{"id":16424,"date":"2019-10-03T15:21:11","date_gmt":"2019-10-03T15:21:11","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/chapter\/7-2-2-adding-and-subtracting-radicals\/"},"modified":"2024-05-02T15:45:55","modified_gmt":"2024-05-02T15:45:55","slug":"7-2-2-adding-and-subtracting-radicals","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/chapter\/7-2-2-adding-and-subtracting-radicals\/","title":{"raw":"Adding and Subtracting Radicals","rendered":"Adding and Subtracting Radicals"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Identify\u00a0radicals that can be added or subtracted<\/li>\r\n \t<li>Add and subtract radical expressions<\/li>\r\n<\/ul>\r\n<\/div>\r\nAdding and subtracting radicals is much like combining like terms with variables. \u00a0We can add and subtract expressions with variables like this:\r\n<p style=\"text-align: center;\">[latex]5x+3y - 4x+7y=x+10y[\/latex]<\/p>\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.\r\n\r\nRemember the index is the degree of the root and the radicand is the term or expression under the radical. In the diagram below, the index is n, and the radicand is [latex]100[\/latex]. \u00a0The radicand is placed under the root symbol and the index is placed outside the root symbol to the left:\r\n\r\n[caption id=\"attachment_5161\" align=\"aligncenter\" width=\"548\"]<img class=\" wp-image-5161\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/117\/2016\/06\/23224212\/Screen-Shot-2016-06-23-at-3.41.42-PM-300x139.png\" alt=\"nth root with 100 as the radicand and the word &quot;radicand&quot; below it has an arrow from it to the number 100, the word index points to the letter n which is in the position of the index of the root\" width=\"548\" height=\"254\" \/> Index and radicand[\/caption]\r\n\r\nIn 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=\"aligncenter wp-image-3200\" 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=\"12 times the cuberoot of x y. 3 is the index and x y is the radicand.\" width=\"511\" height=\"240\" \/>\r\n<div class=\"mceTemp\"><\/div>\r\nPractice identifying radicals that are compatible for addition and subtraction by looking at the index and radicand of the roots in the following example.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nIdentify the roots that have the same index and radicand.\r\n\r\n[latex] 10\\sqrt{6}[\/latex]\r\n\r\n[latex] -1\\sqrt[3]{6}[\/latex]\r\n\r\n[latex] \\sqrt{25}[\/latex]\r\n\r\n[latex] 12\\sqrt{6}[\/latex]\r\n\r\n[latex] \\frac{1}{2}\\sqrt[3]{25}[\/latex]\r\n\r\n[latex] -7\\sqrt[3]{6}[\/latex]\r\n[reveal-answer q=\"332991\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"332991\"]\r\n\r\nLet's start with\u00a0[latex] 10\\sqrt{6}[\/latex]. \u00a0The index is [latex]2[\/latex] because no root was specified, and the radicand is [latex]6[\/latex]. The only other radical that has the same index and radicand is\u00a0[latex] 12\\sqrt{6}[\/latex].\r\n\r\n[latex] -1\\sqrt[3]{6}[\/latex] has an index of [latex]3[\/latex], and a radicand of [latex]6[\/latex]. The only other radical that has the same index and radicand is\u00a0[latex] -7\\sqrt[3]{6}[\/latex].\r\n\r\n[latex] \\sqrt{25}[\/latex] has an index of [latex]2[\/latex] and a radicand of [latex]25[\/latex]. \u00a0There are no other radicals in the list that have the same index and radicand.\r\n\r\n[latex] 12\\sqrt{6}[\/latex] has the same index and radicand as\u00a0[latex]10\\sqrt{6}[\/latex]\r\n\r\n[latex] \\frac{1}{2}\\sqrt[3]{25}[\/latex] has an index of [latex]3[\/latex] and a radicand of [latex]25[\/latex]. \u00a0There are no other radicals in the list that share these.\r\n\r\n[latex] -7\\sqrt[3]{6}[\/latex] has the same index and radicand as\u00a0[latex] -1\\sqrt[3]{6}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\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\nLet\u2019s use this concept to add some 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\"]The two radicals have\u00a0the same index and radicand. 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\n\r\n<h4>Answer<\/h4>\r\n[latex] 3\\sqrt{11}+7\\sqrt{11}=10\\sqrt{11}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nIt may help to think of radical terms with words when you are adding and subtracting them. The last example could be read \"three square roots of eleven plus [latex]7[\/latex] square roots of eleven\".\r\n\r\n&nbsp;\r\n\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\"]Rearrange 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\n\r\n<h4>Answer<\/h4>\r\n[latex] 5\\sqrt{2}+\\sqrt{3}+4\\sqrt{3}+2\\sqrt{2}=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\"]Rearrange 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\n\r\n<h4>Answer<\/h4>\r\n[latex] 3\\sqrt{x}+12\\sqrt[3]{xy}+\\sqrt{x}=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<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question]3493[\/ohm_question]\r\n\r\n<\/div>\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\"]Simplify 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\nAdd.