{"id":1362,"date":"2016-10-24T21:01:43","date_gmt":"2016-10-24T21:01:43","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/waymakercollegealgebra\/?post_type=chapter&#038;p=1362"},"modified":"2017-04-10T19:18:35","modified_gmt":"2017-04-10T19:18:35","slug":"opeartions-on-square-roots","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/ivytech-wmopen-collegealgebra\/chapter\/opeartions-on-square-roots\/","title":{"raw":"Operations on Square Roots","rendered":"Operations on Square Roots"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Objectives<\/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\n<a href=\"https:\/\/s3-us-west-2.amazonaws.com\/oerfiles\/College+Algebra\/calculator.html\" target=\"_blank\"><img class=\"alignnone size-full wp-image-3370\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2017\/02\/13193222\/calculator.png\" alt=\"\" width=\"251\" height=\"46\" \/><\/a>\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\"]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\"]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<iframe id=\"mom1\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=2049&amp;theme=oea&amp;iframe_resize_id=mom1\" width=\"100%\" height=\"350\"><\/iframe>\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\"]Solution[\/reveal-answer]\r\n[hidden-answer a=\"902648\"]\r\nRewrite each term so they have equal radicands.\r\n<div style=\"text-align: center;\">[latex]\\begin{array}{ccc}\\hfill 20\\sqrt{72{a}^{3}{b}^{4}c}&amp; =&amp; 20\\sqrt{9}\\sqrt{4}\\sqrt{2}\\sqrt{a}\\sqrt{{a}^{2}}\\sqrt{{\\left({b}^{2}\\right)}^{2}}\\sqrt{c}\\hfill \\\\ &amp; =&amp; 20\\left(3\\right)\\left(2\\right)|a|{b}^{2}\\sqrt{2ac}\\hfill \\\\ &amp; =&amp; 120|a|{b}^{2}\\sqrt{2ac}\\hfill \\end{array}[\/latex]<\/div>\r\n<div style=\"text-align: center;\">[latex]\\begin{array}{ccc}\\hfill 14\\sqrt{8{a}^{3}{b}^{4}c}&amp; =&amp; 14\\sqrt{2}\\sqrt{4}\\sqrt{a}\\sqrt{{a}^{2}}\\sqrt{{\\left({b}^{2}\\right)}^{2}}\\sqrt{c}\\hfill \\\\ &amp; =&amp; 14\\left(2\\right)|a|{b}^{2}\\sqrt{2ac}\\hfill \\\\ &amp; =&amp; 28|a|{b}^{2}\\sqrt{2ac}\\hfill \\end{array}[\/latex]<\/div>\r\nNow the terms have the same radicand so we can subtract.\r\n<div>[latex]120|a|{b}^{2}\\sqrt{2ac}-28|a|{b}^{2}\\sqrt{2ac}\\text{= }92|a|{b}^{2}\\sqrt{2ac}\\text{ }[\/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\nSubtract [latex]3\\sqrt{80x}-4\\sqrt{45x}[\/latex].\r\n\r\n[reveal-answer q=\"236912\"]Solution[\/reveal-answer]\r\n[hidden-answer a=\"236912\"][latex]0[\/latex][\/hidden-answer]\r\n\r\n<iframe id=\"mom1\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=110419&amp;theme=oea&amp;iframe_resize_id=mom2\" width=\"100%\" height=\"350\"><\/iframe>\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\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]\\frac{\\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]\\frac{2\\sqrt{3}}{3\\sqrt{10}}[\/latex] in simplest form.\r\n\r\n[reveal-answer q=\"982148\"]Solution[\/reveal-answer]\r\n[hidden-answer a=\"982148\"]\r\nThe radical in the denominator is [latex]\\sqrt{10}[\/latex]. So multiply the fraction by [latex]\\frac{\\sqrt{10}}{\\sqrt{10}}[\/latex]. Then simplify.\r\n<div style=\"text-align: center;\">[latex]\\begin{array}{l}\\frac{2\\sqrt{3}}{3\\sqrt{10}}\\cdot \\frac{\\sqrt{10}}{\\sqrt{10}}\\text{ }\\\\ \\frac{2\\sqrt{30}}{30}\\text{ }\\\\ \\frac{\\sqrt{30}}{15}\\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\nWrite [latex]\\frac{12\\sqrt{3}}{\\sqrt{2}}[\/latex] in simplest form.\r\n\r\n[reveal-answer q=\"497322\"]Solution[\/reveal-answer]\r\n[hidden-answer a=\"497322\"][latex]6\\sqrt{6}[\/latex][\/hidden-answer]\r\n<iframe id=\"mom1\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=2765&amp;theme=oea&amp;iframe_resize_id=mom4\" width=\"100%\" height=\"350\"><\/iframe>\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]\\frac{4}{1+\\sqrt{5}}[\/latex] in simplest form.\r\n\r\n[reveal-answer q=\"726340\"]Solution[\/reveal-answer]\r\n[hidden-answer a=\"726340\"]\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]\\frac{1-\\sqrt{5}}{1-\\sqrt{5}}[\/latex].\r\n<div style=\"text-align: center;\">[latex]\\begin{array}{cc}\\frac{4}{1+\\sqrt{5}}\\cdot \\frac{1-\\sqrt{5}}{1-\\sqrt{5}}\\hfill &amp; \\\\ \\frac{4 - 4\\sqrt{5}}{-4}\\hfill &amp; \\text{Use the distributive property}.\\hfill \\\\ \\sqrt{5}-1\\hfill &amp; \\text{Simplify}.\\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\nWrite [latex]\\frac{7}{2+\\sqrt{3}}[\/latex] in simplest form.