{"id":1655,"date":"2016-06-22T13:23:02","date_gmt":"2016-06-22T13:23:02","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/intermediatealgebra\/?post_type=chapter&#038;p=1655"},"modified":"2019-07-24T21:28:08","modified_gmt":"2019-07-24T21:28:08","slug":"read-or-watch-adding-and-subtracting-radicals","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/intermediatealgebra\/chapter\/read-or-watch-adding-and-subtracting-radicals\/","title":{"raw":"Add and Subtract Radical Expressions","rendered":"Add and Subtract Radical Expressions"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Determine when two radicals have the same index and radicand<\/li>\r\n \t<li>Recognize when a radical expression can be simplified either before or after addition or subtraction<\/li>\r\n<\/ul>\r\n<\/div>\r\nThere are two keys to combining radicals by addition or subtraction: look at the <strong>index<\/strong>, and look at the <strong>radicand<\/strong>. If these are the same, then addition and subtraction are possible. If not, then you cannot combine the two radicals. In the graphic below, the index of the\u00a0expression [latex]12\\sqrt[3]{xy}[\/latex] is\u00a0[latex]3[\/latex] and the radicand is [latex]xy[\/latex].\r\n\r\n<img class=\" wp-image-3200 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/07\/29230521\/Screen-Shot-2016-07-29-at-4.04.52-PM-300x141.png\" alt=\"Screen Shot 2016-07-29 at 4.04.52 PM\" width=\"511\" height=\"240\" \/>\r\n\r\nMaking sense of a string of radicals may be difficult. One helpful tip is to think of radicals as variables, and treat them the same way. When you add and subtract variables, you look for like terms, which is the same thing you will do when you add and subtract radicals.\r\n\r\nIn this first example, both radicals have the same radicand and index.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nAdd. [latex] 3\\sqrt{11}+7\\sqrt{11}[\/latex]\r\n[reveal-answer q=\"971281\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"971281\"]\r\n\r\nThe two radicals are the same, [latex] [\/latex]. This means you can combine them as you would combine the terms [latex] 3a+7a[\/latex].\r\n<p style=\"text-align: center;\">[latex] \\text{3}\\sqrt{11}\\text{ + 7}\\sqrt{11}[\/latex]<\/p>\r\nThe answer is [latex]10\\sqrt{11}[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nThis next example contains more addends, or terms that are being added together. Notice how you can combine <i>like<\/i> terms (radicals that have the same root and index), but you cannot combine <i>unlike<\/i> terms.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nAdd. [latex] 5\\sqrt{2}+\\sqrt{3}+4\\sqrt{3}+2\\sqrt{2}[\/latex]\r\n\r\n[reveal-answer q=\"687881\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"687881\"]\r\n\r\nRearrange terms so that like radicals are next to each other. Then add.\r\n<p style=\"text-align: center;\">[latex] 5\\sqrt{2}+2\\sqrt{2}+\\sqrt{3}+4\\sqrt{3}[\/latex]<\/p>\r\nThe answer is [latex]7\\sqrt{2}+5\\sqrt{3}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nNotice that the expression in the previous example is simplified even though it has two terms: [latex] 7\\sqrt{2}[\/latex] and [latex] 5\\sqrt{3}[\/latex]. It would be a mistake to try to combine them further! Some people make the mistake that [latex] 7\\sqrt{2}+5\\sqrt{3}=12\\sqrt{5}[\/latex]. This is incorrect because[latex] \\sqrt{2}[\/latex] and [latex]\\sqrt{3}[\/latex] are not like radicals so they cannot be added.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nAdd. [latex] 3\\sqrt{x}+12\\sqrt[3]{xy}+\\sqrt{x}[\/latex]\r\n\r\n[reveal-answer q=\"885242\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"885242\"]\r\n\r\nRearrange terms so that like radicals are next to each other. Then add.