{"id":681,"date":"2019-04-29T16:36:49","date_gmt":"2019-04-29T20:36:49","guid":{"rendered":"https:\/\/pressbooks.bccampus.ca\/intermediatealgebrakpu\/?post_type=chapter&p=681"},"modified":"2020-02-03T12:52:45","modified_gmt":"2020-02-03T17:52:45","slug":"9-2-reducing-higher-power-roots","status":"publish","type":"chapter","link":"https:\/\/pressbooks.bccampus.ca\/intermediatealgebrakpu\/chapter\/9-2-reducing-higher-power-roots\/","title":{"raw":"9.2 Reducing Higher Power Roots","rendered":"9.2 Reducing Higher Power Roots"},"content":{"raw":"[latexpage]\r\n\r\nWhile square roots are the most common type of radical, there are\u00a0higher roots of numbers as well: cube roots, fourth roots, fifth roots, and so on. The following is a definition of radicals:<\/span>\r\n

\\(\\sqrt[m]{a} = b \\text{ if } b^m = a\\)<\/p>\r\nThe small letter \\(m\\) inside the radical is called the index. It dictates which root you are taking. For square roots, the index is 2, which, since it is the most common root, is not usually written.\r\n

\r\n

Example 9.2.1<\/p>\r\n\r\n<\/header>\r\n

\r\n\r\nHere are several higher powers of positive numbers and their roots:\r\n\r\n\\(\\begin{array}{llllll}\r\n2^2=4 & 2^3=8 & 2^4=16 & 2^5=32 & 2^6=64 & 2^7=128 \\\\\r\n3^2=9 &3^3=27 & 3^4=81 & 3^5=243 & 3^6=729 &3^7=2187 \\\\\r\n4^2=16 & 4^3=64 &4^4=256&4^5=1024&4^6=4096&4^7=16384 \\\\\r\n5^2=25&5^3=125&5^4=625&5^5=3125&5^6=15625&5^7=78125 \\\\\r\n6^2=36&6^3=216&6^4=1296&6^5=7776&6^6=46656&6^7=279936 \\\\\r\n7^2=49&7^3=343&7^4=2401&7^5=16807&7^6=117649&7^7=823543 \\\\\r\n8^2=64&8^3=512&8^4=4096&8^5=32768&8^6=262144&8^7=2097152 \\\\\r\n9^2=81&9^3=729&9^4=6561&9^5=59049&9^6=531441&9^7=4782969 \\\\\r\n10^2=100&10^3=1000&10^4=10000&10^5=100000&10^6=1000000& \\\\ \\\\\r\n2=\\sqrt{4}&2=\\sqrt[3]{8}&2=\\sqrt[4]{16}&2=\\sqrt[5]{32}&2=\\sqrt[6]{64}&2=\\sqrt[7]{128} \\\\\r\n3=\\sqrt{9}&3=\\sqrt[3]{27}&3=\\sqrt[4]{81}&3=\\sqrt[5]{243}&3=\\sqrt[6]{729}&3=\\sqrt[7]{2187} \\\\\r\n4=\\sqrt{16}&4=\\sqrt[3]{64}&4=\\sqrt[4]{256}&4=\\sqrt[5]{1024}&4=\\sqrt[6]{4096}&4=\\sqrt[7]{16384} \\\\\r\n5=\\sqrt{25}&5=\\sqrt[3]{125}&5=\\sqrt[4]{625}&5=\\sqrt[5]{3125}&5=\\sqrt[6]{15625}&5=\\sqrt[7]{78125} \\\\\r\n6=\\sqrt{36}&6=\\sqrt[3]{216}&6=\\sqrt[4]{1296}&6=\\sqrt[5]{7776}&6=\\sqrt[6]{46656}&6=\\sqrt[7]{279936} \\\\\r\n7=\\sqrt{49}&7=\\sqrt[3]{343}&7=\\sqrt[4]{2401}&7=\\sqrt[5]{16807}&7=\\sqrt[6]{117649}&7=\\sqrt[7]{823543} \\\\\r\n8=\\sqrt{64}&8=\\sqrt[3]{512}&8=\\sqrt[4]{4096}&8=\\sqrt[5]{32768}&8=\\sqrt[6]{262144}&8=\\sqrt[7]{2097152} \\\\\r\n9=\\sqrt{81}&9=\\sqrt[3]{729}&9=\\sqrt[4]{6561}&9=\\sqrt[5]{59049}&9=\\sqrt[6]{531441}&9=\\sqrt[7]{4782969} \\\\\r\n10=\\sqrt{100}&10=\\sqrt[3]{1000}&10=\\sqrt[4]{10000}&10=\\sqrt[5]{100000}&10=\\sqrt[6]{1000000}&\r\n\\end{array}\\)\r\n\r\n<\/div>\r\n<\/div>\r\nNote there is a notable distinction between solutions of even roots and of odd roots. For even-powered roots, the solution is always +\/\u2212 or \u00b1. The reason for this can shown in the following examples.\r\n
\r\n

