{"id":6806,"date":"2020-10-09T21:16:01","date_gmt":"2020-10-09T21:16:01","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/slcc-beginalgebra\/?post_type=chapter&p=6806"},"modified":"2026-02-05T08:36:29","modified_gmt":"2026-02-05T08:36:29","slug":"7-2-adding-and-subtracting-radical-expressions","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/slcc-elementaryalgebra\/chapter\/7-2-adding-and-subtracting-radical-expressions\/","title":{"raw":"7.2: Adding and Subtracting Radical Expressions","rendered":"7.2: Adding and Subtracting Radical Expressions"},"content":{"raw":"
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section 7.2 Learning Objectives<\/h3>\r\n7.2: Adding and Subtracting Radical Expressions<\/strong>\r\n
    \r\n \t
  • Identify like radicals<\/li>\r\n \t
  • Simplify an expression by adding and subtracting like radicals<\/li>\r\n \t
  • Add and subtract radicals that must be simplified first<\/li>\r\n<\/ul>\r\n<\/div>\r\n \r\n

    Identify like radicals<\/h2>\r\nThere are two keys to combining radicals by addition or subtraction: look at the index<\/strong>, and look at the 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\"Expression\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
    \r\n

    Example 1<\/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

    [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\n

    Simplify an expression by adding like radicals<\/h2>\r\nThis next example contains more addends, or terms that are being added together. Notice how you can combine like<\/i> terms (radicals that have the same root and index), but you cannot combine unlike<\/i> terms.\r\n
    \r\n

    Example 2<\/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

    [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

    \r\n

    Example 3<\/h3>\r\nAdd. [latex] 3\\sqrt{5}+12\\sqrt[3]{7}+\\sqrt{5}[\/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

    [latex] 3\\sqrt{5}+\\sqrt{5}+12\\sqrt[3]{7}[\/latex]<\/p>\r\nThe answer is [latex]4\\sqrt{5}+12\\sqrt[3]{7}[\/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

    Add radicals that must be simplified first<\/h2>\r\nSometimes you may need to add 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
    \r\n

    Example 4<\/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

    [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

    [latex] =2\\cdot 2\\cdot \\sqrt[3]{5}+3\\cdot \\sqrt[3]{5}[\/latex]<\/p>\r\n

    [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

    \r\n

    Example 5<\/h3>\r\nSimplify: [latex] \\sqrt{12}+\\sqrt{27}[\/latex]<\/span><\/span>\r\n\r\n[reveal-answer q=\"690208\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"690208\"]\r\n
    \r\n\r\nAs written, these two radicals cannot be added because they are not like radicals.\u00a0 We\u00a0need to simplify both radicals first.\u00a0<\/span>\r\n\r\nPrime factor each radicand.<\/span><\/span>\r\n

    [latex] \\sqrt{12}+\\sqrt{27}=\\sqrt{2\\cdot 2\\cdot 3} + \\sqrt{3\\cdot 3\\cdot 3}[\/latex]<\/span><\/span><\/p>\r\nIdentify pairs of identical factors and simplify.<\/span><\/span>\r\n

    [latex] =\\sqrt{2^2\\cdot 3} + \\sqrt{3^2\\cdot 3}[\/latex]<\/span><\/span><\/p>\r\n

    [latex] =2\\sqrt{3} + 3\\sqrt{3}[\/latex]<\/span><\/span><\/p>\r\nIn simplified form, we have like radicals and can add.<\/span><\/span>\r\n

    [latex] =5\\sqrt{3}[\/latex]<\/span><\/span><\/p>\r\n \r\n\r\n<\/div>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n \r\n\r\nThe following video shows more examples of adding radicals that require simplification.\r\n\r\nhttps:\/\/youtu.be\/S3fGUeALy7E\r\n

    Simplify an expression by subtracting like 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.\r\n
    \r\n

    Example 6<\/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

    [latex] 5\\sqrt{13}-3\\sqrt{13}[\/latex]<\/p>\r\n

    The answer is [latex]=2\\sqrt{13}[\/latex].<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n

    \r\n

    Example 7<\/h3>\r\nSubtract. [latex] 4\\sqrt[3]{5}-\\sqrt[3]{3}-2\\sqrt[3]{5}[\/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

    [latex]\\begin{array}{r} 4\\sqrt[3]{5}+(-\\sqrt[3]{3})+(-2\\sqrt[3]{5})\\\\=4\\sqrt[3]{5}+(-2\\sqrt[3]{5})+(-\\sqrt[3]{3})\\end{array}[\/latex]<\/p>\r\nCombine. Although the indices of [latex] 2\\sqrt[3]{5}[\/latex] and [latex] -\\sqrt[3]{3}[\/latex] are the same, the radicands are not\u2014so they cannot be combined.\r\n

