{"id":16343,"date":"2019-10-03T04:18:53","date_gmt":"2019-10-03T04:18:53","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/chapter\/6-1-2-multiplying-and-dividing-rational-expressions\/"},"modified":"2024-05-01T19:03:53","modified_gmt":"2024-05-01T19:03:53","slug":"6-1-2-multiplying-and-dividing-rational-expressions","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/chapter\/6-1-2-multiplying-and-dividing-rational-expressions\/","title":{"raw":"Multiplying and Dividing Rational Expressions","rendered":"Multiplying and Dividing Rational Expressions"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Multiply rational expressions<\/li>\r\n \t<li>Divide rational expressions<\/li>\r\n<\/ul>\r\n<\/div>\r\n&nbsp;\r\n\r\nJust as you can multiply and divide fractions, you can multiply and divide <strong>rational expressions<\/strong>. In fact, you use the same processes for multiplying and dividing rational expressions as you use for multiplying and dividing numeric fractions. The process is the same even though the expressions look different!\r\n\r\n[caption id=\"attachment_5014\" align=\"aligncenter\" width=\"370\"]<img class=\" wp-image-5014\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/117\/2016\/06\/20042018\/Screen-Shot-2016-06-19-at-9.19.45-PM.png\" alt=\"Multiply and Divide\" width=\"370\" height=\"194\" \/> Multiply and Divide[\/caption]\r\n<h3>Multiply Rational Expressions<\/h3>\r\nRemember that there are two ways to multiply numeric fractions.\r\n\r\nOne way is to multiply the numerators and the denominators and then simplify the product, as shown here.\r\n<p style=\"text-align: center;\">[latex] \\displaystyle \\frac{4}{5}\\cdot \\frac{9}{8}=\\frac{36}{40}=\\frac{3\\cdot 3\\cdot 2\\cdot 2}{5\\cdot 2\\cdot 2\\cdot 2}=\\frac{3\\cdot 3\\cdot \\cancel{2}\\cdot\\cancel{2}}{5\\cdot \\cancel{2}\\cdot\\cancel{2}\\cdot 2}=\\frac{3\\cdot 3}{5\\cdot 2}\\cdot 1=\\frac{9}{10}[\/latex]<\/p>\r\nA second way is to factor and simplify the fractions <i>before<\/i> performing the multiplication.\r\n<p style=\"text-align: center;\">[latex]\\frac{4}{5}\\cdot\\frac{9}{8}=\\frac{2\\cdot2}{5}\\cdot\\frac{3\\cdot3}{2\\cdot2\\cdot2}=\\frac{\\cancel{2}\\cdot\\cancel{2}\\cdot3\\cdot3}{\\cancel{2}\\cdot5\\cdot\\cancel{2}\\cdot2}=1\\cdot\\frac{3\\cdot3}{5\\cdot2}=\\frac{9}{10}[\/latex]<\/p>\r\nNotice that both methods result in the same product. In some cases you may find it easier to multiply and then simplify, while in others it may make more sense to simplify fractions before multiplying.\r\n\r\nThe same two approaches can be applied to rational expressions. In the following examples, both techniques are shown. First, let\u2019s multiply and then simplify.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nMultiply.[latex] \\displaystyle \\frac{5{{a}^{2}}}{14}\\cdot \\frac{7}{10{{a}^{3}}}[\/latex]\r\n\r\nState the product in simplest form.\r\n\r\n[reveal-answer q=\"518862\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"518862\"]\r\n\r\nMultiply the numerators, and then multiply the denominators.\r\n<p style=\"text-align: center;\">[latex]\\frac{5a^{2}}{14}\\cdot\\frac{7}{10a^{3}}=\\frac{35a^{2}}{140a^{3}}[\/latex]<\/p>\r\nSimplify by finding common factors in the numerator and denominator. Simplify\u00a0the common factors.\r\n<p style=\"text-align: center;\">[latex]\\large\\begin{array}{l}\\frac{35a^{2}}{140a^{3}}=\\frac{5\\cdot7\\cdot{a}^{2}}{5\\cdot7\\cdot2\\cdot2\\cdot{a}^{2}\\cdot{a}}\\\\\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,=\\frac{\\cancel{5}\\cdot\\cancel{7}\\cdot\\cancel{{a}^{2}}}{\\cancel{5}\\cdot\\cancel{7}\\cdot2\\cdot2\\cdot\\cancel{{a}^{2}}\\cdot{a}}\\\\\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,=\\normalsize1\\cdot\\large\\frac{1}{4a}\\end{array}[\/latex]<\/p>\r\nSimplify.