{"id":689,"date":"2019-04-29T16:38:47","date_gmt":"2019-04-29T20:38:47","guid":{"rendered":"https:\/\/pressbooks.bccampus.ca\/intermediatealgebrakpu\/?post_type=chapter&p=689"},"modified":"2019-12-05T13:23:55","modified_gmt":"2019-12-05T18:23:55","slug":"9-6-radicals-and-rational-exponents","status":"publish","type":"chapter","link":"https:\/\/pressbooks.bccampus.ca\/intermediatealgebrakpu\/chapter\/9-6-radicals-and-rational-exponents\/","title":{"raw":"9.6 Radicals and Rational Exponents","rendered":"9.6 Radicals and Rational Exponents"},"content":{"raw":"[latexpage]\r\n\r\nWhen simplifying radicals that use fractional exponents, the numerator on the exponent is divided by the denominator. All radicals can be shown as having an equivalent fractional exponent. For example:\r\n

\\(\\sqrt{x}=x^{\\frac{1}{2}}\\hspace{0.25in} \\sqrt[3]{x}=x^{\\frac{1}{3}}\\hspace{0.25in} \\sqrt[4]{x}=x^{\\frac{1}{4}}\\hspace{0.25in} \\sqrt[5]{x}=x^{\\frac{1}{5}}\\)<\/p>\r\nRadicals having some exponent value inside the radical can also be written as a fractional exponent. For example:\r\n

\\(\\sqrt{x^3}=x^{\\frac{3}{2}}\\hspace{0.25in} \\sqrt[3]{x^2}=x^{\\frac{2}{3}}\\hspace{0.25in} \\sqrt[4]{x^5}=x^{\\frac{5}{4}}\\hspace{0.25in} \\sqrt[5]{x^9}=x^{\\frac{9}{5}}\\)<\/p>\r\nThe general form that radicals having exponents take is:\r\n

\\(x^{\\frac{b}{a}}=\\sqrt[a]{x^b}\\text{ or }(\\sqrt[a]{x})^b\\)<\/p>\r\nShould the reciprocal of a radical having an exponent, it would look as follows:\r\n

\\(x^{-\\frac{b}{a}}=\\dfrac{1}{\\sqrt[a]{x^b}}\\text{ or }\\dfrac{1}{(\\sqrt[a]{x})^b}\\)<\/p>\r\nIn both cases shown above, the power of the radical is \\(b\\) and the root of the radical is \\(a\\). These are the two forms that a radical having an exponent is commonly written in. It is convenient to work with a radical containing an exponent in one of these two forms.\r\n

\r\n

Example 9.6.1<\/p>\r\n\r\n<\/header>\r\n

\r\n\r\nEvaluate \\(27^{-\\frac{4}{3}}\\).\r\n\r\nConverting to a radical form:\r\n

\\(\\dfrac{1}{\\sqrt[3]{27^4}}\\text{ or }\\dfrac{1}{(\\sqrt[3]{27})^4}\\)<\/p>\r\nFirst, the cube root of 27 will reduce to 3, which leaves:\r\n

\\(\\dfrac{1}{3^4}\\text{ or }\\dfrac{1}{81}\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\nOnce the radical having an exponent is converted into a pure fractional exponent, then the following rules can be used.\r\n

Properties of Exponents<\/h2>\r\n

\\(\\begin{array}{ccc}\r\na^ma^n=a^{m+n}\\hspace{0.25in} &(ab)^m=a^mb^m\\hspace{0.25in} &a^{-m}=\\dfrac{1}{a^m} \\\\ \\\\\r\n\\dfrac{a^m}{a^n}=a^{m-n}&\\left(\\dfrac{a}{b}\\right)=\\dfrac{a^m}{b^m}&\\dfrac{1}{a^{-m}}=a^m \\\\ \\\\\r\n(a^m)^n=a^{mn}&a^0=1&\\left(\\dfrac{a}{b}\\right)^{-m}=\\dfrac{b^m}{a^m}\r\n\\end{array}\\)<\/p>\r\n\r\n

