}}-\\dfrac{1}{{2y}}\\,\\,}{\\,\\,\\normalsize \\dfrac{2}{y}\\normalsize+\\dfrac{7}{{3y}}\\,\\,}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block;\">\n<p><span class=\"show-answer collapsed\" style=\"cursor: pointer;\" data-target=\"q926020\">Show Solution<\/span><\/p>\n<div id=\"q926020\" class=\"hidden-answer\" style=\"display: none;\">\n<p>[latex]\\displaystyle\\dfrac{\\,\\,\\normalsize \\dfrac{7}{{{y}^2}}-\\dfrac{1}{{2y}}\\,\\,}{\\,\\,\\normalsize \\dfrac{2}{y}\\normalsize+\\dfrac{7}{{3y}}\\,\\,}=\\frac{3(14-y)}{26y}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>Watch the video example below for a similar problem.<\/p>\n<p><iframe loading=\"lazy\" src=\"https:\/\/www.youtube.com\/embed\/P5dfmX_FNPk?feature=oembed&amp;rel=0\" width=\"500\" height=\"281\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Simplify.<\/p>\n<p style=\"text-align: center;\">[latex]\\displaystyle\\dfrac{\\,\\,\\normalsize \\dfrac{7}{y}+y\\,\\,}{\\,\\,\\normalsize \\dfrac{5}{y}\\normalsize-y\\,\\,}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block;\">\n<p><span class=\"show-answer collapsed\" style=\"cursor: pointer;\" data-target=\"q344100\">Show Solution<\/span><\/p>\n<div id=\"q344100\" class=\"hidden-answer\" style=\"display: none;\">\n<p>First, find the LCD of the denominators that occur in the numerator of the complex fraction.<\/p>\n<p>If necessary, factor the denominators before finding the LCD. In this problem, there is just one denominator, [latex]y[\/latex]. The LCD of the numerator is [latex]y[\/latex].<\/p>\n<p>Multiply the [latex]y[\/latex] in the numerator by [latex]\\frac{y}{y}[\/latex], so that the denominators match the LCD.<\/p>\n<p>[latex]\\dfrac{\\dfrac{7}{y}+y}{\\dfrac{5}{y}-y}=\\dfrac{\\dfrac{7}{y}+y \\cdot \\dfrac{y}{y}}{\\dfrac{5}{y}-y}=\\dfrac{\\dfrac{7}{y}+\\dfrac{y^2}{y}}{\\dfrac{5}{y}-y}[\/latex]<\/p>\n<p>Add the fractions in the numerator.<\/p>\n<p>[latex]\\dfrac{\\dfrac{7}{y}+\\dfrac{y^2}{y}}{\\dfrac{5}{y}-y}=\\dfrac{\\dfrac{7+y^2}{y}}{\\dfrac{5}{y}-y}[\/latex]<\/p>\n<p>At this point, repeat the preceding steps for the denominator of the complex fraction.<\/p>\n<p>[latex]\\dfrac{\\dfrac{7+y^2}{y}}{\\dfrac{5}{y}-y}=\\dfrac{\\dfrac{7+y^2}{y}}{\\dfrac{5}{y}-y \\cdot \\dfrac{y}{y}}=\\dfrac{\\dfrac{7+y^2}{y}}{\\dfrac{5}{y}-\\dfrac{y^2}{y}}=\\dfrac{\\dfrac{7+y^2}{y}}{\\dfrac{5-y^2}{y}}[\/latex]<\/p>\n<p>You are now ready to divide the numerator by the denominator. To accomplish this, multiply the numerator by the reciprocal of the denominator.<\/p>\n<p>[latex]\\dfrac{\\dfrac{7+y^2}{y}}{\\dfrac{5-y^2}{y}}=\\dfrac{7+y^2}{y} \\cdot \\dfrac{y}{5-y^2}[\/latex]<\/p>\n<p>The [latex]y[\/latex] on the top and bottom cancel, therefore:<br \/>\n[latex]\\dfrac{\\dfrac{7}{y}+y}{\\dfrac{5}{y}-y}=\\dfrac{7+y^2}{5-y^2}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Simplify.<\/p>\n<p style=\"text-align: center;\">[latex]\\displaystyle\\dfrac{\\,\\,\\normalsize \\dfrac{9}{x}-x\\,\\,}{\\,\\,\\normalsize \\dfrac{11}{x}\\normalsize+x\\,\\,}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block;\">\n<p><span class=\"show-answer collapsed\" style=\"cursor: pointer;\" data-target=\"q344200\">Show Solution<\/span><\/p>\n<div id=\"q344200\" class=\"hidden-answer\" style=\"display: none;\">\n<p>[latex]\\dfrac{\\dfrac{9}{x}-x}{\\dfrac{11}{x}+x}=\\dfrac{9-x^2}{x^2+11}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<h2>Summary<\/h2>\n<p>Complex rational expressions are quotients with rational expressions in the divisor, dividend, or both. When written in fraction form, they appear to be fractions within a fraction. These can be simplified by first treating the quotient as a division problem. Then you can rewrite the division as multiplication and take the reciprocal of the divisor. Or you can simplify the complex rational expression by multiplying both the numerator and denominator by a denominator common to all rational expressions within the complex expression. This can help simplify the complex expression even faster.<\/p>\n","protected":false},"author":264444,"menu_order":17,"template":"","meta":{"_candela_citation":"[]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-18389","chapter","type-chapter","status-publish","hentry"],"part":18142,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/pressbooks\/v2\/chapters\/18389","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/wp\/v2\/users\/264444"}],"version-history":[{"count":85,"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/pressbooks\/v2\/chapters\/18389\/revisions"}],"predecessor-version":[{"id":18712,"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/pressbooks\/v2\/chapters\/18389\/revisions\/18712"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/pressbooks\/v2\/parts\/18142"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/pressbooks\/v2\/chapters\/18389\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/wp\/v2\/media?parent=18389"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/pressbooks\/v2\/chapter-type?post=18389"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/wp\/v2\/contributor?post=18389"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/wp\/v2\/license?post=18389"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}