{"id":50,"date":"2023-06-21T13:22:28","date_gmt":"2023-06-21T13:22:28","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/chapter\/opeartions-on-square-roots\/"},"modified":"2023-07-04T04:55:10","modified_gmt":"2023-07-04T04:55:10","slug":"opeartions-on-square-roots","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/chapter\/opeartions-on-square-roots\/","title":{"raw":"\u25aa   Operations on Square Roots","rendered":"\u25aa   Operations on Square Roots"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Add and subtract square roots.<\/li>\r\n \t<li>Rationalize denominators.<\/li>\r\n<\/ul>\r\n<\/div>\r\nWe can add or subtract radical expressions only when they have the same radicand and when they have the same radical type such as square roots. For example, the sum of [latex]\\sqrt{2}[\/latex] and [latex]3\\sqrt{2}[\/latex] is [latex]4\\sqrt{2}[\/latex]. However, it is often possible to simplify radical expressions, and that may change the radicand. The radical expression [latex]\\sqrt{18}[\/latex] can be written with a [latex]2[\/latex] in the radicand, as [latex]3\\sqrt{2}[\/latex], so [latex]\\sqrt{2}+\\sqrt{18}=\\sqrt{2}+3\\sqrt{2}=4\\sqrt{2}[\/latex].\r\n<div class=\"textbox\">\r\n<h3>How To: Given a radical expression requiring addition or subtraction of square roots, solve.<\/h3>\r\n<ol>\r\n \t<li>Simplify each radical expression.<\/li>\r\n \t<li>Add or subtract expressions with equal radicands.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Adding Square Roots<\/h3>\r\nAdd [latex]5\\sqrt{12}+2\\sqrt{3}[\/latex].\r\n\r\n[reveal-answer q=\"742464\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"742464\"]\r\nWe can rewrite [latex]5\\sqrt{12}[\/latex] as [latex]5\\sqrt{4\\cdot 3}[\/latex]. According the product rule, this becomes [latex]5\\sqrt{4}\\sqrt{3}[\/latex]. The square root of [latex]\\sqrt{4}[\/latex] is 2, so the expression becomes [latex]5\\left(2\\right)\\sqrt{3}[\/latex], which is [latex]10\\sqrt{3}[\/latex]. Now we can the terms have the same radicand so we can add.\r\n<p style=\"text-align: center;\">[latex]10\\sqrt{3}+2\\sqrt{3}=12\\sqrt{3}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nAdd [latex]\\sqrt{5}+6\\sqrt{20}[\/latex].\r\n\r\n[reveal-answer q=\"21382\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"21382\"]\r\n\r\n[latex]13\\sqrt{5}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n[embed]https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=2049&amp;theme=oea&amp;iframe_resize_id=mom1[\/embed]\r\n\r\n<\/div>\r\nWatch this video to see more examples of adding roots.\r\nhttps:\/\/youtu.be\/S3fGUeALy7E\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Subtracting Square Roots<\/h3>\r\nSubtract [latex]20\\sqrt{72{a}^{3}{b}^{4}c}-14\\sqrt{8{a}^{3}{b}^{4}c}[\/latex].\r\n\r\n[reveal-answer q=\"902648\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"902648\"]\r\n\r\nRewrite each term so they have equal radicands.\r\n<div style=\"text-align: center;\">\r\n<div style=\"text-align: center;\">[latex]\\begin{align} 20\\sqrt{72{a}^{3}{b}^{4}c}&amp; = 20\\sqrt{9}\\sqrt{4}\\sqrt{2}\\sqrt{a}\\sqrt{{a}^{2}}\\sqrt{{\\left({b}^{2}\\right)}^{2}}\\sqrt{c} \\\\ &amp; = 20\\left(3\\right)\\left(2\\right)a{b}^{2}\\sqrt{2ac} \\\\ &amp; = 120a{b}^{2}\\sqrt{2ac}\\\\ \\text{ } \\end{align}[\/latex]<\/div>\r\n<div><\/div>\r\n<div style=\"text-align: center;\">[latex]\\begin{align} 14\\sqrt{8{a}^{3}{b}^{4}c}&amp; = 14\\sqrt{2}\\sqrt{4}\\sqrt{a}\\sqrt{{a}^{2}}\\sqrt{{\\left({b}^{2}\\right)}^{2}}\\sqrt{c} \\\\ &amp; = 14\\left(2\\right)a{b}^{2}\\sqrt{2ac} \\\\ &amp; = 28a{b}^{2}\\sqrt{2ac} \\end{align}[\/latex]<\/div>\r\n<\/div>\r\nNow the terms have the same radicand so we can subtract.