{"id":273,"date":"2015-09-18T20:26:53","date_gmt":"2015-09-18T20:26:53","guid":{"rendered":"https:\/\/courses.candelalearning.com\/collegealgebra1xmaster\/?post_type=chapter&p=273"},"modified":"2015-11-03T18:52:04","modified_gmt":"2015-11-03T18:52:04","slug":"using-rational-exponents","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/chapter\/using-rational-exponents\/","title":{"raw":"Using Rational Exponents","rendered":"Using Rational Exponents"},"content":{"raw":"Radical expressions can also be written without using the radical symbol. We can use rational (fractional) exponents. The index must be a positive integer. If the index [latex]n[\/latex] is even, then [latex]a[\/latex] cannot be negative.\r\n
[latex]{a}^{\\frac{1}{n}}=\\sqrt[n]{a}[\/latex]<\/div>\r\nWe can also have rational exponents with numerators other than 1. In these cases, the exponent must be a fraction in lowest terms. We raise the base to a power and take an n<\/em>th root. The numerator tells us the power and the denominator tells us the root.\r\n
[latex]{a}^{\\frac{m}{n}}={\\left(\\sqrt[n]{a}\\right)}^{m}=\\sqrt[n]{{a}^{m}}[\/latex]<\/div>\r\nAll of the properties of exponents that we learned for integer exponents also hold for rational exponents.\r\n
\r\n

Example 11: Rational Exponents<\/h3>\r\nRational exponents are another way to express principal n<\/em>th roots. The general form for converting between a radical expression with a radical symbol and one with a rational exponent is\r\n
[latex]{a}^{\\frac{m}{n}}={\\left(\\sqrt[n]{a}\\right)}^{m}=\\sqrt[n]{{a}^{m}}[\/latex]<\/div>\r\n<\/div>\r\n
\r\n

How To: Given an expression with a rational exponent, write the expression as a radical.<\/h3>\r\n
    \r\n\t
  1. Determine the power by looking at the numerator of the exponent.<\/li>\r\n\t
  2. Determine the root by looking at the denominator of the exponent.<\/li>\r\n\t
  3. Using the base as the radicand, raise the radicand to the power and use the root as the index.<\/li>\r\n<\/ol>\r\n<\/div>\r\n
    \r\n

    Example 11: Writing Rational Exponents as Radicals<\/h3>\r\nWrite [latex]{343}^{\\frac{2}{3}}[\/latex] as a radical. Simplify.\r\n\r\n<\/div>\r\n
    \r\n

    Solution<\/h3>\r\nThe 2 tells us the power and the 3 tells us the root.\r\n

    [latex]{343}^{\\frac{2}{3}}={\\left(\\sqrt[3]{343}\\right)}^{2}=\\sqrt[3]{{343}^{2}}[\/latex]<\/p>\r\nWe know that [latex]\\sqrt[3]{343}=7[\/latex] because [latex]{7}^{3}=343[\/latex]. Because the cube root is easy to find, it is easiest to find the cube root before squaring for this problem. In general, it is easier to find the root first and then raise it to a power.\r\n

    [latex]{343}^{\\frac{2}{3}}={\\left(\\sqrt[3]{343}\\right)}^{2}={7}^{2}=49[\/latex]<\/p>\r\n\r\n<\/div>\r\n

    \r\n

    Try It 11<\/h3>\r\nWrite [latex]{9}^{\\frac{5}{2}}[\/latex] as a radical. Simplify.\r\n\r\nSolution<\/a>\r\n\r\n<\/div>\r\n
    \r\n

    Example 12: Writing Radicals as Rational Exponents<\/h3>\r\nWrite [latex]\\frac{4}{\\sqrt[7]{{a}^{2}}}[\/latex] using a rational exponent.\r\n\r\n<\/div>\r\n
    \r\n

    Solution<\/h3>\r\nThe power is 2 and the root is 7, so the rational exponent will be [latex]\\frac{2}{7}[\/latex]. We get [latex]\\frac{4}{{a}^{\\frac{2}{7}}}[\/latex]. Using properties of exponents, we get [latex]\\frac{4}{\\sqrt[7]{{a}^{2}}}=4{a}^{\\frac{-2}{7}}[\/latex].\r\n\r\n<\/div>\r\n
    \r\n