\r\n<p style=\"text-align: center;\">[latex] \\displaystyle \\frac{1}{4a}[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\n[latex] \\displaystyle \\frac{5{{a}^{2}}}{14}\\cdot \\frac{7}{10{{a}^{3}}}=\\frac{1}{4a}[\/latex][latex] \\displaystyle [\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nOkay, that worked. But this time let\u2019s simplify first, then multiply. When using this method, it helps to look for the <strong>greatest common factor<\/strong>. You can factor out <i>any<\/i> common factors, but finding the greatest one will take fewer steps.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nMultiply. \u00a0[latex]\\frac{5a^{2}}{14}\\cdot\\frac{7}{10a^{3}}[\/latex]\r\n\r\nState the product in simplest form.\r\n\r\n[reveal-answer q=\"724339\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"724339\"]\r\n\r\nFactor the numerators and denominators. Look for the greatest common factors.\r\n<p style=\"text-align: center;\">[latex] \\displaystyle \\frac{5\\cdot {{a}^{2}}}{7\\cdot 2}\\cdot \\frac{7}{5\\cdot 2\\cdot {{a}^{2}}\\cdot a}[\/latex]<\/p>\r\nSimplify\u00a0common factors, then multiply.\r\n<p style=\"text-align: center;\">[latex]\\large\\begin{array}{c}\\frac{5\\cdot {{a}^{2}}}{7\\cdot 2}\\cdot \\frac{7}{5\\cdot 2\\cdot {{a}^{2}}\\cdot a}\\\\\\\\=\\frac{\\cancel{5}\\cdot\\cancel{{a}^{2}}}{\\cancel{7}\\cdot 2}\\cdot \\frac{\\cancel{7}}{\\cancel{5}\\cdot 2\\cdot\\cancel{{a}^{2}}\\cdot a}\\\\\\\\=\\frac{1\\cdot1\\cdot1}{2\\cdot2\\cdot{a}}=\\frac{1}{4a}\\end{array}[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\n[latex]\\frac{5a^{2}}{14}\\cdot\\frac{7}{10a^{3}}=\\frac{1}{4a}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nBoth methods produced the same answer.\r\n\r\nAlso, remember that when working with rational expressions, you should get into the habit of identifying any values for the variables that would result in division by [latex]0[\/latex]. These excluded values must be eliminated from the domain, the set of all possible values of the variable. In the example above, [latex] \\displaystyle \\frac{5{{a}^{2}}}{14}\\cdot \\frac{7}{10{{a}^{3}}}[\/latex], the domain is all real numbers where <i>a<\/i> is not equal to 0. When [latex]a=0[\/latex], the denominator of the fraction [latex]\\frac{7}{10a^{3}}[\/latex]\u00a0equals 0, which will make the fraction undefined.\r\n\r\nSome rational expressions contain quadratic expressions and other multi-term polynomials. To multiply these rational expressions, the best approach is to first factor the polynomials and then look for common factors. (Multiplying the terms before factoring will often create complicated polynomials\u2026and then you will have to factor these polynomials anyway! For this reason, it is easier to factor, simplify, and then multiply.) Just take it step by step, like in the examples below.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nMultiply. \u00a0[latex] \\displaystyle \\frac{{{a}^{2}}-a-2}{5a}\\cdot \\frac{10a}{a+1}\\,\\,,\\,\\,\\,\\,\\,\\,a\\,\\ne \\,\\,-1\\,,\\,\\,0[\/latex]\r\n\r\nState the product in simplest form.\r\n\r\n[reveal-answer q=\"794041\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"794041\"]\r\n<p style=\"text-align: center;\">Factor the numerators and denominators.<\/p>\r\n<p style=\"text-align: center;\">[latex]\\frac{\\left(a-2\\right)\\left(a+1\\right)}{5\\cdot{a}}\\cdot\\frac{5\\cdot2\\cdot{a}}{\\left(a+1\\right)}[\/latex]<\/p>\r\nSimplify\u00a0common factors:\r\n<p style=\"text-align: center;\">[latex]\\large\\begin{array}{c}\\frac{\\left(a-2\\right)\\cancel{\\left(a+1\\right)}}{\\cancel{5}\\cdot{\\cancel{a}}}\\cdot\\frac{\\cancel{5}\\cdot2\\cdot{\\cancel{a}}}{\\cancel{\\left(a+1\\right)}}\\\\\\\\=\\frac{a-2}{1}\\cdot\\frac{2}{1}\\end{array}[\/latex]<\/p>\r\nMultiply simplified rational expressions. This expression can be left with the numerator in factored form or multiplied out.\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}\\frac{\\left(a-2\\right)}{1}\\cdot\\frac{2}{1}\\\\\\\\=2\\left(a-2\\right)\\end{array}[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\n[latex] \\displaystyle \\frac{{{a}^{2}}-a-2}{5a}\\cdot \\frac{10a}{a+1}=2a-4[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nMultiply. \u00a0[latex]\\frac{a^{2}+4a+4}{2a^{2}-a-10}\\cdot\\frac{a+5}{a^{2}+2a},\\,\\,\\,a\\neq-2,0,\\frac{5}{2}[\/latex]\r\n\r\nState the product in simplest form.