\r\n<div>[latex]120a{b}^{2}\\sqrt{2ac}-28a{b}^{2}\\sqrt{2ac}=92a{b}^{2}\\sqrt{2ac} \\\\ [\/latex]<\/div>\r\n<div>Note that we do not need an absolute value around the a because the [latex]a^3[\/latex] under the radical means that\u00a0<em>a<\/em> can't be negative.<\/div>\r\n<div>[\/hidden-answer]<\/div>\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nSubtract [latex]3\\sqrt{80x}-4\\sqrt{45x}[\/latex].\r\n\r\n[reveal-answer q=\"236912\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"236912\"]\r\n\r\n[latex]0[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n[embed]https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=110419&amp;theme=oea&amp;iframe_resize_id=mom2[\/embed]\r\n\r\n<\/div>\r\nin the next video we show more examples of how to subtract radicals.\r\nhttps:\/\/youtu.be\/77TR9HsPZ6M\r\n<h2>Rationalize Denominators<\/h2>\r\n<div class=\"textbox examples\">\r\n<h3>Recall the identity property of Multiplication<\/h3>\r\nWe leverage an important and useful identity in this section in a technique commonly used in college algebra:\r\n<p style=\"text-align: center;\"><em> rewriting an expression by multiplying it by a well-chosen form of the number 1.<\/em><\/p>\r\nBecause the multiplicative identity states that\u00a0[latex]a\\cdot1=a[\/latex],\u00a0we are able to multiply the top and bottom of any fraction by the same number without changing its value. We use this idea when we\u00a0<em>rationalize the denominator.<\/em>\r\n\r\n<\/div>\r\nWhen an expression involving square root radicals is written in simplest form, it will not contain a radical in the denominator. We can remove radicals from the denominators of fractions using a process called <em>rationalizing the denominator<\/em>.\r\n\r\nWe know that multiplying by 1 does not change the value of an expression. We use this property of multiplication to change expressions that contain radicals in the denominator. To remove radicals from the denominators of fractions, multiply by the form of 1 that will eliminate the radical.\r\n\r\nFor a denominator containing a single term, multiply by the radical in the denominator over itself. In other words, if the denominator is [latex]b\\sqrt{c}[\/latex], multiply by [latex]\\dfrac{\\sqrt{c}}{\\sqrt{c}}[\/latex].\r\n\r\nFor a denominator containing the sum or difference of a rational and an irrational term, multiply the numerator and denominator by the conjugate of the denominator, which is found by changing the sign of the radical portion of the denominator. If the denominator is [latex]a+b\\sqrt{c}[\/latex], then the conjugate is [latex]a-b\\sqrt{c}[\/latex].\r\n<div class=\"textbox\">\r\n<h3>How To: Given an expression with a single square root radical term in the denominator, rationalize the denominator.<\/h3>\r\n<ol>\r\n \t<li>Multiply the numerator and denominator by the radical in the denominator.<\/li>\r\n \t<li>Simplify.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Rationalizing a Denominator Containing a Single Term<\/h3>\r\nWrite [latex]\\dfrac{2\\sqrt{3}}{3\\sqrt{10}}[\/latex] in simplest form.\r\n\r\n[reveal-answer q=\"982148\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"982148\"]\r\n\r\nThe radical in the denominator is [latex]\\sqrt{10}[\/latex]. So multiply the fraction by [latex]\\dfrac{\\sqrt{10}}{\\sqrt{10}}[\/latex]. Then simplify.\r\n<div style=\"text-align: center;\">[latex]\\begin{align}\\frac{2\\sqrt{3}}{3\\sqrt{10}}\\cdot \\frac{\\sqrt{10}}{\\sqrt{10}} &amp;= \\frac{2\\sqrt{30}}{30} \\\\ &amp;= \\frac{\\sqrt{30}}{15}\\end{align}[\/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\nWrite [latex]\\dfrac{12\\sqrt{3}}{\\sqrt{2}}[\/latex] in simplest form.