{"id":71,"date":"2023-06-21T13:22:30","date_gmt":"2023-06-21T13:22:30","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/chapter\/add-and-subtract-rational-expressions\/"},"modified":"2023-07-04T04:58:25","modified_gmt":"2023-07-04T04:58:25","slug":"add-and-subtract-rational-expressions","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/chapter\/add-and-subtract-rational-expressions\/","title":{"raw":"\u25aa   Adding and Subtracting Rational Expressions","rendered":"\u25aa   Adding and Subtracting Rational Expressions"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Find the least common denominator of two rational expressions.<\/li>\r\n \t<li>Add and subtract rational expressions.<\/li>\r\n \t<li>Simplify complex rational expressions.<\/li>\r\n<\/ul>\r\n<\/div>\r\nAdding and subtracting rational expressions works just like adding and subtracting numerical fractions. To add fractions, we need to find a common denominator. Let\u2019s look at an example of fraction addition.\r\n<div style=\"text-align: center;\">[latex]\\begin{array}{ccc}\\hfill \\dfrac{5}{24}+\\dfrac{1}{40}&amp; =&amp; \\dfrac{25}{120}+\\dfrac{3}{120}\\hfill \\\\ &amp; =&amp; \\dfrac{28}{120}\\hfill \\\\ &amp; =&amp; \\dfrac{7}{30}\\hfill \\end{array}[\/latex]<\/div>\r\n<div>\r\n<div class=\"textbox examples\">\r\n<h3>how did we know what number to use for the denominator?<\/h3>\r\nIn the example above, we rewrote the fractions as equivalent fractions with a common denominator of 120. Recall that we use the least common multiple of the original denominators.\r\n\r\nTo find the LCM of 24 and 40, rewrite 24 and 40 as products of primes, then select the largest set of each prime appearing.\r\n\r\n[latex]24 = 2^3\\cdot3[\/latex]\r\n\r\n[latex]40=2^3\\cdot5[\/latex]\r\n\r\nWe choose [latex]2^3\\cdot3\\cdot5=120[\/latex] as the LCM, since that's the largest number of factors of 2, 3, and 5 we see. The LCM is 120.\r\n\r\nWe multiply each numerator with just enough of the LCM to make each denominator 120 to get the equivalent fractions.\r\n\r\nWhen referring to fractions, we call the LCM the <strong>least common denominator<\/strong>, or the LCD.\r\n\r\n<\/div>\r\n<span style=\"font-size: 1rem; text-align: initial;\">We have to rewrite the fractions so they share a common denominator before we are able to add. We must do the same thing when adding or subtracting rational expressions.<\/span>\r\n\r\n<\/div>\r\nThe easiest common denominator to use will be the <strong>least common denominator<\/strong>\u00a0or LCD. The LCD is the smallest multiple that the denominators have in common. To find the LCD of two rational expressions, we factor the expressions and multiply all of the distinct factors. For instance, if the factored denominators were [latex]\\left(x+3\\right)\\left(x+4\\right)[\/latex] and [latex]\\left(x+4\\right)\\left(x+5\\right)[\/latex], then the LCD would be [latex]\\left(x+3\\right)\\left(x+4\\right)\\left(x+5\\right)[\/latex].\r\n\r\nOnce we find the LCD, we need to multiply each expression by the form of 1 that will change the denominator to the LCD. We would need to multiply the expression with a denominator of [latex]\\left(x+3\\right)\\left(x+4\\right)[\/latex] by [latex]\\dfrac{x+5}{x+5}[\/latex] and the expression with a denominator of [latex]\\left(x+4\\right)\\left(x+5\\right)[\/latex] by [latex]\\dfrac{x+3}{x+3}[\/latex].\r\n<div class=\"textbox\">\r\n<h3>How To: Given two rational expressions, add or subtract them<\/h3>\r\n<ol>\r\n \t<li>Factor the numerator and denominator.<\/li>\r\n \t<li>Find the LCD of the expressions.<\/li>\r\n \t<li>Multiply the expressions by a form of 1 that changes the denominators to the LCD.<\/li>\r\n \t<li>Add or subtract the numerators.<\/li>\r\n \t<li>Simplify.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Adding Rational Expressions<\/h3>\r\nAdd the rational expressions:\r\n<div style=\"text-align: center;\">[latex]\\dfrac{5}{x}+\\dfrac{6}{y}[\/latex]<\/div>\r\n[reveal-answer q=\"232817\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"232817\"]\r\n\r\nFirst, we have to find the LCD. In this case, the LCD will be [latex]xy[\/latex]. We then multiply each expression by the appropriate form of 1 to obtain [latex]xy[\/latex] as the denominator for each fraction.