\r\n<p style=\"text-align: center;\">[latex] 3\\sqrt{x}+\\sqrt{x}+12\\sqrt[3]{xy}[\/latex]<\/p>\r\nThe answer is [latex]4\\sqrt{x}+12\\sqrt[3]{xy}[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nIn the following video, we show more examples of how to identify and add like radicals.\r\n\r\nhttps:\/\/youtu.be\/ihcZhgm3yBg\r\n\r\nSometimes you may need to add <i>and<\/i> simplify the radical. If the radicals are different, try simplifying first\u2014you may end up being able to combine the radicals at the end as shown in these next two examples.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nAdd and simplify. [latex] 2\\sqrt[3]{40}+\\sqrt[3]{135}[\/latex]\r\n\r\n[reveal-answer q=\"638886\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"638886\"]\r\n\r\nSimplify each radical by identifying perfect cubes.\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{r}2\\sqrt[3]{8\\cdot 5}+\\sqrt[3]{27\\cdot 5}\\\\2\\sqrt[3]{{{(2)}^{3}}\\cdot 5}+\\sqrt[3]{{{(3)}^{3}}\\cdot 5}\\\\2\\sqrt[3]{{{(2)}^{3}}}\\cdot \\sqrt[3]{5}+\\sqrt[3]{{{(3)}^{3}}}\\cdot \\sqrt[3]{5}\\end{array}[\/latex]<\/p>\r\nSimplify.\r\n<p style=\"text-align: center;\">[latex] 2\\cdot 2\\cdot \\sqrt[3]{5}+3\\cdot \\sqrt[3]{5}[\/latex]<\/p>\r\n<p style=\"text-align: center;\">[latex]4\\sqrt[3]{5}+3\\sqrt[3]{5}[\/latex]<\/p>\r\nThe answer is [latex]7\\sqrt[3]{5}[\/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\nAdd and simplify. [latex] x\\sqrt[3]{x{{y}^{4}}}+y\\sqrt[3]{{{x}^{4}}y}[\/latex]\r\n\r\n[reveal-answer q=\"95976\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"95976\"]\r\n\r\nSimplify each radical by identifying perfect cubes.\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}x\\sqrt[3]{x\\cdot {{y}^{3}}\\cdot y}+y\\sqrt[3]{{{x}^{3}}\\cdot x\\cdot y}\\\\x\\sqrt[3]{{{y}^{3}}}\\cdot \\sqrt[3]{xy}+y\\sqrt[3]{{{x}^{3}}}\\cdot \\sqrt[3]{xy}\\\\xy\\cdot \\sqrt[3]{xy}+xy\\cdot \\sqrt[3]{xy}\\end{array}[\/latex]<\/p>\r\nAdd like radicals.\r\n<p style=\"text-align: center;\">[latex] xy\\sqrt[3]{xy}+xy\\sqrt[3]{xy}[\/latex]<\/p>\r\nThe answer is [latex]2xy\\sqrt[3]{xy}[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nThe following video shows more examples of adding radicals that require simplification.\r\n\r\nhttps:\/\/youtu.be\/S3fGUeALy7E\r\n<h2>Subtract Radicals<\/h2>\r\nSubtraction of radicals follows the same set of rules and approaches as addition\u2014the radicands and the indices must be the same for two (or more) radicals to be subtracted. In the three examples that follow, subtraction has been rewritten as addition of the opposite.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nSubtract. [latex] 5\\sqrt{13}-3\\sqrt{13}[\/latex]\r\n\r\n[reveal-answer q=\"107411\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"107411\"]\r\n\r\nThe radicands and indices are the same, so these two radicals can be combined.\r\n<p style=\"text-align: center;\">[latex] 5\\sqrt{13}-3\\sqrt{13}[\/latex]<\/p>\r\nThe answer is [latex]2\\sqrt{13}[\/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\nSubtract. [latex] 4\\sqrt[3]{5a}-\\sqrt[3]{3a}-2\\sqrt[3]{5a}[\/latex]\r\n\r\n[reveal-answer q=\"491962\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"491962\"]\r\n\r\nTwo of the radicals have the same index and radicand, so they can be combined. Rewrite the expression so that like radicals are next to each other.\r\n<p style=\"text-align: center;\">[latex] 4\\sqrt[3]{5a}+(-\\sqrt[3]{3a})+(-2\\sqrt[3]{5a})\\\\4\\sqrt[3]{5a}+(-2\\sqrt[3]{5a})+(-\\sqrt[3]{3a})[\/latex]<\/p>\r\nCombine. Although the indices of [latex] 2\\sqrt[3]{5a}[\/latex] and [latex] -\\sqrt[3]{3a}[\/latex] are the same, the radicands are not\u2014so they cannot be combined.\r\n<p style=\"text-align: center;\">[latex] 2\\sqrt[3]{5a}+(-\\sqrt[3]{3a})[\/latex]<\/p>\r\nThe answer is [latex]2\\sqrt[3]{5a}-\\sqrt[3]{3a}[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nIn the following video, we show more examples of subtracting radical expressions when no simplifying is required.