Example 9.2.2<\/p>\r\n\r\n<\/header>\r\n

\r\n\r\nFind the solutions to \u221a4.\r\n\r\nThere are two ways to multiple identical numbers to equal 4:\r\n

\\((2)(2) = 4 \\text{ and } (-2)(-2) = 4\\)<\/p>\r\nThis means that the \u221a4 is either +2 or \u22122, which is often written as \u00b12.\r\n\r\nThe \u00b1 solution occurs for all even roots and can be seen in:\r\n

\\(\\sqrt[4]{16} =\\pm 2 \\text{ and }\\sqrt[6]{64} = \\pm 2 \\text{ and }\\sqrt[8]{256} = \\pm 2\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\nAll roots that have an even index will always have \u00b1 solutions.\r\n\r\nOdd-powered roots do not share this feature and will only maintain the sign of the number that you are taking the root of.\r\n

\r\n

Example 9.2.3<\/p>\r\n\r\n<\/header>\r\n

\r\n\r\nFind the solutions to \\(\\sqrt[3]{8}\\) and \\(\\sqrt[3]{-8}\\).\r\n\r\nThe solution of \\(\\sqrt[3]{8}\\) is 2 and \\(\\sqrt[3]{-8}\\) is \u22122.\r\n\r\nThe reason for this is (2)3<\/sup> = 8 and (\u22122)3<\/sup> = \u22128.\r\n\r\n<\/div>\r\n<\/div>\r\nAll negative-indexed roots will keep the sign of the number being rooted.<\/strong>\r\n\r\nHigher roots can be simplified in much the same way one simplifies square roots: through using the product property of radicals.\r\n

\\(\\text{Product Property of Radicals: }m\\sqrt{ab} = m(\\sqrt{a}\\cdot m\\sqrt{b})\\)<\/p>\r\n\r\n

\r\n

Examples 9.2.4<\/p>\r\n\r\n<\/header>\r\n

\r\n\r\nUse the product property of radicals to simplify the following.\r\n
    \r\n \t
  1. \\(\\sqrt[3]{32}\\)32 can be broken down into 25<\/sup>. Since you are taking the cube root of this number, you can only take out numbers that have a cube root. This means that 32 is broken into 8 \u00d7 4, with the number 8 being the only number that you can take the cube root of.\r\n

    \\[\\sqrt[3]{32}=\\sqrt[3]{8}\\cdot \\sqrt[3]{4}\\]<\/p>\r\nThis simplifies to:\r\n

    \\[\\sqrt[3]{32}=2 \\sqrt[3]{4}\\]<\/p>\r\n<\/li>\r\n \t

  2. \\(\\sqrt[5]{96}\\)\\[\\sqrt[5]{96}=\\sqrt[5]{32}\\cdot \\sqrt[5]{3}\\]96 can be broken down into 25<\/sup> \u00d7 3. Since you are taking the fifth root of this number, you can only take out numbers that have a fifth root. This means that 96 is broken into 32 \u00d7 3, with the number 32 being the only number that you can take the fifth root of.This simplifies to:\\[\\sqrt[5]{96}=2\\sqrt[5]{3}\\]<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\nThis strategy is used to take the higher roots of variables. In this case, only take out whole number multiples of the root index. This is shown in the following examples.\r\n
    \r\n

    Example 9.2.5<\/p>\r\n\r\n<\/header>\r\n

    \r\n\r\nReduce \\(\\sqrt[4]{x^{25}y^{16}z^4}\\).\r\n\r\nFor this root, you will break the exponent into multiples of the index 4.\r\n\r\nThis means that \\(x^{25}y^{16}z^4\\) will be broken up into \\(x^{24}xy^{16}z^4\\).\r\n\r\nThe fourth roots of \\(x^{24}y^{16}z^4\\) are \\(x^6y^4z\\) and the solitary \\(x\\) remains under the fourth root radical. This looks like:\r\n