\r\n\r\n[reveal-answer q=\"497322\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"497322\"]\r\n\r\n[latex]6\\sqrt{6}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n[embed]https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=2765&amp;theme=oea&amp;iframe_resize_id=mom4[\/embed]\r\n\r\n<\/div>\r\n<div class=\"textbox\">\r\n<h3>How To: Given an expression with a radical term and a constant in the denominator, rationalize the denominator.<\/h3>\r\n<ol>\r\n \t<li>Find the conjugate of the denominator.<\/li>\r\n \t<li>Multiply the numerator and denominator by the conjugate.<\/li>\r\n \t<li>Use the distributive property.<\/li>\r\n \t<li>Simplify.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Rationalizing a Denominator Containing Two Terms<\/h3>\r\nWrite [latex]\\dfrac{4}{1+\\sqrt{5}}[\/latex] in simplest form.\r\n\r\n[reveal-answer q=\"726340\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"726340\"]\r\n\r\nBegin by finding the conjugate of the denominator by writing the denominator and changing the sign. So the conjugate of [latex]1+\\sqrt{5}[\/latex] is [latex]1-\\sqrt{5}[\/latex]. Then multiply the fraction by [latex]\\dfrac{1-\\sqrt{5}}{1-\\sqrt{5}}[\/latex].\r\n<div style=\"text-align: center;\">[latex]\\begin{align}\\frac{4}{1+\\sqrt{5}}\\cdot \\frac{1-\\sqrt{5}}{1-\\sqrt{5}} &amp;= \\frac{4 - 4\\sqrt{5}}{-4} &amp;&amp; \\text{Use the distributive property}. \\\\ &amp;=\\sqrt{5}-1 &amp;&amp; \\text{Simplify}. \\end{align}[\/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\nWrite [latex]\\dfrac{7}{2+\\sqrt{3}}[\/latex] in simplest form.\r\n\r\n[reveal-answer q=\"132932\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"132932\"]\r\n\r\n[latex]14 - 7\\sqrt{3}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n[embed]https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=3441&amp;theme=oea&amp;iframe_resize_id=mom5[\/embed]\r\n\r\n<\/div>\r\nhttps:\/\/www.youtube.com\/watch?v=vINRIRgeKqU&amp;feature=youtu.be\r\n\r\n&nbsp;","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Add and subtract square roots.<\/li>\n<li>Rationalize denominators.<\/li>\n<\/ul>\n<\/div>\n<p>We can add or subtract radical expressions only when they have the same radicand and when they have the same radical type such as square roots. For example, the sum of [latex]\\sqrt{2}[\/latex] and [latex]3\\sqrt{2}[\/latex] is [latex]4\\sqrt{2}[\/latex]. However, it is often possible to simplify radical expressions, and that may change the radicand. The radical expression [latex]\\sqrt{18}[\/latex] can be written with a [latex]2[\/latex] in the radicand, as [latex]3\\sqrt{2}[\/latex], so [latex]\\sqrt{2}+\\sqrt{18}=\\sqrt{2}+3\\sqrt{2}=4\\sqrt{2}[\/latex].<\/p>\n<div class=\"textbox\">\n<h3>How To: Given a radical expression requiring addition or subtraction of square roots, solve.<\/h3>\n<ol>\n<li>Simplify each radical expression.<\/li>\n<li>Add or subtract expressions with equal radicands.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Adding Square Roots<\/h3>\n<p>Add [latex]5\\sqrt{12}+2\\sqrt{3}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q742464\">Show Solution<\/span><\/p>\n<div id=\"q742464\" class=\"hidden-answer\" style=\"display: none\">\nWe can rewrite [latex]5\\sqrt{12}[\/latex] as [latex]5\\sqrt{4\\cdot 3}[\/latex]. According the product rule, this becomes [latex]5\\sqrt{4}\\sqrt{3}[\/latex]. The square root of [latex]\\sqrt{4}[\/latex] is 2, so the expression becomes [latex]5\\left(2\\right)\\sqrt{3}[\/latex], which is [latex]10\\sqrt{3}[\/latex]. Now we can the terms have the same radicand so we can add.