{"id":266,"date":"2015-09-18T20:25:02","date_gmt":"2015-09-18T20:25:02","guid":{"rendered":"https:\/\/courses.candelalearning.com\/collegealgebra1xmaster\/?post_type=chapter&p=266"},"modified":"2015-11-03T01:09:31","modified_gmt":"2015-11-03T01:09:31","slug":"adding-and-subtracting-square-roots","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/ivytech-collegealgebra\/chapter\/adding-and-subtracting-square-roots\/","title":{"raw":"Adding and Subtracting Square Roots","rendered":"Adding and Subtracting Square Roots"},"content":{"raw":"We can add or subtract radical expressions only when they have the same radicand and when they have the same radical type such as square roots. For example, the sum of [latex]\\sqrt{2}[\/latex] and [latex]3\\sqrt{2}[\/latex] is [latex]4\\sqrt{2}[\/latex]. However, it is often possible to simplify radical expressions, and that may change the radicand. The radical expression [latex]\\sqrt{18}[\/latex] can be written with a [latex]2[\/latex] in the radicand, as [latex]3\\sqrt{2}[\/latex], so [latex]\\sqrt{2}+\\sqrt{18}=\\sqrt{2}+3\\sqrt{2}=4\\sqrt{2}[\/latex].\r\n
\r\n

How To: Given a radical expression requiring addition or subtraction of square roots, solve.<\/h3>\r\n
    \r\n\t
  1. Simplify each radical expression.<\/li>\r\n\t
  2. Add or subtract expressions with equal radicands.<\/li>\r\n<\/ol>\r\n<\/div>\r\n
    \r\n

    Example 6: Adding Square Roots<\/h3>\r\nAdd [latex]5\\sqrt{12}+2\\sqrt{3}\\\\[\/latex].\r\n\r\n<\/div>\r\n
    \r\n

    Solution<\/h3>\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

    [latex]10\\sqrt{3}+2\\sqrt{3}=12\\sqrt{3}[\/latex]<\/p>\r\n\r\n<\/div>\r\n

    \r\n

    Try It 6<\/h3>\r\nAdd [latex]\\sqrt{5}+6\\sqrt{20}[\/latex].\r\n\r\nSolution<\/a>\r\n\r\n<\/div>\r\n
    \r\n

    Example 7: 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<\/div>\r\n
    \r\n

    Solution<\/h3>\r\nRewrite each term so they have equal radicands.\r\n
    [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>\r\n
    [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>\r\nNow the terms have the same radicand so we can subtract.\r\n
    [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>\r\n
    \r\n

    Try It 7<\/h3>\r\nSubtract [latex]3\\sqrt{80x}-4\\sqrt{45x}[\/latex].\r\n\r\nSolution<\/a>\r\n\r\n<\/div>","rendered":"

    We can add or subtract radical expressions only when they have the same radicand and when they have the same radical type such as square roots. For example, the sum of [latex]\\sqrt{2}[\/latex] and [latex]3\\sqrt{2}[\/latex] is [latex]4\\sqrt{2}[\/latex]. However, it is often possible to simplify radical expressions, and that may change the radicand. The radical expression [latex]\\sqrt{18}[\/latex] can be written with a [latex]2[\/latex] in the radicand, as [latex]3\\sqrt{2}[\/latex], so [latex]\\sqrt{2}+\\sqrt{18}=\\sqrt{2}+3\\sqrt{2}=4\\sqrt{2}[\/latex].<\/p>\n

    \n

    How To: Given a radical expression requiring addition or subtraction of square roots, solve.<\/h3>\n
      \n
    1. Simplify each radical expression.<\/li>\n
    2. Add or subtract expressions with equal radicands.<\/li>\n<\/ol>\n<\/div>\n
      \n

      Example 6: Adding Square Roots<\/h3>\n

      Add [latex]5\\sqrt{12}+2\\sqrt{3}\\\\[\/latex].<\/p>\n<\/div>\n

      \n

      Solution<\/h3>\n

      We 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

      [latex]10\\sqrt{3}+2\\sqrt{3}=12\\sqrt{3}[\/latex]<\/p>\n<\/div>\n

      \n

      Try It 6<\/h3>\n

      Add [latex]\\sqrt{5}+6\\sqrt{20}[\/latex].<\/p>\n

      Solution<\/a><\/p>\n<\/div>\n

      \n

      Example 7: Subtracting Square Roots<\/h3>\n

      Subtract [latex]20\\sqrt{72{a}^{3}{b}^{4}c}-14\\sqrt{8{a}^{3}{b}^{4}c}[\/latex].<\/p>\n<\/div>\n

      \n

      Solution<\/h3>\n

      Rewrite each term so they have equal radicands.<\/p>\n

      [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
      [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

      Now the terms have the same radicand so we can subtract.<\/p>\n

      [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>\n
      \n

      Try It 7<\/h3>\n

      Subtract [latex]3\\sqrt{80x}-4\\sqrt{45x}[\/latex].<\/p>\n

      Solution<\/a><\/p>\n<\/div>\n\n\t\t\t

      \n\t\t\t

      Candela Citations<\/h3>\n\t\t\t\t\t
      \n\t\t\t\t\t\t
      \n\t\t\t\t\t\t\t
      CC licensed content, Specific attribution<\/div>