\r\n<div style=\"text-align: center;\">[latex]\\begin{array}{l}\\dfrac{5}{x}\\cdot \\dfrac{y}{y}+\\dfrac{6}{y}\\cdot \\dfrac{x}{x}\\\\ \\dfrac{5y}{xy}+\\dfrac{6x}{xy}\\end{array}[\/latex]<\/div>\r\nNow that the expressions have the same denominator, we simply add the numerators to find the sum.\r\n<div style=\"text-align: center;\">[latex]\\dfrac{6x+5y}{xy}[\/latex]<\/div>\r\n&nbsp;\r\n<h4>Analysis of the Solution<\/h4>\r\nMultiplying by [latex]\\dfrac{y}{y}[\/latex] or [latex]\\dfrac{x}{x}[\/latex] does not change the value of the original expression because any number divided by itself is 1, and multiplying an expression by 1 gives the original expression.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question]110918-110919[\/ohm_question]\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Subtracting Rational Expressions<\/h3>\r\nSubtract the rational expressions:\r\n<div style=\"text-align: center;\">[latex]\\dfrac{6}{{x}^{2}+4x+4}-\\dfrac{2}{{x}^{2}-4}[\/latex]<\/div>\r\n[reveal-answer q=\"122137\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"122137\"]\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{cc}\\dfrac{6}{{\\left(x+2\\right)}^{2}}-\\dfrac{2}{\\left(x+2\\right)\\left(x - 2\\right)}\\hfill &amp; \\text{Factor}.\\hfill \\\\ \\dfrac{6}{{\\left(x+2\\right)}^{2}}\\cdot \\dfrac{x - 2}{x - 2}-\\dfrac{2}{\\left(x+2\\right)\\left(x - 2\\right)}\\cdot \\dfrac{x+2}{x+2}\\hfill &amp; \\text{Multiply each fraction to get the LCD as the denominator}.\\hfill \\\\ \\dfrac{6\\left(x - 2\\right)}{{\\left(x+2\\right)}^{2}\\left(x - 2\\right)}-\\dfrac{2\\left(x+2\\right)}{{\\left(x+2\\right)}^{2}\\left(x - 2\\right)}\\hfill &amp; \\text{Multiply}.\\hfill \\\\ \\dfrac{6x - 12-\\left(2x+4\\right)}{{\\left(x+2\\right)}^{2}\\left(x - 2\\right)}\\hfill &amp; \\text{Apply distributive property}.\\hfill \\\\ \\dfrac{4x - 16}{{\\left(x+2\\right)}^{2}\\left(x - 2\\right)}\\hfill &amp; \\text{Subtract}.\\hfill \\\\ \\dfrac{4\\left(x - 4\\right)}{{\\left(x+2\\right)}^{2}\\left(x - 2\\right)}\\hfill &amp; \\text{Simplify}.\\hfill \\end{array}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox\">\r\n<h3>Q &amp; A<\/h3>\r\n<strong>Do we have to use the LCD to add or subtract rational expressions?<\/strong>\r\n\r\n<em>No. Any common denominator will work, but it is easiest to use the LCD.<\/em>\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nSubtract the rational expressions: [latex]\\dfrac{3}{x+5}-\\dfrac{1}{x - 3}[\/latex].\r\n\r\n[reveal-answer q=\"820348\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"820348\"]\r\n\r\n[latex]\\dfrac{2\\left(x - 7\\right)}{\\left(x+5\\right)\\left(x - 3\\right)}[\/latex][\/hidden-answer]\r\n\r\n[ohm_question]39519[\/ohm_question]\r\n\r\n<\/div>\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>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Find the least common denominator of two rational expressions.<\/li>\n<li>Add and subtract rational expressions.<\/li>\n<li>Simplify complex rational expressions.<\/li>\n<\/ul>\n<\/div>\n<p>Adding and subtracting rational expressions works just like adding and subtracting numerical fractions. To add fractions, we need to find a common denominator. Let\u2019s look at an example of fraction addition.<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{array}{ccc}\\hfill \\dfrac{5}{24}+\\dfrac{1}{40}& =& \\dfrac{25}{120}+\\dfrac{3}{120}\\hfill \\\\ & =& \\dfrac{28}{120}\\hfill \\\\ & =& \\dfrac{7}{30}\\hfill \\end{array}[\/latex]<\/div>\n<div>\n<div class=\"textbox examples\">\n<h3>how did we know what number to use for the denominator?<\/h3>\n<p>In the example above, we rewrote the fractions as equivalent fractions with a common denominator of 120. Recall that we use the least common multiple of the original denominators.<\/p>\n<p>To find the LCM of 24 and 40, rewrite 24 and 40 as products of primes, then select the largest set of each prime appearing.