\r\n\r\n[reveal-answer q=\"980309\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"980309\"]\r\n\r\nFactor the numerators and denominators.\r\n<p style=\"text-align: center;\">[latex]\\frac{\\left(a+2\\right)\\left(a+2\\right)}{\\left(2a-5\\right)\\left(a+2\\right)}\\cdot\\frac{a+5}{a\\left(a+2\\right)}[\/latex]<\/p>\r\nSimplify\u00a0common factors.\r\n<p style=\"text-align: center;\">[latex]\\large\\frac{\\cancel{\\left(a+2\\right)}\\cancel{\\left(a+2\\right)}}{\\left(2a-5\\right)\\cancel{\\left(a+2\\right)}}\\cdot\\frac{a+5}{a\\cancel{\\left(a+2\\right)}}[\/latex]<\/p>\r\nMultiply simplified rational expressions. This expression can be left with the denominator in factored form or multiplied out.\r\n<p style=\"text-align: center;\">[latex]\\frac{1}{\\left(2a-5\\right)}\\cdot\\frac{a+5}{a}=\\frac{a+5}{a\\left(2a-5\\right)}[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\n[latex]\\frac{a^{2}+4a+4}{2a^{2}-a-10}\\cdot\\frac{a+5}{a^{2}+2a}=\\frac{a+5}{a\\left(2a-5\\right)}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nNote that in the answer above, you cannot simplify the rational expression any further. It may be tempting to express the [latex]5[\/latex]\u2019s in the numerator and denominator as the fraction [latex]\\frac{5}{5}[\/latex], but these [latex]5[\/latex]\u2019s are terms because they are being added or subtracted. Remember that only common factors, not terms, can be regrouped to form factors of [latex]1[\/latex]!\r\n\r\nIn the following video we present another example of multiplying rational expressions.\r\n\r\nhttps:\/\/www.youtube.com\/watch?v=Hj6gF1SNttk&amp;feature=youtu.be\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question]1458[\/ohm_question]\r\n\r\n<\/div>\r\n<h2>Divide Rational Expressions<\/h2>\r\nYou've seen that you multiply rational expressions as you multiply numeric fractions. It should come as no surprise that you also divide rational expressions the same way you divide numeric fractions. Specifically, to divide rational expressions, keep the first rational expression, change the division sign to multiplication, and then take the reciprocal of the second rational expression.\r\n\r\nLet\u2019s begin by recalling division of numerical fractions.\r\n<p style=\"text-align: center;\">[latex]\\frac{2}{3}\\div\\frac{5}{9}=\\frac{2}{3}\\cdot\\frac{9}{5}=\\frac{18}{15}=\\frac{6}{5}[\/latex]<\/p>\r\nUse the same process to divide rational expressions. You can think of division as multiplication by the reciprocal, and then use what you know about multiplication to simplify.\r\n\r\n[caption id=\"attachment_5013\" align=\"aligncenter\" width=\"496\"]<img class=\" wp-image-5013\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/117\/2016\/06\/20041101\/Screen-Shot-2016-06-19-at-9.10.18-PM-300x225.png\" alt=\"Reciprocal Architecture\" width=\"496\" height=\"374\" \/> Reciprocal Architecture[\/caption]\r\n\r\nYou do still need to think about the domain, specifically the variable values that would make either denominator equal zero. But there's a new consideration this time\u2014because you divide by multiplying by the reciprocal of one of the rational expressions, you also need to find the values that would make the <i>numerator <\/i>of that expression equal zero. Have a look.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nIdentify the domain of the expression. \u00a0[latex]\\frac{5x^{2}}{9}\\div\\frac{15x^{3}}{27}[\/latex]\r\n\r\n[reveal-answer q=\"988831\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"988831\"]\r\n\r\n<span style=\"text-decoration: underline;\">State the Domain:<\/span>\r\n\r\nFind excluded values. Notice the two denominators, [latex]9[\/latex] and\u00a0[latex]27[\/latex], can never equal\u00a0[latex]0[\/latex].\r\n\r\nBecause [latex]15x^{3}[\/latex]\u00a0becomes the denominator in the reciprocal of [latex] \\displaystyle \\frac{15{{x}^{3}}}{27}[\/latex], you must find the values of <i>x<\/i> that would make [latex]15x^{3}[\/latex] equal 0.