\r\n

Example 9.6.2<\/p>\r\n\r\n<\/header>\r\n

\r\n\r\nSimplify \\((x^2y^{\\frac{4}{3}})(x^{-1}y^\\frac{2}{3})\\).\r\n\r\nFirst, you need to separate the different variables:\r\n

\\((x^2y^{\\frac{4}{3}})(x^{-1}y^\\frac{2}{3})\\) becomes \\(x^2\\cdot x^{-1}\\cdot y^{\\frac{4}{3}}\\cdot y^{\\frac{2}{3}}\\)<\/p>\r\nCombining the exponents yields:\r\n

\\(x^{2 - 1}\\cdot y^{\\frac{4}{3}+\\frac{2}{3}}\\)<\/p>\r\nWhich results in:\r\n

\\(x^1\\cdot y^{\\frac{6}{3}}\\)<\/p>\r\nWhich simplifies to:\r\n

\\(xy^2\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

\r\n

Example 9.6.3<\/p>\r\n\r\n<\/header>\r\n

\r\n\r\nSimplify \\(\\dfrac{ab^{\\frac{2}{3}}3b^{-\\frac{5}{3}}}{5a^{-\\frac{3}{2}}b^{-\\frac{4}{3}}}\\).\r\n\r\nFirst, separate the different variables:\r\n

\\(\\dfrac{ab^{\\frac{2}{3}}3b^{-\\frac{5}{3}}}{5a^{-\\frac{3}{2}}b^{-\\frac{4}{3}}}\\) becomes \\(3\\cdot 5^{-1}\\cdot a \\cdot a^{\\frac{3}{2}}\\cdot b^{\\frac{2}{3}}\\cdot b^{-\\frac{5}{3}}\\cdot b^{\\frac{4}{3}}\\)[footnote]When we divide by an exponent, we subtract powers.[\/footnote]<\/p>\r\nCombining the exponents yields:\r\n

\\(3\\cdot 5^{-1}\\cdot a^{1+\\frac{3}{2}}\\cdot b^{\\frac{2}{3}-\\frac{5}{3}+\\frac{4}{3}}\\)<\/p>\r\n

Which gives:<\/p>\r\n

\\(3\\cdot 5^{-1}\\cdot a^{\\frac{5}{2}}\\cdot b^{\\frac{1}{3}}\\)<\/p>\r\n

Which simplifies to:<\/p>\r\n

\\(\\dfrac{3\\cdot a^{\\frac{5}{2}}\\cdot b^{\\frac{1}{3}}}{5}\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

