{"id":271,"date":"2015-09-18T20:26:18","date_gmt":"2015-09-18T20:26:18","guid":{"rendered":"https:\/\/courses.candelalearning.com\/collegealgebra1xmaster\/?post_type=chapter&p=271"},"modified":"2017-03-31T18:05:39","modified_gmt":"2017-03-31T18:05:39","slug":"understanding-nth-roots","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/chapter\/understanding-nth-roots\/","title":{"raw":"Understanding nth Roots","rendered":"Understanding nth Roots"},"content":{"raw":"Suppose we know that [latex]{a}^{3}=8[\/latex]. We want to find what number raised to the 3rd power is equal to 8. Since [latex]{2}^{3}=8[\/latex], we say that 2 is the cube root of 8.\r\n\r\nThe n<\/em>th<\/sup> root of [latex]a[\/latex] is a number that, when raised to the n<\/em>th<\/sup> power, gives [latex]a[\/latex]. For example, [latex]-3[\/latex] is the 5th root of [latex]-243[\/latex] because [latex]{\\left(-3\\right)}^{5}=-243[\/latex]. If [latex]a[\/latex] is a real number with at least one n<\/em>th<\/sup> root, then the principal n<\/em>th<\/sup> root<\/strong> of [latex]a[\/latex] is the number with the same sign as [latex]a[\/latex] that, when raised to the n<\/em>th<\/sup> power, equals [latex]a[\/latex].\r\n\r\nThe principal n<\/em>th<\/sup> root of [latex]a[\/latex] is written as [latex]\\sqrt[n]{a}[\/latex], where [latex]n[\/latex] is a positive integer greater than or equal to 2. In the radical expression, [latex]n[\/latex] is called the index<\/strong> of the radical.\r\n
\r\n

A General Note: Principal n<\/em>th Root<\/h3>\r\nIf [latex]a[\/latex] is a real number with at least one n<\/em>th<\/sup> root, then the principal n<\/em>th<\/sup> root<\/strong> of [latex]a[\/latex], written as [latex]\\sqrt[n]{a}[\/latex], is the number with the same sign as [latex]a[\/latex] that, when raised to the n<\/em>th<\/sup> power, equals [latex]a[\/latex]. The index<\/strong> of the radical is [latex]n[\/latex].\r\n\r\n<\/div>\r\n
\r\n

Example 10: Simplifying n<\/em>th Roots<\/h3>\r\nSimplify each of the following:\r\n
    \r\n \t
  1. [latex]\\sqrt[5]{-32}[\/latex]<\/li>\r\n \t
  2. [latex]\\sqrt[4]{4}\\cdot \\sqrt[4]{1,024}[\/latex]<\/li>\r\n \t
  3. [latex]-\\sqrt[3]{\\frac{8{x}^{6}}{125}}[\/latex]<\/li>\r\n \t
  4. [latex]8\\sqrt[4]{3}-\\sqrt[4]{48}[\/latex]<\/li>\r\n<\/ol>\r\n<\/div>\r\n
    \r\n

    Solution<\/h3>\r\n
      \r\n \t
    1. [latex]\\sqrt[5]{-32}=-2[\/latex] because [latex]{\\left(-2\\right)}^{5}=-32 \\\\ \\text{ }[\/latex]<\/li>\r\n \t
    2. First, express the product as a single radical expression. [latex]\\sqrt[4]{4,096}=8[\/latex] because [latex]{8}^{4}=4,096 \\\\[\/latex]<\/li>\r\n \t
    3. [latex]\\begin{array}{cc}\\\\ \\frac{-\\sqrt[3]{8{x}^{6}}}{\\sqrt[3]{125}}\\hfill & \\text{Write as quotient of two radical expressions}.\\hfill \\\\ \\frac{-2{x}^{2}}{5}\\hfill & \\text{Simplify}.\\hfill \\\\ \\end{array}[\/latex]<\/li>\r\n \t
    4. [latex]\\begin{array}{cc}\\\\ 8\\sqrt[4]{3}-2\\sqrt[4]{3}\\hfill & \\text{Simplify to get equal radicands}.\\hfill \\\\ 6\\sqrt[4]{3} \\hfill & \\text{Add}.\\hfill \\\\ \\end{array}[\/latex]<\/li>\r\n<\/ol>\r\n<\/div>\r\n
      \r\n