\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}15x^{3}=0\\\\x=0\\,\\text{is an excluded value}.\\end{array}[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\nThe domain is all real numbers except \u00a0[latex]0[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nKnowing how to find the domain may seem unimportant here, but it will help you when you learn how to solve rational equations. To divide, multiply by the reciprocal.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nDivide. \u00a0[latex]\\frac{5x^{2}}{9}\\div\\frac{15x^{3}}{27}[\/latex]\r\n\r\nState the quotient in simplest form.\r\n\r\n[reveal-answer q=\"688236\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"688236\"]\r\n\r\nRewrite 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\nSimplify\u00a0common factors.\r\n\r\nSimplify.\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\r\n<h4>Answer<\/h4>\r\n[latex] \\displaystyle \\frac{5{{x}^{2}}}{9}\\div \\frac{15{{x}^{3}}}{27}=\\frac{1}{x},x\\ne 0[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nDivide. \u00a0[latex]\\frac{3x^{2}}{x+2}\\div\\frac{6x^{4}}{\\left(x^{2}+5x+6\\right)}[\/latex]\r\n\r\nState the quotient in simplest form, and express the domain of the expression.\r\n\r\n[reveal-answer q=\"53255\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"53255\"]\r\n\r\nDetermine the excluded values that make the denominators and the numerator of the divisor equal to [latex]0[\/latex].\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}\\left(x+2\\right)=0\\,\\,\\,\\,\\,\\\\x=-2\\\\\\left({{x}^{2}}+5x+6 \\right)=0\\,\\,\\,\\,\\,\\\\\\left(x+3\\right)\\left(x+2\\right)=0\\,\\,\\,\\,\\,\\\\x=-3\\,\\,\\,\\,\\text{or}\\,\\,\\,\\,-2\\\\6x^{4}=0\\,\\,\\,\\,\\,\\\\x=0\\,\\,\\,\\,\\,\\end{array}[\/latex]<\/p>\r\nDomain is all real numbers except [latex]0[\/latex], [latex]\u22122[\/latex], and [latex]\u22123[\/latex].\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\nSimplify\u00a0common factors\r\n<p style=\"text-align: center;\">[latex]\\large\\frac{\\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\r\n<h4>Answer<\/h4>\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\nThe domain is all real numbers except [latex]0[\/latex], [latex]\u22122[\/latex], and [latex]\u22123[\/latex].\r\n\r\n[\/hidden-answer]\r\n\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\nhttps:\/\/www.youtube.com\/watch?v=B1tigfgs268&amp;feature=youtu.be\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question]38288[\/ohm_question]\r\n\r\n<\/div>\r\n<h2>Summary<\/h2>\r\nRational expressions are multiplied and divided the same way as numeric fractions. To multiply, first find the greatest common factors of the numerator and denominator. Next, cancel common factors by regrouping the factors to make fractions equivalent to one. Then, multiply any remaining factors. To divide, first rewrite the division as multiplication by the reciprocal of the denominator. The steps are then the same as for multiplication.\r\n\r\nWhen expressing a product or quotient, it is important to state the excluded values. These are all values of a variable that would make a denominator equal zero at any step in the calculations.","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Multiply rational expressions<\/li>\n<li>Divide rational expressions<\/li>\n<\/ul>\n<\/div>\n<p>&nbsp;<\/p>\n<p>Just as you can multiply and divide fractions, you can multiply and divide <strong>rational expressions<\/strong>. In fact, you use the same processes for multiplying and dividing rational expressions as you use for multiplying and dividing numeric fractions. The process is the same even though the expressions look different!<\/p>\n<div id=\"attachment_5014\" style=\"width: 380px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-5014\" class=\"wp-image-5014\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/117\/2016\/06\/20042018\/Screen-Shot-2016-06-19-at-9.19.45-PM.png\" alt=\"Multiply and Divide\" width=\"370\" height=\"194\" \/><\/p>\n<p id=\"caption-attachment-5014\" class=\"wp-caption-text\">Multiply and Divide<\/p>\n<\/div>\n<h3>Multiply Rational Expressions<\/h3>\n<p>Remember that there are two ways to multiply numeric fractions.