    [latex] =2\\sqrt[3]{5}+(-\\sqrt[3]{3})[\/latex]<\/p>\r\nThe answer is [latex]2\\sqrt[3]{5}-\\sqrt[3]{3}[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n

    \r\n

    Example 8<\/h3>\r\nSimplify:\u00a0[latex] 5\\sqrt{7}-8\\sqrt[4]{11}+\\sqrt{7}+3\\sqrt[4]{11}[\/latex]\r\n\r\n[reveal-answer q=\"780228\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"780228\"]\r\n\r\nNotice, the radical terms are already in simplest form.<\/span><\/span>\u00a0<\/span>\r\n

    [latex] 5\\sqrt{7}-8\\sqrt[4]{11}+\\sqrt{7}+3\\sqrt[4]{11}[\/latex]<\/p>\r\nCombine like radicals; Note that [latex]\\sqrt{7}=1\\sqrt{7}[\/latex].<\/span><\/span>\r\n

    [latex] =6\\sqrt{7}-5\\sqrt[4]{11}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n \r\n\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

    Subtract radicals that must be simplified first<\/h2>\r\n
    \r\n

    Example 9<\/h3>\r\nSimplify:\u00a0[latex] 3\\sqrt{45}-8\\sqrt{20}[\/latex]\r\n\r\n[reveal-answer q=\"667355\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"667355\"]\r\n\r\nPrime factor each radicand and identify pairs of identical factors.\r\n

    [latex] 3\\sqrt{45}-8\\sqrt{20}= 3\\sqrt{3\\cdot 3\\cdot 5}-8\\sqrt{2\\cdot 2\\cdot 5}[\/latex]<\/p>\r\n

    [latex] =3\\sqrt{3^2\\cdot 5}-8\\sqrt{2^2\\cdot 5}[\/latex]<\/p>\r\n

    [latex] =3\\cdot 3\\sqrt{5}-8\\cdot 2\\sqrt{5}[\/latex]<\/p>\r\nMultiply factors outside each radical to simplify.<\/span><\/span>\r\n

    [latex] =9\\sqrt{5}-16\\sqrt{5}[\/latex]<\/p>\r\nNow that the radicals have been simplified, notice that we can now subtract like radicals.\r\n

    [latex] =-7\\sqrt{5}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n \r\n\r\nIn our last video, we show more examples of subtracting radicals that require simplifying.\r\n\r\nhttps:\/\/youtu.be\/6MogonN1PRQ\r\n

    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":"
    \n

    section 7.2 Learning Objectives<\/h3>\n

    7.2: Adding and Subtracting Radical Expressions<\/strong><\/p>\n

      \n
    • Identify like radicals<\/li>\n
    • Simplify an expression by adding and subtracting like radicals<\/li>\n
    • Add and subtract radicals that must be simplified first<\/li>\n<\/ul>\n<\/div>\n

       <\/p>\n

      Identify like radicals<\/h2>\n

      There are two keys to combining radicals by addition or subtraction: look at the index<\/strong>, and look at the 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

      \"Expression<\/p>\n

      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

      In this first example, both radicals have the same radicand and index.<\/p>\n

      \n

      Example 1<\/h3>\n

      Add. [latex]3\\sqrt{11}+7\\sqrt{11}[\/latex]<\/p>\n

      Show Solution<\/span><\/p>\n
      \n

      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

      [latex]\\text{3}\\sqrt{11}\\text{ + 7}\\sqrt{11}[\/latex]<\/p>\n

      The answer is [latex]10\\sqrt{11}[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n

      Simplify an expression by adding like radicals<\/h2>\n

      This next example contains more addends, or terms that are being added together. Notice how you can combine like<\/i> terms (radicals that have the same root and index), but you cannot combine unlike<\/i> terms.<\/p>\n

      \n

      Example 2<\/h3>\n

      Add. [latex]5\\sqrt{2}+\\sqrt{3}+4\\sqrt{3}+2\\sqrt{2}[\/latex]<\/p>\n

      Show Solution<\/span><\/p>\n
      \n

      Rearrange terms so that like radicals are next to each other. Then add.<\/p>\n

      [latex]5\\sqrt{2}+2\\sqrt{2}+\\sqrt{3}+4\\sqrt{3}[\/latex]<\/p>\n

      The answer is [latex]7\\sqrt{2}+5\\sqrt{3}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n

      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

      \n

      Example 3<\/h3>\n

      Add. [latex]3\\sqrt{5}+12\\sqrt[3]{7}+\\sqrt{5}[\/latex]<\/p>\n

      Show Solution<\/span><\/p>\n
      \n

      Rearrange terms so that like radicals are next to each other. Then add.<\/p>\n

      [latex]3\\sqrt{5}+\\sqrt{5}+12\\sqrt[3]{7}[\/latex]<\/p>\n

      The answer is [latex]4\\sqrt{5}+12\\sqrt[3]{7}[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n

      In the following video, we show more examples of how to identify and add like radicals.<\/p>\n