Questions<\/h1>\r\nWrite each of the following fractional exponents in radical form.\r\n
    \r\n \t
  1. \\(m^{\\frac{3}{5}}\\)<\/li>\r\n \t
  2. \\((10r)^{-\\frac{3}{4}}\\)<\/li>\r\n \t
  3. \\((7x)^{\\frac{3}{2}}\\)<\/li>\r\n \t
  4. \\((6b)^{-\\frac{4}{3}}\\)<\/li>\r\n \t
  5. \\((2x+3)^{-\\frac{3}{2}}\\)<\/li>\r\n \t
  6. \\((x-3y)^{\\frac{3}{4}}\\)<\/li>\r\n<\/ol>\r\nWrite each of the following radicals in exponential form.\r\n
      \r\n \t
    1. \\(\\sqrt[3]{5}\\)<\/li>\r\n \t
    2. \\(\\sqrt[5]{2^3}\\)<\/li>\r\n \t
    3. \\(\\sqrt[3]{ab^5}\\)<\/li>\r\n \t
    4. \\(\\sqrt[5]{x^3}\\)<\/li>\r\n \t
    5. \\(\\sqrt[3]{(a+5)^2}\\)<\/li>\r\n \t
    6. \\(\\sqrt[5]{(a-2)^3}\\)<\/li>\r\n<\/ol>\r\nEvaluate the following.\r\n
        \r\n \t
      1. \\(8^{\\frac{2}{3}}\\)<\/li>\r\n \t
      2. \\(16^{\\frac{1}{4}}\\)<\/li>\r\n \t
      3. \\(\\sqrt[3]{4^6}\\)<\/li>\r\n \t
      4. \\(\\sqrt[5]{32^2}\\)<\/li>\r\n<\/ol>\r\nSimplify. Your answer should only contain positive exponents.\r\n
          \r\n \t
        1. \\((xy^{\\frac{1}{3}})(xy^{\\frac{2}{3}})\\)<\/li>\r\n \t
        2. \\((4v^{\\frac{2}{3}})(v^{-1})\\)<\/li>\r\n \t
        3. \\((a^{\\frac{1}{2}}b^{\\frac{1}{2}})^{-1}\\)<\/li>\r\n \t
        4. \\((x^{\\frac{5}{3}}y^{-2})^0\\)<\/li>\r\n \t
        5. \\(\\dfrac{a^2b^0}{3a^4}\\)<\/li>\r\n \t
        6. \\(\\dfrac{2x^{\\frac{1}{2}}y^{\\frac{1}{3}}}{2x^{\\frac{4}{3}}y^{\\frac{7}{4}}}\\)<\/li>\r\n \t
        7. \\(\\dfrac{a^{\\frac{3}{4}}b^{-1}b^{\\frac{7}{4}}}{3b^{-1}}\\)<\/li>\r\n \t
        8. \\(\\dfrac{2x^{-2}y^{\\frac{5}{3}}}{x^{-\\frac{5}{4}}y^{-\\frac{5}{3}}xy^{\\frac{1}{2}}}\\)<\/li>\r\n \t
        9. \\(\\dfrac{3y^{-\\frac{5}{4}}}{y^{-1}2y^{-\\frac{1}{3}}}\\)<\/li>\r\n \t
        10. \\(\\dfrac{ab^{\\frac{1}{3}}2b^{-\\frac{5}{4}}}{4a^{-\\frac{1}{2}}b^{-\\frac{2}{3}}}\\)<\/li>\r\n<\/ol>\r\nAnswer Key 9.6<\/a>","rendered":"

          When simplifying radicals that use fractional exponents, the numerator on the exponent is divided by the denominator. All radicals can be shown as having an equivalent fractional exponent. For example:<\/p>\n

          \"\sqrt{x}=x^{\frac{1}{2}}\hspace{0.25in} \sqrt[3]{x}=x^{\frac{1}{3}}\hspace{0.25in} \sqrt[4]{x}=x^{\frac{1}{4}}\hspace{0.25in} \sqrt[5]{x}=x^{\frac{1}{5}}\"<\/p>\n

          Radicals having some exponent value inside the radical can also be written as a fractional exponent. For example:<\/p>\n

          \"\sqrt{x^3}=x^{\frac{3}{2}}\hspace{0.25in} \sqrt[3]{x^2}=x^{\frac{2}{3}}\hspace{0.25in} \sqrt[4]{x^5}=x^{\frac{5}{4}}\hspace{0.25in} \sqrt[5]{x^9}=x^{\frac{9}{5}}\"<\/p>\n

          The general form that radicals having exponents take is:<\/p>\n

          \"x^{\frac{b}{a}}=\sqrt[a]{x^b}\text{ or }(\sqrt[a]{x})^b\"<\/p>\n

          Should the reciprocal of a radical having an exponent, it would look as follows:<\/p>\n

          \"x^{-\frac{b}{a}}=\dfrac{1}{\sqrt[a]{x^b}}\text{ or }\dfrac{1}{(\sqrt[a]{x})^b}\"<\/p>\n

          In both cases shown above, the power of the radical is \"b\" and the root of the radical is \"a\". These are the two forms that a radical having an exponent is commonly written in. It is convenient to work with a radical containing an exponent in one of these two forms.<\/p>\n