<\/p>\n<p>One way is to multiply the numerators and the denominators and then simplify the product, as shown here.<\/p>\n<p style=\"text-align: center;\">[latex]\\displaystyle \\frac{4}{5}\\cdot \\frac{9}{8}=\\frac{36}{40}=\\frac{3\\cdot 3\\cdot 2\\cdot 2}{5\\cdot 2\\cdot 2\\cdot 2}=\\frac{3\\cdot 3\\cdot \\cancel{2}\\cdot\\cancel{2}}{5\\cdot \\cancel{2}\\cdot\\cancel{2}\\cdot 2}=\\frac{3\\cdot 3}{5\\cdot 2}\\cdot 1=\\frac{9}{10}[\/latex]<\/p>\n<p>A second way is to factor and simplify the fractions <i>before<\/i> performing the multiplication.<\/p>\n<p style=\"text-align: center;\">[latex]\\frac{4}{5}\\cdot\\frac{9}{8}=\\frac{2\\cdot2}{5}\\cdot\\frac{3\\cdot3}{2\\cdot2\\cdot2}=\\frac{\\cancel{2}\\cdot\\cancel{2}\\cdot3\\cdot3}{\\cancel{2}\\cdot5\\cdot\\cancel{2}\\cdot2}=1\\cdot\\frac{3\\cdot3}{5\\cdot2}=\\frac{9}{10}[\/latex]<\/p>\n<p>Notice that both methods result in the same product. In some cases you may find it easier to multiply and then simplify, while in others it may make more sense to simplify fractions before multiplying.<\/p>\n<p>The same two approaches can be applied to rational expressions. In the following examples, both techniques are shown. First, let\u2019s multiply and then simplify.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Multiply.[latex]\\displaystyle \\frac{5{{a}^{2}}}{14}\\cdot \\frac{7}{10{{a}^{3}}}[\/latex]<\/p>\n<p>State the product in simplest form.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q518862\">Show Solution<\/span><\/p>\n<div id=\"q518862\" class=\"hidden-answer\" style=\"display: none\">\n<p>Multiply the numerators, and then multiply the denominators.<\/p>\n<p style=\"text-align: center;\">[latex]\\frac{5a^{2}}{14}\\cdot\\frac{7}{10a^{3}}=\\frac{35a^{2}}{140a^{3}}[\/latex]<\/p>\n<p>Simplify by finding common factors in the numerator and denominator. Simplify\u00a0the common factors.<\/p>\n<p style=\"text-align: center;\">[latex]\\large\\begin{array}{l}\\frac{35a^{2}}{140a^{3}}=\\frac{5\\cdot7\\cdot{a}^{2}}{5\\cdot7\\cdot2\\cdot2\\cdot{a}^{2}\\cdot{a}}\\\\\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,=\\frac{\\cancel{5}\\cdot\\cancel{7}\\cdot\\cancel{{a}^{2}}}{\\cancel{5}\\cdot\\cancel{7}\\cdot2\\cdot2\\cdot\\cancel{{a}^{2}}\\cdot{a}}\\\\\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,=\\normalsize1\\cdot\\large\\frac{1}{4a}\\end{array}[\/latex]<\/p>\n<p>Simplify.<\/p>\n<p style=\"text-align: center;\">[latex]\\displaystyle \\frac{1}{4a}[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex]\\displaystyle \\frac{5{{a}^{2}}}{14}\\cdot \\frac{7}{10{{a}^{3}}}=\\frac{1}{4a}[\/latex][latex]\\displaystyle[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>Okay, that worked. But this time let\u2019s simplify first, then multiply. When using this method, it helps to look for the <strong>greatest common factor<\/strong>. You can factor out <i>any<\/i> common factors, but finding the greatest one will take fewer steps.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Multiply. \u00a0[latex]\\frac{5a^{2}}{14}\\cdot\\frac{7}{10a^{3}}[\/latex]<\/p>\n<p>State the product in simplest form.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q724339\">Show Solution<\/span><\/p>\n<div id=\"q724339\" class=\"hidden-answer\" style=\"display: none\">\n<p>Factor the numerators and denominators. Look for the greatest common factors.<\/p>\n<p style=\"text-align: center;\">[latex]\\displaystyle \\frac{5\\cdot {{a}^{2}}}{7\\cdot 2}\\cdot \\frac{7}{5\\cdot 2\\cdot {{a}^{2}}\\cdot a}[\/latex]<\/p>\n<p>Simplify\u00a0common factors, then multiply.<\/p>\n<p style=\"text-align: center;\">[latex]\\large\\begin{array}{c}\\frac{5\\cdot {{a}^{2}}}{7\\cdot 2}\\cdot \\frac{7}{5\\cdot 2\\cdot {{a}^{2}}\\cdot a}\\\\\\\\=\\frac{\\cancel{5}\\cdot\\cancel{{a}^{2}}}{\\cancel{7}\\cdot 2}\\cdot \\frac{\\cancel{7}}{\\cancel{5}\\cdot 2\\cdot\\cancel{{a}^{2}}\\cdot a}\\\\\\\\=\\frac{1\\cdot1\\cdot1}{2\\cdot2\\cdot{a}}=\\frac{1}{4a}\\end{array}[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex]\\frac{5a^{2}}{14}\\cdot\\frac{7}{10a^{3}}=\\frac{1}{4a}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>Both methods produced the same answer.