    \\(\\sqrt[4]{x^{25}y^{16}z^4}=\\sqrt[4]{x^{24}}\\cdot \\sqrt[4]{x}\\cdot \\sqrt[4]{y^{16}}\\cdot \\sqrt[4]{z^4}\\)<\/p>\r\nWhich simplifies to:\r\n

    \\(x^6y^4z\\sqrt[4]{x}\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

    \r\n

    Example 9.2.6<\/p>\r\n\r\n<\/header>\r\n

    \r\n\r\nReduce\u00a0 \\(\\sqrt[5]{64x^{25}y^{16}z^4}\\).\r\n\r\nFor this root, you will break the exponent into multiples of the index 5.\r\n\r\nThis means that \\(x^{25}y^{16}z^4\\) will be broken up into \\(x^{25}y^{15}yz^4\\) and 64 broken up into 32\u00a0\u00d7 2.\r\n\r\nThe fifth roots of \\(32x^{25}y^{15}\\) are \\(2x^5y^3\\) and the remainder \\(2yz^4\\) remains under the fifth root radical.\r\n\r\nThis looks like:\r\n

    \\(\\sqrt[5]{64x^{25}y^{16}z^4}=\\sqrt[5]{32}\\cdot \\sqrt[5]{2}\\cdot \\sqrt[5]{x^{25}}\\cdot \\sqrt[5]{y^{15}}\\cdot \\sqrt[5]{y}\\cdot \\sqrt[5]{z^4}\\)<\/p>\r\nWhich simplifies to:\r\n

    \\(2x^5y^3\\sqrt[5]{2yz^4}\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

    Questions<\/h1>\r\nSimplify the following radicals.\r\n
      \r\n \t
    1. \\(\\sqrt[3]{64}\\)<\/li>\r\n \t
    2. \\(\\sqrt[3]{-125}\\)<\/li>\r\n \t
    3. \\(\\sqrt[3]{625}\\)<\/li>\r\n \t
    4. \\(\\sqrt[3]{250}\\)<\/li>\r\n \t
    5. \\(\\sqrt[3]{192}\\)<\/li>\r\n \t
    6. \\(\\sqrt[3]{-24}\\)<\/li>\r\n \t
    7. \\(-4\\sqrt[4]{96}\\)<\/li>\r\n \t
    8. \\(-8\\sqrt[4]{48}\\)<\/li>\r\n \t
    9. \\(6\\sqrt[4]{112}\\)<\/li>\r\n \t
    10. \\(5\\sqrt[4]{243}\\)<\/li>\r\n \t
    11. \\(6\\sqrt[4]{648x^5y^7z^2}\\)<\/li>\r\n \t
    12. \\(-6\\sqrt[4]{405a^5b^8c}\\)<\/li>\r\n \t
    13. \\(\\sqrt[5]{224n^3p^7q^5}\\)<\/li>\r\n \t
    14. \\(\\sqrt[5]{-96x^3y^6z^5}\\)<\/li>\r\n \t
    15. \\(\\sqrt[5]{224p^5q^{10}r^{15}}\\)<\/li>\r\n \t
    16. \\(\\sqrt[6]{256x^6y^6z^7}\\)<\/li>\r\n \t
    17. \\(-3\\sqrt[7]{896rs^7t^{14}}\\)<\/li>\r\n \t
    18. \\(-8\\sqrt[7]{384b^8c^7d^6}\\)<\/li>\r\n<\/ol>\r\nAnswer Key 9.2<\/a>","rendered":"

      While square roots are the most common type of radical, there are\u00a0higher roots of numbers as well: cube roots, fourth roots, fifth roots, and so on. The following is a definition of radicals:<\/span><\/p>\n

      \"\sqrt[m]{a} = b \text{ if } b^m = a\"<\/p>\n

      The small letter \"m\" inside the radical is called the index. It dictates which root you are taking. For square roots, the index is 2, which, since it is the most common root, is not usually written.<\/p>\n