          \n
          \n

          Example 9.6.1<\/p>\n<\/header>\n

          \n

          Evaluate \"27^{-\frac{4}{3}}\".<\/p>\n

          Converting to a radical form:<\/p>\n

          \"\dfrac{1}{\sqrt[3]{27^4}}\text{ or }\dfrac{1}{(\sqrt[3]{27})^4}\"<\/p>\n

          First, the cube root of 27 will reduce to 3, which leaves:<\/p>\n

          \"\dfrac{1}{3^4}\text{ or }\dfrac{1}{81}\"<\/p>\n<\/div>\n<\/div>\n

          Once the radical having an exponent is converted into a pure fractional exponent, then the following rules can be used.<\/p>\n

          Properties of Exponents<\/h2>\n

          \"\begin{array}{ccc}<\/p>\n

          \n
          \n

          Example 9.6.2<\/p>\n<\/header>\n

          \n

          Simplify \"(x^2y^{\frac{4}{3}})(x^{-1}y^\frac{2}{3})\".<\/p>\n

          First, you need to separate the different variables:<\/p>\n

          \"(x^2y^{\frac{4}{3}})(x^{-1}y^\frac{2}{3})\" becomes \"x^2\cdot x^{-1}\cdot y^{\frac{4}{3}}\cdot y^{\frac{2}{3}}\"<\/p>\n

          Combining the exponents yields:<\/p>\n

          \"x^{2 - 1}\cdot y^{\frac{4}{3}+\frac{2}{3}}\"<\/p>\n

          Which results in:<\/p>\n

          \"x^1\cdot y^{\frac{6}{3}}\"<\/p>\n

          Which simplifies to:<\/p>\n

          \"xy^2\"<\/p>\n<\/div>\n<\/div>\n

          \n
          \n

          Example 9.6.3<\/p>\n<\/header>\n

          \n

          Simplify \"\dfrac{ab^{\frac{2}{3}}3b^{-\frac{5}{3}}}{5a^{-\frac{3}{2}}b^{-\frac{4}{3}}}\".<\/p>\n

          First, separate the different variables:<\/p>\n

          \"\dfrac{ab^{\frac{2}{3}}3b^{-\frac{5}{3}}}{5a^{-\frac{3}{2}}b^{-\frac{4}{3}}}\" becomes \"3\cdot 5^{-1}\cdot a \cdot a^{\frac{3}{2}}\cdot b^{\frac{2}{3}}\cdot b^{-\frac{5}{3}}\cdot b^{\frac{4}{3}}\"[1]<\/sup><\/a><\/p>\n

          Combining the exponents yields:<\/p>\n

          \"3\cdot 5^{-1}\cdot a^{1+\frac{3}{2}}\cdot b^{\frac{2}{3}-\frac{5}{3}+\frac{4}{3}}\"<\/p>\n

          Which gives:<\/p>\n

          \"3\cdot 5^{-1}\cdot a^{\frac{5}{2}}\cdot b^{\frac{1}{3}}\"<\/p>\n

          Which simplifies to:<\/p>\n

          \"\dfrac{3\cdot a^{\frac{5}{2}}\cdot b^{\frac{1}{3}}}{5}\"<\/p>\n<\/div>\n<\/div>\n

          Questions<\/h1>\n

          Write each of the following fractional exponents in radical form.<\/p>\n

            \n
          1. \"m^{\frac{3}{5}}\"<\/li>\n
          2. \"(10r)^{-\frac{3}{4}}\"<\/li>\n
          3. \"(7x)^{\frac{3}{2}}\"<\/li>\n
          4. \"(6b)^{-\frac{4}{3}}\"<\/li>\n
          5. \"(2x+3)^{-\frac{3}{2}}\"<\/li>\n
          6. \"(x-3y)^{\frac{3}{4}}\"<\/li>\n<\/ol>\n

            Write each of the following radicals in exponential form.<\/p>\n

              \n
            1. \"\sqrt[3]{5}\"<\/li>\n
            2. \"\sqrt[5]{2^3}\"<\/li>\n
            3. \"\sqrt[3]{ab^5}\"<\/li>\n
            4. \"\sqrt[5]{x^3}\"<\/li>\n
            5. \"\sqrt[3]{(a+5)^2}\"<\/li>\n
            6. \"\sqrt[5]{(a-2)^3}\"<\/li>\n<\/ol>\n