\r\n\r\nhttps:\/\/youtu.be\/77TR9HsPZ6M\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nSubtract and simplify. [latex] 5\\sqrt[4]{{{a}^{5}}b}-a\\sqrt[4]{16ab}[\/latex], where [latex]a\\ge 0[\/latex] and [latex]b\\ge 0[\/latex]\r\n\r\n[reveal-answer q=\"802638\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"802638\"]\r\n\r\nSimplify each radical by identifying and pulling out powers of\u00a0[latex]4[\/latex].\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}5\\sqrt[4]{{{a}^{4}}\\cdot a\\cdot b}-a\\sqrt[4]{{{(2)}^{4}}\\cdot a\\cdot b}\\\\5\\cdot a\\sqrt[4]{a\\cdot b}-a\\cdot 2\\sqrt[4]{a\\cdot b}\\\\5a\\sqrt[4]{ab}-2a\\sqrt[4]{ab}\\end{array}[\/latex]<\/p>\r\nThe answer is [latex]3a\\sqrt[4]{ab}[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nIn our last video, we show more examples of subtracting radicals that require simplifying.\r\n\r\nhttps:\/\/youtu.be\/6MogonN1PRQ\r\n<h2>Summary<\/h2>\r\nCombining radicals is possible when the index and the radicand of two or more radicals are the same. Radicals with the same index and radicand are known as like radicals. It is often helpful to treat radicals just as you would treat variables: like radicals can be added and subtracted in the same way that like variables can be added and subtracted. Sometimes, you will need to simplify a radical expression before it is possible to add or subtract like terms.","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Determine when two radicals have the same index and radicand<\/li>\n<li>Recognize when a radical expression can be simplified either before or after addition or subtraction<\/li>\n<\/ul>\n<\/div>\n<p>There are two keys to combining radicals by addition or subtraction: look at the <strong>index<\/strong>, and look at the <strong>radicand<\/strong>. If these are the same, then addition and subtraction are possible. If not, then you cannot combine the two radicals. In the graphic below, the index of the\u00a0expression [latex]12\\sqrt[3]{xy}[\/latex] is\u00a0[latex]3[\/latex] and the radicand is [latex]xy[\/latex].<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-3200 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/07\/29230521\/Screen-Shot-2016-07-29-at-4.04.52-PM-300x141.png\" alt=\"Screen Shot 2016-07-29 at 4.04.52 PM\" width=\"511\" height=\"240\" \/><\/p>\n<p>Making sense of a string of radicals may be difficult. One helpful tip is to think of radicals as variables, and treat them the same way. When you add and subtract variables, you look for like terms, which is the same thing you will do when you add and subtract radicals.<\/p>\n<p>In this first example, both radicals have the same radicand and index.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Add. [latex]3\\sqrt{11}+7\\sqrt{11}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q971281\">Show Solution<\/span><\/p>\n<div id=\"q971281\" class=\"hidden-answer\" style=\"display: none\">\n<p>The two radicals are the same, [latex][\/latex]. This means you can combine them as you would combine the terms [latex]3a+7a[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\text{3}\\sqrt{11}\\text{ + 7}\\sqrt{11}[\/latex]<\/p>\n<p>The answer is [latex]10\\sqrt{11}[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>This next example contains more addends, or terms that are being added together. Notice how you can combine <i>like<\/i> terms (radicals that have the same root and index), but you cannot combine <i>unlike<\/i> terms.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Add. [latex]5\\sqrt{2}+\\sqrt{3}+4\\sqrt{3}+2\\sqrt{2}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q687881\">Show Solution<\/span><\/p>\n<div id=\"q687881\" class=\"hidden-answer\" style=\"display: none\">\n<p>Rearrange terms so that like radicals are next to each other. Then add.