\r\n<p style=\"text-align: center;\">[latex]4\\sqrt[3]{5}+3\\sqrt[3]{5}[\/latex]<\/p>\r\n&nbsp;\r\n<h4>Answer<\/h4>\r\n[latex] 2\\sqrt[3]{40}+\\sqrt[3]{135}=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\"]Simplify 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\n\r\n<h4>Answer<\/h4>\r\n[latex] x\\sqrt[3]{x{{y}^{4}}}+y\\sqrt[3]{{{x}^{4}}y}=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>Subtracting Radicals<\/h2>\r\nSubtraction of radicals follows the same set of rules and approaches as addition\u2014the radicands and the indices (plural of index) must be the same for two (or more) radicals to be subtracted.\u00a0\u00a0In the 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\"]The 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\n\r\n<h4>Answer<\/h4>\r\n[latex] 5\\sqrt{13}-3\\sqrt{13}=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\"]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.\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\n\r\n<h4>Answer<\/h4>\r\n[latex] 4\\sqrt[3]{5a}-\\sqrt[3]{3a}-2\\sqrt[3]{5a}=2\\sqrt[3]{5a}-\\sqrt[3]{3a}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nIn the video examples that follow, we show more examples of how to add and subtract radicals that don't need to be simplified beforehand.\r\n\r\nhttps:\/\/youtu.be\/5pVc44dEsTI\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 videos, we show more examples of subtracting radicals that require simplifying.\r\n\r\nhttps:\/\/youtu.be\/6MogonN1PRQ\r\n\r\n&nbsp;\r\n\r\nhttps:\/\/youtu.be\/tJk6_7lbrlw\r\n<h3>Summary<\/h3>\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>Identify\u00a0radicals that can be added or subtracted<\/li>\n<li>Add and subtract radical expressions<\/li>\n<\/ul>\n<\/div>\n<p>Adding and subtracting radicals is much like combining like terms with variables. \u00a0We can add and subtract expressions with variables like this:<\/p>\n<p style=\"text-align: center;\">[latex]5x+3y - 4x+7y=x+10y[\/latex]<\/p>\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.<\/p>\n<p>Remember the index is the degree of the root and the radicand is the term or expression under the radical. In the diagram below, the index is n, and the radicand is [latex]100[\/latex]. \u00a0The radicand is placed under the root symbol and the index is placed outside the root symbol to the left:<\/p>\n<div id=\"attachment_5161\" style=\"width: 558px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-5161\" class=\"wp-image-5161\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/117\/2016\/06\/23224212\/Screen-Shot-2016-06-23-at-3.41.42-PM-300x139.png\" alt=\"nth root with 100 as the radicand and the word &quot;radicand&quot; below it has an arrow from it to the number 100, the word index points to the letter n which is in the position of the index of the root\" width=\"548\" height=\"254\" \/><\/p>\n<p id=\"caption-attachment-5161\" class=\"wp-caption-text\">Index and radicand<\/p>\n<\/div>\n<p>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=\"aligncenter wp-image-3200\" 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=\"12 times the cuberoot of x y. 3 is the index and x y is the radicand.\" width=\"511\" height=\"240\" \/><\/p>\n<div class=\"mceTemp\"><\/div>\n<p>Practice identifying radicals that are compatible for addition and subtraction by looking at the index and radicand of the roots in the following example.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Identify the roots that have the same index and radicand.<\/p>\n<p>[latex]10\\sqrt{6}[\/latex]<\/p>\n<p>[latex]-1\\sqrt[3]{6}[\/latex]<\/p>\n<p>[latex]\\sqrt{25}[\/latex]<\/p>\n<p>[latex]12\\sqrt{6}[\/latex]<\/p>\n<p>[latex]\\frac{1}{2}\\sqrt[3]{25}[\/latex]<\/p>\n<p>[latex]-7\\sqrt[3]{6}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q332991\">Show Solution<\/span><\/p>\n<div id=\"q332991\" class=\"hidden-answer\" style=\"display: none\">\n<p>Let&#8217;s start with\u00a0[latex]10\\sqrt{6}[\/latex]. \u00a0The index is [latex]2[\/latex] because no root was specified, and the radicand is [latex]6[\/latex]. The only other radical that has the same index and radicand is\u00a0[latex]12\\sqrt{6}[\/latex].<\/p>\n<p>[latex]-1\\sqrt[3]{6}[\/latex] has an index of [latex]3[\/latex], and a radicand of [latex]6[\/latex]. The only other radical that has the same index and radicand is\u00a0[latex]-7\\sqrt[3]{6}[\/latex].<\/p>\n<p>[latex]\\sqrt{25}[\/latex] has an index of [latex]2[\/latex] and a radicand of [latex]25[\/latex]. \u00a0There are no other radicals in the list that have the same index and radicand.<\/p>\n<p>[latex]12\\sqrt{6}[\/latex] has the same index and radicand as\u00a0[latex]10\\sqrt{6}[\/latex]<\/p>\n<p>[latex]\\frac{1}{2}\\sqrt[3]{25}[\/latex] has an index of [latex]3[\/latex] and a radicand of [latex]25[\/latex]. \u00a0There are no other radicals in the list that share these.<\/p>\n<p>[latex]-7\\sqrt[3]{6}[\/latex] has the same index and radicand as\u00a0[latex]-1\\sqrt[3]{6}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\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>Let\u2019s use this concept to add some 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\">The two radicals have\u00a0the same index and radicand. 