      Try It 10<\/h3>\r\nSimplify.\r\n
        \r\n \t
      1. [latex]\\sqrt[3]{-216}[\/latex]<\/li>\r\n \t
      2. [latex]\\frac{3\\sqrt[4]{80}}{\\sqrt[4]{5}}[\/latex]<\/li>\r\n \t
      3. [latex]6\\sqrt[3]{9,000}+7\\sqrt[3]{576}[\/latex]<\/li>\r\n<\/ol>\r\nSolution<\/a>\r\n\r\n<\/div>","rendered":"

        Suppose we know that [latex]{a}^{3}=8[\/latex]. We want to find what number raised to the 3rd power is equal to 8. Since [latex]{2}^{3}=8[\/latex], we say that 2 is the cube root of 8.<\/p>\n

        The n<\/em>th<\/sup> root of [latex]a[\/latex] is a number that, when raised to the n<\/em>th<\/sup> power, gives [latex]a[\/latex]. For example, [latex]-3[\/latex] is the 5th root of [latex]-243[\/latex] because [latex]{\\left(-3\\right)}^{5}=-243[\/latex]. If [latex]a[\/latex] is a real number with at least one n<\/em>th<\/sup> root, then the principal n<\/em>th<\/sup> root<\/strong> of [latex]a[\/latex] is the number with the same sign as [latex]a[\/latex] that, when raised to the n<\/em>th<\/sup> power, equals [latex]a[\/latex].<\/p>\n

        The principal n<\/em>th<\/sup> root of [latex]a[\/latex] is written as [latex]\\sqrt[n]{a}[\/latex], where [latex]n[\/latex] is a positive integer greater than or equal to 2. In the radical expression, [latex]n[\/latex] is called the index<\/strong> of the radical.<\/p>\n

        \n

        A General Note: Principal n<\/em>th Root<\/h3>\n

        If [latex]a[\/latex] is a real number with at least one n<\/em>th<\/sup> root, then the principal n<\/em>th<\/sup> root<\/strong> of [latex]a[\/latex], written as [latex]\\sqrt[n]{a}[\/latex], is the number with the same sign as [latex]a[\/latex] that, when raised to the n<\/em>th<\/sup> power, equals [latex]a[\/latex]. The index<\/strong> of the radical is [latex]n[\/latex].<\/p>\n<\/div>\n

        \n

        Example 10: Simplifying n<\/em>th Roots<\/h3>\n

        Simplify each of the following:<\/p>\n

          \n
        1. [latex]\\sqrt[5]{-32}[\/latex]<\/li>\n
        2. [latex]\\sqrt[4]{4}\\cdot \\sqrt[4]{1,024}[\/latex]<\/li>\n
        3. [latex]-\\sqrt[3]{\\frac{8{x}^{6}}{125}}[\/latex]<\/li>\n
        4. [latex]8\\sqrt[4]{3}-\\sqrt[4]{48}[\/latex]<\/li>\n<\/ol>\n<\/div>\n
          \n

          Solution<\/h3>\n
            \n
          1. [latex]\\sqrt[5]{-32}=-2[\/latex] because [latex]{\\left(-2\\right)}^{5}=-32 \\\\ \\text{ }[\/latex]<\/li>\n
          2. First, express the product as a single radical expression. [latex]\\sqrt[4]{4,096}=8[\/latex] because [latex]{8}^{4}=4,096 \\\\[\/latex]<\/li>\n
          3. [latex]\\begin{array}{cc}\\\\ \\frac{-\\sqrt[3]{8{x}^{6}}}{\\sqrt[3]{125}}\\hfill & \\text{Write as quotient of two radical expressions}.\\hfill \\\\ \\frac{-2{x}^{2}}{5}\\hfill & \\text{Simplify}.\\hfill \\\\ \\end{array}[\/latex]<\/li>\n
          4. [latex]\\begin{array}{cc}\\\\ 8\\sqrt[4]{3}-2\\sqrt[4]{3}\\hfill & \\text{Simplify to get equal radicands}.\\hfill \\\\ 6\\sqrt[4]{3} \\hfill & \\text{Add}.\\hfill \\\\ \\end{array}[\/latex]<\/li>\n<\/ol>\n<\/div>\n
            \n

            Try It 10<\/h3>\n

            Simplify.<\/p>\n

              \n
            1. [latex]\\sqrt[3]{-216}[\/latex]<\/li>\n
            2. [latex]\\frac{3\\sqrt[4]{80}}{\\sqrt[4]{5}}[\/latex]<\/li>\n
            3. [latex]6\\sqrt[3]{9,000}+7\\sqrt[3]{576}[\/latex]<\/li>\n<\/ol>\n

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

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