      \n
      \n

      Example 9.2.1<\/p>\n<\/header>\n

      \n

      Here are several higher powers of positive numbers and their roots:<\/p>\n

      \"\begin{array}{llllll}<\/p>\n<\/div>\n<\/div>\n

      Note there is a notable distinction between solutions of even roots and of odd roots. For even-powered roots, the solution is always +\/\u2212 or \u00b1. The reason for this can shown in the following examples.<\/p>\n

      \n
      \n

      Example 9.2.2<\/p>\n<\/header>\n

      \n

      Find the solutions to \u221a4.<\/p>\n

      There are two ways to multiple identical numbers to equal 4:<\/p>\n

      \"(2)(2) = 4 \text{ and } (-2)(-2) = 4\"<\/p>\n

      This means that the \u221a4 is either +2 or \u22122, which is often written as \u00b12.<\/p>\n

      The \u00b1 solution occurs for all even roots and can be seen in:<\/p>\n

      \"\sqrt[4]{16} =\pm 2 \text{ and }\sqrt[6]{64} = \pm 2 \text{ and }\sqrt[8]{256} = \pm 2\"<\/p>\n<\/div>\n<\/div>\n

      All roots that have an even index will always have \u00b1 solutions.<\/p>\n

      Odd-powered roots do not share this feature and will only maintain the sign of the number that you are taking the root of.<\/p>\n

      \n
      \n

      Example 9.2.3<\/p>\n<\/header>\n

      \n

      Find the solutions to \"\sqrt[3]{8}\" and \"\sqrt[3]{-8}\".<\/p>\n

      The solution of \"\sqrt[3]{8}\" is 2 and \"\sqrt[3]{-8}\" is \u22122.<\/p>\n

      The reason for this is (2)3<\/sup> = 8 and (\u22122)3<\/sup> = \u22128.<\/p>\n<\/div>\n<\/div>\n

      All negative-indexed roots will keep the sign of the number being rooted.<\/strong><\/p>\n

      Higher roots can be simplified in much the same way one simplifies square roots: through using the product property of radicals.<\/p>\n

      \"\text{Product Property of Radicals: }m\sqrt{ab} = m(\sqrt{a}\cdot m\sqrt{b})\"<\/p>\n

      \n
      \n

      Examples 9.2.4<\/p>\n<\/header>\n

      \n

      Use the product property of radicals to simplify the following.<\/p>\n

        \n
      1. \"\sqrt[3]{32}\"32 can be broken down into 25<\/sup>. Since you are taking the cube root of this number, you can only take out numbers that have a cube root. This means that 32 is broken into 8 \u00d7 4, with the number 8 being the only number that you can take the cube root of.\n

        \n

          <\/span>   <\/span>\"\[\sqrt[3]{32}=\sqrt[3]{8}\cdot \sqrt[3]{4}\]\"<\/p>\n

        This simplifies to:<\/p>\n

        \n

          <\/span>   <\/span>\"\[\sqrt[3]{32}=2 \sqrt[3]{4}\]\"<\/p>\n<\/li>\n

      2. \"\sqrt[5]{96}\"\n

          <\/span>   <\/span>\"\[\sqrt[5]{96}=\sqrt[5]{32}\cdot \sqrt[5]{3}\]\"<\/p>\n

        96 can be broken down into 25<\/sup> \u00d7 3. Since you are taking the fifth root of this number, you can only take out numbers that have a fifth root. This means that 96 is broken into 32 \u00d7 3, with the number 32 being the only number that you can take the fifth root of.This simplifies to:<\/p>\n

          <\/span>   <\/span>\"\[\sqrt[5]{96}=2\sqrt[5]{3}\]\"<\/p>\n<\/li>\n<\/ol>\n<\/div>\n<\/div>\n

        This strategy is used to take the higher roots of variables. In this case, only take out whole number multiples of the root index. This is shown in the following examples.<\/p>\n

        \n
        \n

        Example 9.2.5<\/p>\n<\/header>\n

        \n

        Reduce \"\sqrt[4]{x^{25}y^{16}z^4}\".<\/p>\n

        For this root, you will break the exponent into multiples of the index 4.<\/p>\n

        This means that \"x^{25}y^{16}z^4\" will be broken up into \"x^{24}xy^{16}z^4\".<\/p>\n

        The fourth roots of \"x^{24}y^{16}z^4\" are \"x^6y^4z\" and the solitary \"x\" remains under the fourth root radical. This looks like:<\/p>\n