<\/p>\n<p style=\"text-align: center;\">[latex]10\\sqrt{3}+2\\sqrt{3}=12\\sqrt{3}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Add [latex]\\sqrt{5}+6\\sqrt{20}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q21382\">Show Solution<\/span><\/p>\n<div id=\"q21382\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]13\\sqrt{5}[\/latex]<\/p>\n<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"ohm2049\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=2049&#38;theme=oea&#38;iframe_resize_id=ohm2049&#38;show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>Watch this video to see more examples of adding roots.<br \/>\n<iframe loading=\"lazy\" id=\"oembed-1\" title=\"Adding Radicals That Requires Simplifying\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/S3fGUeALy7E?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<div class=\"textbox exercises\">\n<h3>Example: Subtracting Square Roots<\/h3>\n<p>Subtract [latex]20\\sqrt{72{a}^{3}{b}^{4}c}-14\\sqrt{8{a}^{3}{b}^{4}c}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q902648\">Show Solution<\/span><\/p>\n<div id=\"q902648\" class=\"hidden-answer\" style=\"display: none\">\n<p>Rewrite each term so they have equal radicands.<\/p>\n<div style=\"text-align: center;\">\n<div style=\"text-align: center;\">[latex]\\begin{align} 20\\sqrt{72{a}^{3}{b}^{4}c}& = 20\\sqrt{9}\\sqrt{4}\\sqrt{2}\\sqrt{a}\\sqrt{{a}^{2}}\\sqrt{{\\left({b}^{2}\\right)}^{2}}\\sqrt{c} \\\\ & = 20\\left(3\\right)\\left(2\\right)a{b}^{2}\\sqrt{2ac} \\\\ & = 120a{b}^{2}\\sqrt{2ac}\\\\ \\text{ } \\end{align}[\/latex]<\/div>\n<div><\/div>\n<div style=\"text-align: center;\">[latex]\\begin{align} 14\\sqrt{8{a}^{3}{b}^{4}c}& = 14\\sqrt{2}\\sqrt{4}\\sqrt{a}\\sqrt{{a}^{2}}\\sqrt{{\\left({b}^{2}\\right)}^{2}}\\sqrt{c} \\\\ & = 14\\left(2\\right)a{b}^{2}\\sqrt{2ac} \\\\ & = 28a{b}^{2}\\sqrt{2ac} \\end{align}[\/latex]<\/div>\n<\/div>\n<p>Now the terms have the same radicand so we can subtract.<\/p>\n<div>[latex]120a{b}^{2}\\sqrt{2ac}-28a{b}^{2}\\sqrt{2ac}=92a{b}^{2}\\sqrt{2ac} \\\\[\/latex]<\/div>\n<div>Note that we do not need an absolute value around the a because the [latex]a^3[\/latex] under the radical means that\u00a0<em>a<\/em> can&#8217;t be negative.<\/div>\n<div><\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Subtract [latex]3\\sqrt{80x}-4\\sqrt{45x}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q236912\">Show Solution<\/span><\/p>\n<div id=\"q236912\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]0[\/latex]<\/p>\n<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"ohm110419\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=110419&#38;theme=oea&#38;iframe_resize_id=ohm110419&#38;show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>in the next video we show more examples of how to subtract radicals.<br \/>\n<iframe loading=\"lazy\" id=\"oembed-2\" title=\"Subtracting Radicals (Basic With No Simplifying)\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/77TR9HsPZ6M?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h2>Rationalize Denominators<\/h2>\n<div class=\"textbox examples\">\n<h3>Recall the identity property of Multiplication<\/h3>\n<p>We leverage an important and useful identity in this section in a technique commonly used in college algebra:<\/p>\n<p style=\"text-align: center;\"><em> rewriting an expression by multiplying it by a well-chosen form of the number 1.<\/em><\/p>\n<p>Because the multiplicative identity states that\u00a0[latex]a\\cdot1=a[\/latex],\u00a0we are able to multiply the top and bottom of any fraction by the same number without changing its value. We use this idea when we\u00a0<em>rationalize the denominator.<\/em><\/p>\n<\/div>\n<p>When an expression involving square root radicals is written in simplest form, it will not contain a radical in the denominator. We can remove radicals from the denominators of fractions using a process called <em>rationalizing the denominator<\/em>.