<\/p>\n<p>Also, remember that when working with rational expressions, you should get into the habit of identifying any values for the variables that would result in division by [latex]0[\/latex]. These excluded values must be eliminated from the domain, the set of all possible values of the variable. In the example above, [latex]\\displaystyle \\frac{5{{a}^{2}}}{14}\\cdot \\frac{7}{10{{a}^{3}}}[\/latex], the domain is all real numbers where <i>a<\/i> is not equal to 0. When [latex]a=0[\/latex], the denominator of the fraction [latex]\\frac{7}{10a^{3}}[\/latex]\u00a0equals 0, which will make the fraction undefined.<\/p>\n<p>Some rational expressions contain quadratic expressions and other multi-term polynomials. To multiply these rational expressions, the best approach is to first factor the polynomials and then look for common factors. (Multiplying the terms before factoring will often create complicated polynomials\u2026and then you will have to factor these polynomials anyway! For this reason, it is easier to factor, simplify, and then multiply.) Just take it step by step, like in the examples below.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Multiply. \u00a0[latex]\\displaystyle \\frac{{{a}^{2}}-a-2}{5a}\\cdot \\frac{10a}{a+1}\\,\\,,\\,\\,\\,\\,\\,\\,a\\,\\ne \\,\\,-1\\,,\\,\\,0[\/latex]<\/p>\n<p>State the product in simplest form.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q794041\">Show Solution<\/span><\/p>\n<div id=\"q794041\" class=\"hidden-answer\" style=\"display: none\">\n<p style=\"text-align: center;\">Factor the numerators and denominators.<\/p>\n<p style=\"text-align: center;\">[latex]\\frac{\\left(a-2\\right)\\left(a+1\\right)}{5\\cdot{a}}\\cdot\\frac{5\\cdot2\\cdot{a}}{\\left(a+1\\right)}[\/latex]<\/p>\n<p>Simplify\u00a0common factors:<\/p>\n<p style=\"text-align: center;\">[latex]\\large\\begin{array}{c}\\frac{\\left(a-2\\right)\\cancel{\\left(a+1\\right)}}{\\cancel{5}\\cdot{\\cancel{a}}}\\cdot\\frac{\\cancel{5}\\cdot2\\cdot{\\cancel{a}}}{\\cancel{\\left(a+1\\right)}}\\\\\\\\=\\frac{a-2}{1}\\cdot\\frac{2}{1}\\end{array}[\/latex]<\/p>\n<p>Multiply simplified rational expressions. This expression can be left with the numerator in factored form or multiplied out.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}\\frac{\\left(a-2\\right)}{1}\\cdot\\frac{2}{1}\\\\\\\\=2\\left(a-2\\right)\\end{array}[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex]\\displaystyle \\frac{{{a}^{2}}-a-2}{5a}\\cdot \\frac{10a}{a+1}=2a-4[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Multiply. \u00a0[latex]\\frac{a^{2}+4a+4}{2a^{2}-a-10}\\cdot\\frac{a+5}{a^{2}+2a},\\,\\,\\,a\\neq-2,0,\\frac{5}{2}[\/latex]<\/p>\n<p>State the product in simplest form.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q980309\">Show Solution<\/span><\/p>\n<div id=\"q980309\" class=\"hidden-answer\" style=\"display: none\">\n<p>Factor the numerators and denominators.<\/p>\n<p style=\"text-align: center;\">[latex]\\frac{\\left(a+2\\right)\\left(a+2\\right)}{\\left(2a-5\\right)\\left(a+2\\right)}\\cdot\\frac{a+5}{a\\left(a+2\\right)}[\/latex]<\/p>\n<p>Simplify\u00a0common factors.<\/p>\n<p style=\"text-align: center;\">[latex]\\large\\frac{\\cancel{\\left(a+2\\right)}\\cancel{\\left(a+2\\right)}}{\\left(2a-5\\right)\\cancel{\\left(a+2\\right)}}\\cdot\\frac{a+5}{a\\cancel{\\left(a+2\\right)}}[\/latex]<\/p>\n<p>Multiply simplified rational expressions. This expression can be left with the denominator in factored form or multiplied out.