<\/p>\n<p style=\"text-align: center;\">[latex]5\\sqrt{2}+2\\sqrt{2}+\\sqrt{3}+4\\sqrt{3}[\/latex]<\/p>\n<p>The answer is [latex]7\\sqrt{2}+5\\sqrt{3}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>Notice that the expression in the previous example is simplified even though it has two terms: [latex]7\\sqrt{2}[\/latex] and [latex]5\\sqrt{3}[\/latex]. It would be a mistake to try to combine them further! Some people make the mistake that [latex]7\\sqrt{2}+5\\sqrt{3}=12\\sqrt{5}[\/latex]. This is incorrect because[latex]\\sqrt{2}[\/latex] and [latex]\\sqrt{3}[\/latex] are not like radicals so they cannot be added.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Add. [latex]3\\sqrt{x}+12\\sqrt[3]{xy}+\\sqrt{x}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q885242\">Show Solution<\/span><\/p>\n<div id=\"q885242\" class=\"hidden-answer\" style=\"display: none\">\n<p>Rearrange terms so that like radicals are next to each other. Then add.<\/p>\n<p style=\"text-align: center;\">[latex]3\\sqrt{x}+\\sqrt{x}+12\\sqrt[3]{xy}[\/latex]<\/p>\n<p>The answer is [latex]4\\sqrt{x}+12\\sqrt[3]{xy}[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>In the following video, we show more examples of how to identify and add like radicals.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Adding Radicals (Basic With No Simplifying)\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/ihcZhgm3yBg?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>Sometimes you may need to add <i>and<\/i> simplify the radical. If the radicals are different, try simplifying first\u2014you may end up being able to combine the radicals at the end as shown in these next two examples.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Add and simplify. [latex]2\\sqrt[3]{40}+\\sqrt[3]{135}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q638886\">Show Solution<\/span><\/p>\n<div id=\"q638886\" class=\"hidden-answer\" style=\"display: none\">\n<p>Simplify each radical by identifying perfect cubes.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}2\\sqrt[3]{8\\cdot 5}+\\sqrt[3]{27\\cdot 5}\\\\2\\sqrt[3]{{{(2)}^{3}}\\cdot 5}+\\sqrt[3]{{{(3)}^{3}}\\cdot 5}\\\\2\\sqrt[3]{{{(2)}^{3}}}\\cdot \\sqrt[3]{5}+\\sqrt[3]{{{(3)}^{3}}}\\cdot \\sqrt[3]{5}\\end{array}[\/latex]<\/p>\n<p>Simplify.<\/p>\n<p style=\"text-align: center;\">[latex]2\\cdot 2\\cdot \\sqrt[3]{5}+3\\cdot \\sqrt[3]{5}[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]4\\sqrt[3]{5}+3\\sqrt[3]{5}[\/latex]<\/p>\n<p>The answer is [latex]7\\sqrt[3]{5}[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Add and simplify. [latex]x\\sqrt[3]{x{{y}^{4}}}+y\\sqrt[3]{{{x}^{4}}y}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q95976\">Show Solution<\/span><\/p>\n<div id=\"q95976\" class=\"hidden-answer\" style=\"display: none\">\n<p>Simplify each radical by identifying perfect cubes.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}x\\sqrt[3]{x\\cdot {{y}^{3}}\\cdot y}+y\\sqrt[3]{{{x}^{3}}\\cdot x\\cdot y}\\\\x\\sqrt[3]{{{y}^{3}}}\\cdot \\sqrt[3]{xy}+y\\sqrt[3]{{{x}^{3}}}\\cdot \\sqrt[3]{xy}\\\\xy\\cdot \\sqrt[3]{xy}+xy\\cdot \\sqrt[3]{xy}\\end{array}[\/latex]<\/p>\n<p>Add like radicals.<\/p>\n<p style=\"text-align: center;\">[latex]xy\\sqrt[3]{xy}+xy\\sqrt[3]{xy}[\/latex]<\/p>\n<p>The answer is [latex]2xy\\sqrt[3]{xy}[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>The following video shows more examples of adding radicals that require simplification.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-2\" 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<h2>Subtract Radicals<\/h2>\n<p>Subtraction of radicals follows the same set of rules and approaches as addition\u2014the radicands and the indices must be the same for two (or more) radicals to be subtracted. In the three examples that follow, subtraction has been rewritten as addition of the opposite.