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<h4>Answer<\/h4>\n<p>[latex]3\\sqrt{11}+7\\sqrt{11}=10\\sqrt{11}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>It may help to think of radical terms with words when you are adding and subtracting them. The last example could be read &#8220;three square roots of eleven plus [latex]7[\/latex] square roots of eleven&#8221;.<\/p>\n<p>&nbsp;<\/p>\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\">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<h4>Answer<\/h4>\n<p>[latex]5\\sqrt{2}+\\sqrt{3}+4\\sqrt{3}+2\\sqrt{2}=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\">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<h4>Answer<\/h4>\n<p>[latex]3\\sqrt{x}+12\\sqrt[3]{xy}+\\sqrt{x}=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<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm3493\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=3493&theme=oea&iframe_resize_id=ohm3493&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\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\">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>Add.<\/p>\n<p style=\"text-align: center;\">[latex]4\\sqrt[3]{5}+3\\sqrt[3]{5}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<h4>Answer<\/h4>\n<p>[latex]2\\sqrt[3]{40}+\\sqrt[3]{135}=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\">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<h4>Answer<\/h4>\n<p>[latex]x\\sqrt[3]{x{{y}^{4}}}+y\\sqrt[3]{{{x}^{4}}y}=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>Subtracting Radicals<\/h2>\n<p>Subtraction of radicals follows the same set of rules and approaches as addition\u2014the radicands and the indices (plural of index) must be the same for two (or more) radicals to be subtracted.\u00a0\u00a0In the 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\">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<h4>Answer<\/h4>\n<p>[latex]5\\sqrt{13}-3\\sqrt{13}=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\">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<h4>Answer<\/h4>\n<p>[latex]4\\sqrt[3]{5a}-\\sqrt[3]{3a}-2\\sqrt[3]{5a}=2\\sqrt[3]{5a}-\\sqrt[3]{3a}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>In the video examples that follow, we show more examples of how to add and subtract radicals that don&#8217;t need to be simplified beforehand.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-3\" title=\"Ex:  Add and Subtract Radicals - No Simplifying\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/5pVc44dEsTI?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-4\" 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 videos, we show more examples of subtracting radicals that require simplifying.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-5\" 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<p>&nbsp;<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-6\" title=\"Ex:  Add and Subtract Square Roots\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/tJk6_7lbrlw?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h3>Summary<\/h3>\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-16424\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Original<\/div><ul class=\"citation-list\"><li>Screenshot: keys. <strong>Provided by<\/strong>: Lumen Learning. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Screenshot: Index and radicand. <strong>Provided by<\/strong>: Lumen Learning. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Revision and Adaptation. <strong>Provided by<\/strong>: Lumen Learning. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Unit 16: Radical Expressions and Quadratic Equations, from Developmental Math: An Open Program. <strong>Provided by<\/strong>: Monterey Institute of Technology and Education. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/nrocnetwork.org\/resources\/downloads\/nroc-math-open-textbook-units-1-12-pdf-and-word-formats\/\">http:\/\/nrocnetwork.org\/resources\/downloads\/nroc-math-open-textbook-units-1-12-pdf-and-word-formats\/<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Ex: Add and Subtract Radicals - No Simplifying. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/5pVc44dEsTI\">https:\/\/youtu.be\/5pVc44dEsTI<\/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>Ex: Add and Subtract Square Roots. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) . <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/tJk6_7lbrlw\">https:\/\/youtu.be\/tJk6_7lbrlw<\/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":169554,"menu_order":11,"template":"","meta":{"_candela_citation":"[{\"type\":\"original\",\"description\":\"Screenshot: keys\",\"author\":\"\",\"organization\":\"Lumen Learning\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"Screenshot: Index and radicand\",\"author\":\"\",\"organization\":\"Lumen Learning\",\"url\":\"\",\"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\":\"Unit 16: Radical Expressions and Quadratic Equations, from Developmental Math: An Open Program\",\"author\":\"\",\"organization\":\"Monterey Institute of Technology and Education\",\"url\":\"http:\/\/nrocnetwork.org\/resources\/downloads\/nroc-math-open-textbook-units-1-12-pdf-and-word-formats\/\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Ex: Add and Subtract Radicals - 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