<\/p>\n<p>[latex]24 = 2^3\\cdot3[\/latex]<\/p>\n<p>[latex]40=2^3\\cdot5[\/latex]<\/p>\n<p>We choose [latex]2^3\\cdot3\\cdot5=120[\/latex] as the LCM, since that&#8217;s the largest number of factors of 2, 3, and 5 we see. The LCM is 120.<\/p>\n<p>We multiply each numerator with just enough of the LCM to make each denominator 120 to get the equivalent fractions.<\/p>\n<p>When referring to fractions, we call the LCM the <strong>least common denominator<\/strong>, or the LCD.<\/p>\n<\/div>\n<p><span style=\"font-size: 1rem; text-align: initial;\">We have to rewrite the fractions so they share a common denominator before we are able to add. We must do the same thing when adding or subtracting rational expressions.<\/span><\/p>\n<\/div>\n<p>The easiest common denominator to use will be the <strong>least common denominator<\/strong>\u00a0or LCD. The LCD is the smallest multiple that the denominators have in common. To find the LCD of two rational expressions, we factor the expressions and multiply all of the distinct factors. For instance, if the factored denominators were [latex]\\left(x+3\\right)\\left(x+4\\right)[\/latex] and [latex]\\left(x+4\\right)\\left(x+5\\right)[\/latex], then the LCD would be [latex]\\left(x+3\\right)\\left(x+4\\right)\\left(x+5\\right)[\/latex].<\/p>\n<p>Once we find the LCD, we need to multiply each expression by the form of 1 that will change the denominator to the LCD. We would need to multiply the expression with a denominator of [latex]\\left(x+3\\right)\\left(x+4\\right)[\/latex] by [latex]\\dfrac{x+5}{x+5}[\/latex] and the expression with a denominator of [latex]\\left(x+4\\right)\\left(x+5\\right)[\/latex] by [latex]\\dfrac{x+3}{x+3}[\/latex].<\/p>\n<div class=\"textbox\">\n<h3>How To: Given two rational expressions, add or subtract them<\/h3>\n<ol>\n<li>Factor the numerator and denominator.<\/li>\n<li>Find the LCD of the expressions.<\/li>\n<li>Multiply the expressions by a form of 1 that changes the denominators to the LCD.<\/li>\n<li>Add or subtract the numerators.<\/li>\n<li>Simplify.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Adding Rational Expressions<\/h3>\n<p>Add the rational expressions:<\/p>\n<div style=\"text-align: center;\">[latex]\\dfrac{5}{x}+\\dfrac{6}{y}[\/latex]<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q232817\">Show Solution<\/span><\/p>\n<div id=\"q232817\" class=\"hidden-answer\" style=\"display: none\">\n<p>First, we have to find the LCD. In this case, the LCD will be [latex]xy[\/latex]. We then multiply each expression by the appropriate form of 1 to obtain [latex]xy[\/latex] as the denominator for each fraction.<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{array}{l}\\dfrac{5}{x}\\cdot \\dfrac{y}{y}+\\dfrac{6}{y}\\cdot \\dfrac{x}{x}\\\\ \\dfrac{5y}{xy}+\\dfrac{6x}{xy}\\end{array}[\/latex]<\/div>\n<p>Now that the expressions have the same denominator, we simply add the numerators to find the sum.<\/p>\n<div style=\"text-align: center;\">[latex]\\dfrac{6x+5y}{xy}[\/latex]<\/div>\n<p>&nbsp;<\/p>\n<h4>Analysis of the Solution<\/h4>\n<p>Multiplying by [latex]\\dfrac{y}{y}[\/latex] or [latex]\\dfrac{x}{x}[\/latex] does not change the value of the original expression because any number divided by itself is 1, and multiplying an expression by 1 gives the original expression.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm110918\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=110918-110919&theme=oea&iframe_resize_id=ohm110918&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Subtracting Rational Expressions<\/h3>\n<p>Subtract the rational expressions:<\/p>\n<div style=\"text-align: center;\">[latex]\\dfrac{6}{{x}^{2}+4x+4}-\\dfrac{2}{{x}^{2}-4}[\/latex]<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q122137\">Show Solution<\/span><\/p>\n<div id=\"q122137\" class=\"hidden-answer\" style=\"display: none\">\n<p style=\"text-align: center;\">[latex]\\begin{array}{cc}\\dfrac{6}{{\\left(x+2\\right)}^{2}}-\\dfrac{2}{\\left(x+2\\right)\\left(x - 2\\right)}\\hfill & \\text{Factor}.