    Try It 12<\/h3>\r\nWrite [latex]x\\sqrt{{\\left(5y\\right)}^{9}}[\/latex] using a rational exponent.\r\n\r\nSolution<\/a>\r\n\r\n<\/div>\r\n
    \r\n

    Example 13: Simplifying Rational Exponents<\/h3>\r\nSimplify:\r\n
      \r\n\t
    1. [latex]5\\left(2{x}^{\\frac{3}{4}}\\right)\\left(3{x}^{\\frac{1}{5}}\\right)[\/latex]<\/li>\r\n\t
    2. [latex]{\\left(\\frac{16}{9}\\right)}^{-\\frac{1}{2}}[\/latex]<\/li>\r\n<\/ol>\r\n<\/div>\r\n
      \r\n

      Solution<\/h3>\r\n
        \r\n\t
      1. [latex]\\begin{array}{cc}30{x}^{\\frac{3}{4}}{x}^{\\frac{1}{5}}\\hfill & \\text{Multiply the coefficients}.\\\\hfill \\\\ 30{x}^{\\frac{3}{4}+\\frac{1}{5}}\\hfill & \\text{Use properties of exponents}.\\hfill \\\\ 30{x}^{\\frac{19}{20}}\\hfill & \\text{Simplify}.\\hfill \\end{array}[\/latex]<\/li>\r\n\t
      2. \r\n
        \u00a0[latex]\\begin{array}{cc}{\\left(\\frac{9}{16}\\right)}^{\\frac{1}{2}}\\hfill & \\text{ }\\text{Use definition of negative exponents}.\\hfill \\\\ \\sqrt{\\frac{9}{16}}\\hfill & \\text{ }\\text{Rewrite as a radical}.\\hfill \\\\ \\frac{\\sqrt{9}}{\\sqrt{16}}\\hfill & \\text{ }\\text{Use the quotient rule}.\\hfill \\\\ \\frac{3}{4}\\hfill & \\text{ }\\text{Simplify}.\\hfill \\end{array}[\/latex]<\/div><\/li>\r\n<\/ol>\r\n<\/div>\r\n
        \r\n

        Try It 13<\/h3>\r\nSimplify [latex]{\\left(8x\\right)}^{\\frac{1}{3}}\\left(14{x}^{\\frac{6}{5}}\\right)[\/latex].\r\n\r\nSolution<\/a>\r\n\r\n<\/div>","rendered":"

        Radical expressions can also be written without using the radical symbol. We can use rational (fractional) exponents. The index must be a positive integer. If the index [latex]n[\/latex] is even, then [latex]a[\/latex] cannot be negative.<\/p>\n

        [latex]{a}^{\\frac{1}{n}}=\\sqrt[n]{a}[\/latex]<\/div>\n

        We can also have rational exponents with numerators other than 1. In these cases, the exponent must be a fraction in lowest terms. We raise the base to a power and take an n<\/em>th root. The numerator tells us the power and the denominator tells us the root.<\/p>\n

        [latex]{a}^{\\frac{m}{n}}={\\left(\\sqrt[n]{a}\\right)}^{m}=\\sqrt[n]{{a}^{m}}[\/latex]<\/div>\n

        All of the properties of exponents that we learned for integer exponents also hold for rational exponents.<\/p>\n

        \n

        Example 11: Rational Exponents<\/h3>\n

        Rational exponents are another way to express principal n<\/em>th roots. The general form for converting between a radical expression with a radical symbol and one with a rational exponent is<\/p>\n

        [latex]{a}^{\\frac{m}{n}}={\\left(\\sqrt[n]{a}\\right)}^{m}=\\sqrt[n]{{a}^{m}}[\/latex]<\/div>\n<\/div>\n
        \n

        How To: Given an expression with a rational exponent, write the expression as a radical.<\/h3>\n
          \n
        1. Determine the power by looking at the numerator of the exponent.<\/li>\n
        2. Determine the root by looking at the denominator of the exponent.<\/li>\n
        3. Using the base as the radicand, raise the radicand to the power and use the root as the index.<\/li>\n<\/ol>\n<\/div>\n
          \n