<\/p>\n<p style=\"text-align: center;\">[latex]\\frac{1}{\\left(2a-5\\right)}\\cdot\\frac{a+5}{a}=\\frac{a+5}{a\\left(2a-5\\right)}[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex]\\frac{a^{2}+4a+4}{2a^{2}-a-10}\\cdot\\frac{a+5}{a^{2}+2a}=\\frac{a+5}{a\\left(2a-5\\right)}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>Note that in the answer above, you cannot simplify the rational expression any further. It may be tempting to express the [latex]5[\/latex]\u2019s in the numerator and denominator as the fraction [latex]\\frac{5}{5}[\/latex], but these [latex]5[\/latex]\u2019s are terms because they are being added or subtracted. Remember that only common factors, not terms, can be regrouped to form factors of [latex]1[\/latex]!<\/p>\n<p>In the following video we present another example of multiplying rational expressions.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Multiply Rational Expressions and Give the Domain\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/Hj6gF1SNttk?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm1458\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=1458&theme=oea&iframe_resize_id=ohm1458&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<h2>Divide Rational Expressions<\/h2>\n<p>You&#8217;ve seen that you multiply rational expressions as you multiply numeric fractions. It should come as no surprise that you also divide rational expressions the same way you divide numeric fractions. Specifically, to divide rational expressions, keep the first rational expression, change the division sign to multiplication, and then take the reciprocal of the second rational expression.<\/p>\n<p>Let\u2019s begin by recalling division of numerical fractions.<\/p>\n<p style=\"text-align: center;\">[latex]\\frac{2}{3}\\div\\frac{5}{9}=\\frac{2}{3}\\cdot\\frac{9}{5}=\\frac{18}{15}=\\frac{6}{5}[\/latex]<\/p>\n<p>Use the same process to divide rational expressions. You can think of division as multiplication by the reciprocal, and then use what you know about multiplication to simplify.<\/p>\n<div id=\"attachment_5013\" style=\"width: 506px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-5013\" class=\"wp-image-5013\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/117\/2016\/06\/20041101\/Screen-Shot-2016-06-19-at-9.10.18-PM-300x225.png\" alt=\"Reciprocal Architecture\" width=\"496\" height=\"374\" \/><\/p>\n<p id=\"caption-attachment-5013\" class=\"wp-caption-text\">Reciprocal Architecture<\/p>\n<\/div>\n<p>You do still need to think about the domain, specifically the variable values that would make either denominator equal zero. But there&#8217;s a new consideration this time\u2014because you divide by multiplying by the reciprocal of one of the rational expressions, you also need to find the values that would make the <i>numerator <\/i>of that expression equal zero. Have a look.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Identify the domain of the expression. \u00a0[latex]\\frac{5x^{2}}{9}\\div\\frac{15x^{3}}{27}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q988831\">Show Solution<\/span><\/p>\n<div id=\"q988831\" class=\"hidden-answer\" style=\"display: none\">\n<p><span style=\"text-decoration: underline;\">State the Domain:<\/span><\/p>\n<p>Find excluded values. Notice the two denominators, [latex]9[\/latex] and\u00a0[latex]27[\/latex], can never equal\u00a0[latex]0[\/latex].<\/p>\n<p>Because [latex]15x^{3}[\/latex]\u00a0becomes the denominator in the reciprocal of [latex]\\displaystyle \\frac{15{{x}^{3}}}{27}[\/latex], you must find the values of <i>x<\/i> that would make [latex]15x^{3}[\/latex] equal 0.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}15x^{3}=0\\\\x=0\\,\\text{is an excluded value}.\\end{array}[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>The domain is all real numbers except \u00a0[latex]0[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>Knowing how to find the domain may seem unimportant here, but it will help you when you learn how to solve rational equations. To divide, multiply by the reciprocal.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Divide. \u00a0[latex]\\frac{5x^{2}}{9}\\div\\frac{15x^{3}}{27}[\/latex]<\/p>\n<p>State the quotient in simplest form.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><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>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>Simplify\u00a0common factors.