\r\n\r\n[reveal-answer q=\"132932\"]Solution[\/reveal-answer]\r\n[hidden-answer a=\"132932\"][latex]14 - 7\\sqrt{3}[\/latex][\/hidden-answer]\r\n<iframe id=\"mom1\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=3441&amp;theme=oea&amp;iframe_resize_id=mom5\" width=\"100%\" height=\"350\"><\/iframe>\r\n\r\n<\/div>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Objectives<\/h3>\n<ul>\n<li>Add and subtract square roots<\/li>\n<li>Rationalize denominators<\/li>\n<\/ul>\n<\/div>\n<p><a href=\"https:\/\/s3-us-west-2.amazonaws.com\/oerfiles\/College+Algebra\/calculator.html\" target=\"_blank\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-3370\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2017\/02\/13193222\/calculator.png\" alt=\"\" width=\"251\" height=\"46\" \/><\/a><br \/>\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].<\/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\">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\">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=\"mom1\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=2049&amp;theme=oea&amp;iframe_resize_id=mom1\" width=\"100%\" height=\"350\"><\/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\">Solution<\/span><\/p>\n<div id=\"q902648\" class=\"hidden-answer\" style=\"display: none\">\nRewrite each term so they have equal radicands.<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{array}{ccc}\\hfill 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}\\hfill \\\\ & =& 20\\left(3\\right)\\left(2\\right)|a|{b}^{2}\\sqrt{2ac}\\hfill \\\\ & =& 120|a|{b}^{2}\\sqrt{2ac}\\hfill \\end{array}[\/latex]<\/div>\n<div style=\"text-align: center;\">[latex]\\begin{array}{ccc}\\hfill 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}\\hfill \\\\ & =& 14\\left(2\\right)|a|{b}^{2}\\sqrt{2ac}\\hfill \\\\ & =& 28|a|{b}^{2}\\sqrt{2ac}\\hfill \\end{array}[\/latex]<\/div>\n<p>Now the terms have the same radicand so we can subtract.<\/p>\n<div>[latex]120|a|{b}^{2}\\sqrt{2ac}-28|a|{b}^{2}\\sqrt{2ac}\\text{= }92|a|{b}^{2}\\sqrt{2ac}\\text{ }[\/latex]<\/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\">Solution<\/span><\/p>\n<div id=\"q236912\" class=\"hidden-answer\" style=\"display: none\">[latex]0[\/latex]<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"mom1\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=110419&amp;theme=oea&amp;iframe_resize_id=mom2\" width=\"100%\" height=\"350\"><\/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<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]\\frac{\\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]\\frac{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\">Solution<\/span><\/p>\n<div id=\"q982148\" class=\"hidden-answer\" style=\"display: none\">\nThe radical in the denominator is [latex]\\sqrt{10}[\/latex]. So multiply the fraction by [latex]\\frac{\\sqrt{10}}{\\sqrt{10}}[\/latex]. Then simplify.<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{array}{l}\\frac{2\\sqrt{3}}{3\\sqrt{10}}\\cdot \\frac{\\sqrt{10}}{\\sqrt{10}}\\text{ }\\\\ \\frac{2\\sqrt{30}}{30}\\text{ }\\\\ \\frac{\\sqrt{30}}{15}\\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>Write [latex]\\frac{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\">Solution<\/span><\/p>\n<div id=\"q497322\" class=\"hidden-answer\" style=\"display: none\">[latex]6\\sqrt{6}[\/latex]<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"mom1\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=2765&amp;theme=oea&amp;iframe_resize_id=mom4\" width=\"100%\" height=\"350\"><\/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]\\frac{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\">Solution<\/span><\/p>\n<div id=\"q726340\" class=\"hidden-answer\" style=\"display: none\">\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]\\frac{1-\\sqrt{5}}{1-\\sqrt{5}}[\/latex].<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{array}{cc}\\frac{4}{1+\\sqrt{5}}\\cdot \\frac{1-\\sqrt{5}}{1-\\sqrt{5}}\\hfill & \\\\ \\frac{4 - 4\\sqrt{5}}{-4}\\hfill & \\text{Use the distributive property}.\\hfill \\\\ \\sqrt{5}-1\\hfill & \\text{Simplify}.\\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>Write [latex]\\frac{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\">Solution<\/span><\/p>\n<div id=\"q132932\" class=\"hidden-answer\" style=\"display: none\">[latex]14 - 7\\sqrt{3}[\/latex]<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"mom1\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=3441&amp;theme=oea&amp;iframe_resize_id=mom5\" width=\"100%\" height=\"350\"><\/iframe><\/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-1362\">\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><\/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":21,"menu_order":14,"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 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