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Subtract. [latex]5\\sqrt{13}-3\\sqrt{13}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q107411\">Show Solution<\/span><\/p>\n<div id=\"q107411\" class=\"hidden-answer\" style=\"display: none\">\n<p>The radicands and indices are the same, so these two radicals can be combined.<\/p>\n<p style=\"text-align: center;\">[latex]5\\sqrt{13}-3\\sqrt{13}[\/latex]<\/p>\n<p>The answer is [latex]2\\sqrt{13}[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Subtract. [latex]4\\sqrt[3]{5a}-\\sqrt[3]{3a}-2\\sqrt[3]{5a}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q491962\">Show Solution<\/span><\/p>\n<div id=\"q491962\" class=\"hidden-answer\" style=\"display: none\">\n<p>Two of the radicals have the same index and radicand, so they can be combined. Rewrite the expression so that like radicals are next to each other.<\/p>\n<p style=\"text-align: center;\">[latex]4\\sqrt[3]{5a}+(-\\sqrt[3]{3a})+(-2\\sqrt[3]{5a})\\\\4\\sqrt[3]{5a}+(-2\\sqrt[3]{5a})+(-\\sqrt[3]{3a})[\/latex]<\/p>\n<p>Combine. Although the indices of [latex]2\\sqrt[3]{5a}[\/latex] and [latex]-\\sqrt[3]{3a}[\/latex] are the same, the radicands are not\u2014so they cannot be combined.<\/p>\n<p style=\"text-align: center;\">[latex]2\\sqrt[3]{5a}+(-\\sqrt[3]{3a})[\/latex]<\/p>\n<p>The answer is [latex]2\\sqrt[3]{5a}-\\sqrt[3]{3a}[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>In the following video, we show more examples of subtracting radical expressions when no simplifying is required.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-3\" 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<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Subtract and simplify. [latex]5\\sqrt[4]{{{a}^{5}}b}-a\\sqrt[4]{16ab}[\/latex], where [latex]a\\ge 0[\/latex] and [latex]b\\ge 0[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q802638\">Show Solution<\/span><\/p>\n<div id=\"q802638\" class=\"hidden-answer\" style=\"display: none\">\n<p>Simplify each radical by identifying and pulling out powers of\u00a0[latex]4[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}5\\sqrt[4]{{{a}^{4}}\\cdot a\\cdot b}-a\\sqrt[4]{{{(2)}^{4}}\\cdot a\\cdot b}\\\\5\\cdot a\\sqrt[4]{a\\cdot b}-a\\cdot 2\\sqrt[4]{a\\cdot b}\\\\5a\\sqrt[4]{ab}-2a\\sqrt[4]{ab}\\end{array}[\/latex]<\/p>\n<p>The answer is [latex]3a\\sqrt[4]{ab}[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>In our last video, we show more examples of subtracting radicals that require simplifying.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-4\" title=\"Subtracting Radicals That Requires Simplifying\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/6MogonN1PRQ?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h2>Summary<\/h2>\n<p>Combining radicals is possible when the index and the radicand of two or more radicals are the same. Radicals with the same index and radicand are known as like radicals. It is often helpful to treat radicals just as you would treat variables: like radicals can be added and subtracted in the same way that like variables can be added and subtracted. Sometimes, you will need to simplify a radical expression before it is possible to add or subtract like terms.<\/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-1655\">\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>Adding Radicals (Basic With No Simplifying). <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/ihcZhgm3yBg\">https:\/\/youtu.be\/ihcZhgm3yBg<\/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>Adding Radicals That Requires Simplifying. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <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 (Basic With No Simplifying). <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <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>Subtracting Radicals That Requires Simplifying. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/6MogonN1PRQ\">https:\/\/youtu.be\/6MogonN1PRQ<\/a>. 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