\\hfill \\\\ \\dfrac{6}{{\\left(x+2\\right)}^{2}}\\cdot \\dfrac{x - 2}{x - 2}-\\dfrac{2}{\\left(x+2\\right)\\left(x - 2\\right)}\\cdot \\dfrac{x+2}{x+2}\\hfill & \\text{Multiply each fraction to get the LCD as the denominator}.\\hfill \\\\ \\dfrac{6\\left(x - 2\\right)}{{\\left(x+2\\right)}^{2}\\left(x - 2\\right)}-\\dfrac{2\\left(x+2\\right)}{{\\left(x+2\\right)}^{2}\\left(x - 2\\right)}\\hfill & \\text{Multiply}.\\hfill \\\\ \\dfrac{6x - 12-\\left(2x+4\\right)}{{\\left(x+2\\right)}^{2}\\left(x - 2\\right)}\\hfill & \\text{Apply distributive property}.\\hfill \\\\ \\dfrac{4x - 16}{{\\left(x+2\\right)}^{2}\\left(x - 2\\right)}\\hfill & \\text{Subtract}.\\hfill \\\\ \\dfrac{4\\left(x - 4\\right)}{{\\left(x+2\\right)}^{2}\\left(x - 2\\right)}\\hfill & \\text{Simplify}.\\hfill \\end{array}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox\">\n<h3>Q &amp; A<\/h3>\n<p><strong>Do we have to use the LCD to add or subtract rational expressions?<\/strong><\/p>\n<p><em>No. Any common denominator will work, but it is easiest to use the LCD.<\/em><\/p>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Subtract the rational expressions: [latex]\\dfrac{3}{x+5}-\\dfrac{1}{x - 3}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q820348\">Show Solution<\/span><\/p>\n<div id=\"q820348\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]\\dfrac{2\\left(x - 7\\right)}{\\left(x+5\\right)\\left(x - 3\\right)}[\/latex]<\/p><\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"ohm39519\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=39519&theme=oea&iframe_resize_id=ohm39519&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\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\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-71\">\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>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><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>College Algebra. <strong>Authored by<\/strong>: Abramson, Jay et al.. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2\">http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: Download for free at http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2<\/li><li>Question ID 110918, 110919. <strong>Authored 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>. <strong>License Terms<\/strong>: IMathAS Community License CC- BY + GPL<\/li><li>Question ID 39519. <strong>Authored by<\/strong>: Roy Shahbazian. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: IMathAS Community License CC- BY + GPL<\/li><li>Question ID 3078, 3080. <strong>Authored by<\/strong>: Tophe Anderson. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: IMathAS Community License CC- BY + GPL<\/li><li>Question ID 59554. <strong>Authored by<\/strong>: Gary Parker. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: IMathAS Community License CC- BY + GPL<\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Specific attribution<\/div><ul class=\"citation-list\"><li>College Algebra. <strong>Authored by<\/strong>: OpenStax College Algebra. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@3.278:1\/Preface\">http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@3.278:1\/Preface<\/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 <\/section>","protected":false},"author":395986,"menu_order":17,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc-attribution\",\"description\":\"College Algebra\",\"author\":\"OpenStax College Algebra\",\"organization\":\"OpenStax\",\"url\":\"http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@3.278:1\/Preface\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"Revision and Adaptation\",\"author\":\"\",\"organization\":\"Lumen 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Anderson\",\"organization\":\"\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"IMathAS Community License CC- BY + GPL\"},{\"type\":\"cc\",\"description\":\"Question ID 59554\",\"author\":\"Gary Parker\",\"organization\":\"\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"IMathAS Community License CC- BY + 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