              Evaluate the following.<\/p>\n

                \n
              1. \"8^{\frac{2}{3}}\"<\/li>\n
              2. \"16^{\frac{1}{4}}\"<\/li>\n
              3. \"\sqrt[3]{4^6}\"<\/li>\n
              4. \"\sqrt[5]{32^2}\"<\/li>\n<\/ol>\n

                Simplify. Your answer should only contain positive exponents.<\/p>\n

                  \n
                1. \"(xy^{\frac{1}{3}})(xy^{\frac{2}{3}})\"<\/li>\n
                2. \"(4v^{\frac{2}{3}})(v^{-1})\"<\/li>\n
                3. \"(a^{\frac{1}{2}}b^{\frac{1}{2}})^{-1}\"<\/li>\n
                4. \"(x^{\frac{5}{3}}y^{-2})^0\"<\/li>\n
                5. \"\dfrac{a^2b^0}{3a^4}\"<\/li>\n
                6. \"\dfrac{2x^{\frac{1}{2}}y^{\frac{1}{3}}}{2x^{\frac{4}{3}}y^{\frac{7}{4}}}\"<\/li>\n
                7. \"\dfrac{a^{\frac{3}{4}}b^{-1}b^{\frac{7}{4}}}{3b^{-1}}\"<\/li>\n
                8. \"\dfrac{2x^{-2}y^{\frac{5}{3}}}{x^{-\frac{5}{4}}y^{-\frac{5}{3}}xy^{\frac{1}{2}}}\"<\/li>\n
                9. \"\dfrac{3y^{-\frac{5}{4}}}{y^{-1}2y^{-\frac{1}{3}}}\"<\/li>\n
                10. \"\dfrac{ab^{\frac{1}{3}}2b^{-\frac{5}{4}}}{4a^{-\frac{1}{2}}b^{-\frac{2}{3}}}\"<\/li>\n<\/ol>\n

                  Answer Key 9.6<\/a><\/p>\n

                  1. When we divide by an exponent, we subtract powers. ↵<\/a><\/li><\/ol><\/div>","protected":false},"author":540,"menu_order":23,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-689","chapter","type-chapter","status-publish","hentry"],"part":389,"_links":{"self":[{"href":"https:\/\/pressbooks.bccampus.ca\/intermediatealgebrakpu\/wp-json\/pressbooks\/v2\/chapters\/689","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/pressbooks.bccampus.ca\/intermediatealgebrakpu\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/pressbooks.bccampus.ca\/intermediatealgebrakpu\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/intermediatealgebrakpu\/wp-json\/wp\/v2\/users\/540"}],"version-history":[{"count":7,"href":"https:\/\/pressbooks.bccampus.ca\/intermediatealgebrakpu\/wp-json\/pressbooks\/v2\/chapters\/689\/revisions"}],"predecessor-version":[{"id":3642,"href":"https:\/\/pressbooks.bccampus.ca\/intermediatealgebrakpu\/wp-json\/pressbooks\/v2\/chapters\/689\/revisions\/3642"}],"part":[{"href":"https:\/\/pressbooks.bccampus.ca\/intermediatealgebrakpu\/wp-json\/pressbooks\/v2\/parts\/389"}],"metadata":[{"href":"https:\/\/pressbooks.bccampus.ca\/intermediatealgebrakpu\/wp-json\/pressbooks\/v2\/chapters\/689\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/pressbooks.bccampus.ca\/intermediatealgebrakpu\/wp-json\/wp\/v2\/media?parent=689"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/intermediatealgebrakpu\/wp-json\/pressbooks\/v2\/chapter-type?post=689"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/intermediatealgebrakpu\/wp-json\/wp\/v2\/contributor?post=689"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/intermediatealgebrakpu\/wp-json\/wp\/v2\/license?post=689"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}