<\/p>\n<p>We know that multiplying by 1 does not change the value of an expression. We use this property of multiplication to change expressions that contain radicals in the denominator. To remove radicals from the denominators of fractions, multiply by the form of 1 that will eliminate the radical.<\/p>\n<p>For a denominator containing a single term, multiply by the radical in the denominator over itself. In other words, if the denominator is [latex]b\\sqrt{c}[\/latex], multiply by [latex]\\dfrac{\\sqrt{c}}{\\sqrt{c}}[\/latex].<\/p>\n<p>For a denominator containing the sum or difference of a rational and an irrational term, multiply the numerator and denominator by the conjugate of the denominator, which is found by changing the sign of the radical portion of the denominator. If the denominator is [latex]a+b\\sqrt{c}[\/latex], then the conjugate is [latex]a-b\\sqrt{c}[\/latex].<\/p>\n<div class=\"textbox\">\n<h3>How To: Given an expression with a single square root radical term in the denominator, rationalize the denominator.<\/h3>\n<ol>\n<li>Multiply the numerator and denominator by the radical in the denominator.<\/li>\n<li>Simplify.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Rationalizing a Denominator Containing a Single Term<\/h3>\n<p>Write [latex]\\dfrac{2\\sqrt{3}}{3\\sqrt{10}}[\/latex] in simplest form.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q982148\">Show Solution<\/span><\/p>\n<div id=\"q982148\" class=\"hidden-answer\" style=\"display: none\">\n<p>The radical in the denominator is [latex]\\sqrt{10}[\/latex]. So multiply the fraction by [latex]\\dfrac{\\sqrt{10}}{\\sqrt{10}}[\/latex]. Then simplify.<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{align}\\frac{2\\sqrt{3}}{3\\sqrt{10}}\\cdot \\frac{\\sqrt{10}}{\\sqrt{10}} &= \\frac{2\\sqrt{30}}{30} \\\\ &= \\frac{\\sqrt{30}}{15}\\end{align}[\/latex]<\/div>\n<div><\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Write [latex]\\dfrac{12\\sqrt{3}}{\\sqrt{2}}[\/latex] in simplest form.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q497322\">Show Solution<\/span><\/p>\n<div id=\"q497322\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]6\\sqrt{6}[\/latex]<\/p>\n<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"ohm2765\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=2765&#38;theme=oea&#38;iframe_resize_id=ohm2765&#38;show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<div class=\"textbox\">\n<h3>How To: Given an expression with a radical term and a constant in the denominator, rationalize the denominator.<\/h3>\n<ol>\n<li>Find the conjugate of the denominator.<\/li>\n<li>Multiply the numerator and denominator by the conjugate.<\/li>\n<li>Use the distributive property.<\/li>\n<li>Simplify.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Rationalizing a Denominator Containing Two Terms<\/h3>\n<p>Write [latex]\\dfrac{4}{1+\\sqrt{5}}[\/latex] in simplest form.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q726340\">Show Solution<\/span><\/p>\n<div id=\"q726340\" class=\"hidden-answer\" style=\"display: none\">\n<p>Begin by finding the conjugate of the denominator by writing the denominator and changing the sign. So the conjugate of [latex]1+\\sqrt{5}[\/latex] is [latex]1-\\sqrt{5}[\/latex]. Then multiply the fraction by [latex]\\dfrac{1-\\sqrt{5}}{1-\\sqrt{5}}[\/latex].<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{align}\\frac{4}{1+\\sqrt{5}}\\cdot \\frac{1-\\sqrt{5}}{1-\\sqrt{5}} &= \\frac{4 - 4\\sqrt{5}}{-4} && \\text{Use the distributive property}. \\\\ &=\\sqrt{5}-1 && \\text{Simplify}. \\end{align}[\/latex]<\/div>\n<div><\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Write [latex]\\dfrac{7}{2+\\sqrt{3}}[\/latex] in simplest form.