          Example 11: Writing Rational Exponents as Radicals<\/h3>\n

          Write [latex]{343}^{\\frac{2}{3}}[\/latex] as a radical. Simplify.<\/p>\n<\/div>\n

          \n

          Solution<\/h3>\n

          The 2 tells us the power and the 3 tells us the root.<\/p>\n

          [latex]{343}^{\\frac{2}{3}}={\\left(\\sqrt[3]{343}\\right)}^{2}=\\sqrt[3]{{343}^{2}}[\/latex]<\/p>\n

          We know that [latex]\\sqrt[3]{343}=7[\/latex] because [latex]{7}^{3}=343[\/latex]. Because the cube root is easy to find, it is easiest to find the cube root before squaring for this problem. In general, it is easier to find the root first and then raise it to a power.<\/p>\n

          [latex]{343}^{\\frac{2}{3}}={\\left(\\sqrt[3]{343}\\right)}^{2}={7}^{2}=49[\/latex]<\/p>\n<\/div>\n

          \n

          Try It 11<\/h3>\n

          Write [latex]{9}^{\\frac{5}{2}}[\/latex] as a radical. Simplify.<\/p>\n

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

          \n

          Example 12: Writing Radicals as Rational Exponents<\/h3>\n

          Write [latex]\\frac{4}{\\sqrt[7]{{a}^{2}}}[\/latex] using a rational exponent.<\/p>\n<\/div>\n

          \n

          Solution<\/h3>\n

          The power is 2 and the root is 7, so the rational exponent will be [latex]\\frac{2}{7}[\/latex]. We get [latex]\\frac{4}{{a}^{\\frac{2}{7}}}[\/latex]. Using properties of exponents, we get [latex]\\frac{4}{\\sqrt[7]{{a}^{2}}}=4{a}^{\\frac{-2}{7}}[\/latex].<\/p>\n<\/div>\n

          \n

          Try It 12<\/h3>\n

          Write [latex]x\\sqrt{{\\left(5y\\right)}^{9}}[\/latex] using a rational exponent.<\/p>\n

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

          \n

          Example 13: Simplifying Rational Exponents<\/h3>\n

          Simplify:<\/p>\n

            \n
          1. [latex]5\\left(2{x}^{\\frac{3}{4}}\\right)\\left(3{x}^{\\frac{1}{5}}\\right)[\/latex]<\/li>\n
          2. [latex]{\\left(\\frac{16}{9}\\right)}^{-\\frac{1}{2}}[\/latex]<\/li>\n<\/ol>\n<\/div>\n
            \n

            Solution<\/h3>\n
              \n
            1. [latex]\\begin{array}{cc}30{x}^{\\frac{3}{4}}{x}^{\\frac{1}{5}}\\hfill & \\text{Multiply the coefficients}.\\\\hfill \\\\ 30{x}^{\\frac{3}{4}+\\frac{1}{5}}\\hfill & \\text{Use properties of exponents}.\\hfill \\\\ 30{x}^{\\frac{19}{20}}\\hfill & \\text{Simplify}.\\hfill \\end{array}[\/latex]<\/li>\n
            2. \n
              \u00a0[latex]\\begin{array}{cc}{\\left(\\frac{9}{16}\\right)}^{\\frac{1}{2}}\\hfill & \\text{ }\\text{Use definition of negative exponents}.\\hfill \\\\ \\sqrt{\\frac{9}{16}}\\hfill & \\text{ }\\text{Rewrite as a radical}.\\hfill \\\\ \\frac{\\sqrt{9}}{\\sqrt{16}}\\hfill & \\text{ }\\text{Use the quotient rule}.\\hfill \\\\ \\frac{3}{4}\\hfill & \\text{ }\\text{Simplify}.\\hfill \\end{array}[\/latex]<\/div>\n<\/li>\n<\/ol>\n<\/div>\n
              \n

              Try It 13<\/h3>\n

              Simplify [latex]{\\left(8x\\right)}^{\\frac{1}{3}}\\left(14{x}^{\\frac{6}{5}}\\right)[\/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
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