        \"\sqrt[4]{x^{25}y^{16}z^4}=\sqrt[4]{x^{24}}\cdot \sqrt[4]{x}\cdot \sqrt[4]{y^{16}}\cdot \sqrt[4]{z^4}\"<\/p>\n

        Which simplifies to:<\/p>\n

        \"x^6y^4z\sqrt[4]{x}\"<\/p>\n<\/div>\n<\/div>\n

        \n
        \n

        Example 9.2.6<\/p>\n<\/header>\n

        \n

        Reduce\u00a0 \"\sqrt[5]{64x^{25}y^{16}z^4}\".<\/p>\n

        For this root, you will break the exponent into multiples of the index 5.<\/p>\n

        This means that \"x^{25}y^{16}z^4\" will be broken up into \"x^{25}y^{15}yz^4\" and 64 broken up into 32\u00a0\u00d7 2.<\/p>\n

        The fifth roots of \"32x^{25}y^{15}\" are \"2x^5y^3\" and the remainder \"2yz^4\" remains under the fifth root radical.<\/p>\n

        This looks like:<\/p>\n

        \"\sqrt[5]{64x^{25}y^{16}z^4}=\sqrt[5]{32}\cdot \sqrt[5]{2}\cdot \sqrt[5]{x^{25}}\cdot \sqrt[5]{y^{15}}\cdot \sqrt[5]{y}\cdot \sqrt[5]{z^4}\"<\/p>\n

        Which simplifies to:<\/p>\n

        \"2x^5y^3\sqrt[5]{2yz^4}\"<\/p>\n<\/div>\n<\/div>\n

        Questions<\/h1>\n

        Simplify the following radicals.<\/p>\n

          \n
        1. \"\sqrt[3]{64}\"<\/li>\n
        2. \"\sqrt[3]{-125}\"<\/li>\n
        3. \"\sqrt[3]{625}\"<\/li>\n
        4. \"\sqrt[3]{250}\"<\/li>\n
        5. \"\sqrt[3]{192}\"<\/li>\n
        6. \"\sqrt[3]{-24}\"<\/li>\n
        7. \"-4\sqrt[4]{96}\"<\/li>\n
        8. \"-8\sqrt[4]{48}\"<\/li>\n
        9. \"6\sqrt[4]{112}\"<\/li>\n
        10. \"5\sqrt[4]{243}\"<\/li>\n
        11. \"6\sqrt[4]{648x^5y^7z^2}\"<\/li>\n
        12. \"-6\sqrt[4]{405a^5b^8c}\"<\/li>\n
        13. \"\sqrt[5]{224n^3p^7q^5}\"<\/li>\n
        14. \"\sqrt[5]{-96x^3y^6z^5}\"<\/li>\n
        15. \"\sqrt[5]{224p^5q^{10}r^{15}}\"<\/li>\n
        16. \"\sqrt[6]{256x^6y^6z^7}\"<\/li>\n
        17. \"-3\sqrt[7]{896rs^7t^{14}}\"<\/li>\n
        18. \"-8\sqrt[7]{384b^8c^7d^6}\"<\/li>\n<\/ol>\n

          Answer Key 9.2<\/a><\/p>\n","protected":false},"author":540,"menu_order":19,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-681","chapter","type-chapter","status-publish","hentry"],"part":389,"_links":{"self":[{"href":"https:\/\/pressbooks.bccampus.ca\/intermediatealgebrakpu\/wp-json\/pressbooks\/v2\/chapters\/681","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":14,"href":"https:\/\/pressbooks.bccampus.ca\/intermediatealgebrakpu\/wp-json\/pressbooks\/v2\/chapters\/681\/revisions"}],"predecessor-version":[{"id":3778,"href":"https:\/\/pressbooks.bccampus.ca\/intermediatealgebrakpu\/wp-json\/pressbooks\/v2\/chapters\/681\/revisions\/3778"}],"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\/681\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/pressbooks.bccampus.ca\/intermediatealgebrakpu\/wp-json\/wp\/v2\/media?parent=681"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/intermediatealgebrakpu\/wp-json\/pressbooks\/v2\/chapter-type?post=681"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/intermediatealgebrakpu\/wp-json\/wp\/v2\/contributor?post=681"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/intermediatealgebrakpu\/wp-json\/wp\/v2\/license?post=681"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}