<\/p>\n<p>Simplify.<\/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<h4>Answer<\/h4>\n<p>[latex]\\displaystyle \\frac{5{{x}^{2}}}{9}\\div \\frac{15{{x}^{3}}}{27}=\\frac{1}{x},x\\ne 0[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Divide. \u00a0[latex]\\frac{3x^{2}}{x+2}\\div\\frac{6x^{4}}{\\left(x^{2}+5x+6\\right)}[\/latex]<\/p>\n<p>State the quotient in simplest form, and express the domain of the expression.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><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>Determine the excluded values that make the denominators and the numerator of the divisor equal to [latex]0[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}\\left(x+2\\right)=0\\,\\,\\,\\,\\,\\\\x=-2\\\\\\left({{x}^{2}}+5x+6 \\right)=0\\,\\,\\,\\,\\,\\\\\\left(x+3\\right)\\left(x+2\\right)=0\\,\\,\\,\\,\\,\\\\x=-3\\,\\,\\,\\,\\text{or}\\,\\,\\,\\,-2\\\\6x^{4}=0\\,\\,\\,\\,\\,\\\\x=0\\,\\,\\,\\,\\,\\end{array}[\/latex]<\/p>\n<p>Domain is all real numbers except [latex]0[\/latex], [latex]\u22122[\/latex], and [latex]\u22123[\/latex].<\/p>\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>Simplify\u00a0common factors<\/p>\n<p style=\"text-align: center;\">[latex]\\large\\frac{\\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<h4>Answer<\/h4>\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<p>The domain is all real numbers except [latex]0[\/latex], [latex]\u22122[\/latex], and [latex]\u22123[\/latex].<\/p>\n<\/div>\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\" id=\"oembed-2\" title=\"Divide Rational Expressions and Give the Domain\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/B1tigfgs268?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm38288\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=38288&theme=oea&iframe_resize_id=ohm38288&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<h2>Summary<\/h2>\n<p>Rational expressions are multiplied and divided the same way as numeric fractions. To multiply, first find the greatest common factors of the numerator and denominator. Next, cancel common factors by regrouping the factors to make fractions equivalent to one. Then, multiply any remaining factors. To divide, first rewrite the division as multiplication by the reciprocal of the denominator. The steps are then the same as for multiplication.<\/p>\n<p>When expressing a product or quotient, it is important to state the excluded values. These are all values of a variable that would make a denominator equal zero at any step in the calculations.<\/p>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-16343\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Original<\/div><ul class=\"citation-list\"><li>Screenshot: Multiply and Divide. <strong>Provided by<\/strong>: Lumen Learning. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Revision and Adaptation. <strong>Provided by<\/strong>: Lumen Learning. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Screenshot: Reciprocal Architecture. <strong>Provided by<\/strong>: Lumen Learning. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Multiply Rational Expressions and Give the Domain. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/Hj6gF1SNttk\">https:\/\/youtu.be\/Hj6gF1SNttk<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Divide Rational Expressions and Give the Domain. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/B1tigfgs268\">https:\/\/youtu.be\/B1tigfgs268<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Unit 15: Rational Expressions, from Developmental Math: An Open Program. <strong>Provided by<\/strong>: Monterey Institute of Technology and Education. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/nrocnetwork.org\/resources\/downloads\/nroc-math-open-textbook-units-1-12-pdf-and-word-formats\/\">http:\/\/nrocnetwork.org\/resources\/downloads\/nroc-math-open-textbook-units-1-12-pdf-and-word-formats\/<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t 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