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q132932\">Show Solution<\/span><\/p>\n<div id=\"q132932\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]14 - 7\\sqrt{3}[\/latex]<\/p>\n<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"ohm3441\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=3441&#38;theme=oea&#38;iframe_resize_id=ohm3441&#38;show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"oembed-3\" title=\"Ex:  Rationalize the Denominator of a Radical Expression - Conjugate\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/vINRIRgeKqU?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>&nbsp;<\/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-50\">\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>Adding Radicals Requiring Simplification. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com). <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/S3fGUeALy7E\">https:\/\/youtu.be\/S3fGUeALy7E<\/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>Subtracting Radicals. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com). <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/77TR9HsPZ6M\">https:\/\/youtu.be\/77TR9HsPZ6M<\/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>Question ID 2049. <strong>Authored by<\/strong>: Lawrence Morales. <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 110419. <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 2765. <strong>Authored by<\/strong>: Bryan Johns. <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 3441. <strong>Authored by<\/strong>: Jessica Reidel. <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>Ex: Rationalize the Denominator of a Radical Expression - Conjugate. <strong>Authored by<\/strong>: James Sousa. <strong>Provided by<\/strong>: Mathispower4u. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/www.youtube.com\/watch?v=vINRIRgeKqU&#038;feature=youtu.be\">https:\/\/www.youtube.com\/watch?v=vINRIRgeKqU&#038;feature=youtu.be<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by-sa\/4.0\/\">CC BY-SA: Attribution-ShareAlike<\/a><\/em><\/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":19,"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 Learning\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"College Algebra\",\"author\":\"Abramson, Jay et al.\",\"organization\":\"OpenStax\",\"url\":\"http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"Download for free at http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2\"},{\"type\":\"cc\",\"description\":\"Adding Radicals Requiring Simplification\",\"author\":\"James Sousa (Mathispower4u.com)\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/S3fGUeALy7E\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Subtracting Radicals\",\"author\":\"James Sousa (Mathispower4u.com)\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/77TR9HsPZ6M\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Question ID 2049\",\"author\":\"Lawrence Morales\",\"organization\":\"\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"IMathAS Community License, CC-BY + GPL\"},{\"type\":\"cc\",\"description\":\"Question ID 110419\",\"author\":\"Lumen Learning\",\"organization\":\"\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"IMathAS Community License, CC-BY + GPL\"},{\"type\":\"cc\",\"description\":\"Question ID 2765\",\"author\":\"Bryan Johns\",\"organization\":\"\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"IMathAS Community License, CC-BY + GPL\"},{\"type\":\"cc\",\"description\":\"Question ID 3441\",\"author\":\"Jessica Reidel\",\"organization\":\"\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"IMathAS Community License, CC-BY + GPL\"},{\"type\":\"cc\",\"description\":\"Ex: Rationalize the Denominator of a Radical Expression - 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