{"id":242,"date":"2015-09-18T20:15:34","date_gmt":"2015-09-18T20:15:34","guid":{"rendered":"https:\/\/courses.candelalearning.com\/collegealgebra1xmaster\/?post_type=chapter&p=242"},"modified":"2015-10-30T22:40:48","modified_gmt":"2015-10-30T22:40:48","slug":"using-the-power-rule-of-exponents","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/ivytech-collegealgebra\/chapter\/using-the-power-rule-of-exponents\/","title":{"raw":"Using the Power Rule of Exponents","rendered":"Using the Power Rule of Exponents"},"content":{"raw":"Suppose an exponential expression is raised to some power. Can we simplify the result? Yes. To do this, we use the power rule of exponents<\/em>. Consider the expression [latex]{\\left({x}^{2}\\right)}^{3}[\/latex]. The expression inside the parentheses is multiplied twice because it has an exponent of 2. Then the result is multiplied three times because the entire expression has an exponent of 3.\r\n
[latex]\\begin{array}{ccc}\\hfill {\\left({x}^{2}\\right)}^{3}& =& \\stackrel{{3\\text{ factors}}}{{{\\left({x}^{2}\\right)\\cdot \\left({x}^{2}\\right)\\cdot \\left({x}^{2}\\right)}}}\\hfill \\\\ & =& \\hfill \\stackrel{{3\\text{ factors}}}{{{\\left(\\stackrel{{2\\text{ factors}}}{{\\overbrace{x\\cdot x}}}\\right)\\cdot \\left(\\stackrel{{2\\text{ factors}}}{{\\overbrace{x\\cdot x}}}\\right)\\cdot \\left(\\stackrel{{2\\text{ factors}}}{{\\overbrace{x\\cdot x}}}\\right)}}}\\\\ & =& x\\cdot x\\cdot x\\cdot x\\cdot x\\cdot x\\hfill \\\\ & =& {x}^{6}\\hfill \\end{array}[\/latex]<\/div>\r\nThe exponent of the answer is the product of the exponents: [latex]{\\left({x}^{2}\\right)}^{3}={x}^{2\\cdot 3}={x}^{6}[\/latex]. In other words, when raising an exponential expression to a power, we write the result with the common base and the product of the exponents.\r\n
[latex]{\\left({a}^{m}\\right)}^{n}={a}^{m\\cdot n}[\/latex]<\/div>\r\nBe careful to distinguish between uses of the product rule and the power rule. When using the product rule, different terms with the same bases are raised to exponents. In this case, you add the exponents. When using the power rule, a term in exponential notation is raised to a power. In this case, you multiply the exponents.\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
Product Rule<\/th>\r\nPower Rule<\/th>\r\n<\/tr>\r\n<\/thead>\r\n
[latex]5^{3}\\cdot5^{4}[\/latex]<\/td>\r\n=<\/td>\r\n\u00a0[latex]5^{3+4}[\/latex]<\/td>\r\n=<\/td>\r\n[latex]5^{7}[\/latex]<\/td>\r\nbut<\/td>\r\n[latex]\\left(5^{3}\\right)^{4}[\/latex]<\/td>\r\n=<\/td>\r\n[latex]5^{3\\cdot4}[\/latex]<\/td>\r\n=<\/td>\r\n[latex]5^{12}[\/latex]<\/td>\r\n<\/tr>\r\n
[latex]x^{5}\\cdot x^{2}[\/latex]<\/td>\r\n=<\/td>\r\n[latex]x^{5+2}[\/latex]<\/td>\r\n=<\/td>\r\n[latex]x^{7}[\/latex]<\/td>\r\nbut<\/td>\r\n[latex]\\left(x^{5}\\right)^{2}[\/latex]<\/td>\r\n=<\/td>\r\n\u00a0[latex]x^{5\\cdot2}[\/latex]<\/td>\r\n=<\/td>\r\n[latex]x^{10}[\/latex]<\/td>\r\n<\/tr>\r\n
[latex]\\left(3a\\right)^{7}\\cdot\\left(3a\\right)^{10} [\/latex]<\/td>\r\n=<\/td>\r\n[latex]\\left(3a\\right)^{7+1-} [\/latex]<\/td>\r\n=<\/td>\r\n[latex]\\left(3a\\right)^{17}[\/latex]<\/td>\r\nbut<\/td>\r\n[latex]\\left(\\left(3a\\right)^{7}\\right)^{10} [\/latex]<\/td>\r\n=<\/td>\r\n[latex]\\left(3a\\right)^{7\\cdot10} [\/latex]<\/td>\r\n=<\/td>\r\n[latex]\\left(3a\\right)^{70}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n
<\/div>\r\n
\r\n

A General Note: The Power Rule of Exponents<\/h3>\r\nFor any real number [latex]a[\/latex] and positive integers [latex]m[\/latex] and [latex]n[\/latex], the power rule of exponents states that\r\n
[latex]{\\left({a}^{m}\\right)}^{n}={a}^{m\\cdot n}[\/latex]<\/div>\r\n<\/div>\r\n
\r\n

Example 3: Using the Power Rule<\/h3>\r\nWrite each of the following products with a single base. Do not simplify further.\r\n
    \r\n\t
  1. [latex]{\\left({x}^{2}\\right)}^{7}[\/latex]<\/li>\r\n\t
  2. [latex]{\\left({\\left(2t\\right)}^{5}\\right)}^{3}[\/latex]<\/li>\r\n\t
  3. [latex]{\\left({\\left(-3\\right)}^{5}\\right)}^{11}[\/latex]<\/li>\r\n<\/ol>\r\n<\/div>\r\n
    \r\n

    Solution<\/h3>\r\nUse the power rule to simplify each expression.\r\n
      \r\n\t
    1. [latex]{\\left({x}^{2}\\right)}^{7}={x}^{2\\cdot 7}={x}^{14}[\/latex]<\/li>\r\n\t
    2. [latex]{\\left({\\left(2t\\right)}^{5}\\right)}^{3}={\\left(2t\\right)}^{5\\cdot 3}={\\left(2t\\right)}^{15}[\/latex]<\/li>\r\n\t
    3. [latex]{\\left({\\left(-3\\right)}^{5}\\right)}^{11}={\\left(-3\\right)}^{5\\cdot 11}={\\left(-3\\right)}^{55}[\/latex]<\/li>\r\n<\/ol>\r\n<\/div>\r\n
      \r\n

      Try It 3<\/h3>\r\nWrite each of the following products with a single base. Do not simplify further.\r\n
        \r\n\t
      1. [latex]{\\left({\\left(3y\\right)}^{8}\\right)}^{3}[\/latex]<\/li>\r\n\t
      2. [latex]{\\left({t}^{5}\\right)}^{7}[\/latex]<\/li>\r\n\t
      3. [latex]{\\left({\\left(-g\\right)}^{4}\\right)}^{4}[\/latex]<\/li>\r\n<\/ol>\r\nSolution<\/a>\r\n\r\n<\/div>","rendered":"

        Suppose an exponential expression is raised to some power. Can we simplify the result? Yes. To do this, we use the power rule of exponents<\/em>. Consider the expression [latex]{\\left({x}^{2}\\right)}^{3}[\/latex]. The expression inside the parentheses is multiplied twice because it has an exponent of 2. Then the result is multiplied three times because the entire expression has an exponent of 3.<\/p>\n

        [latex]\\begin{array}{ccc}\\hfill {\\left({x}^{2}\\right)}^{3}& =& \\stackrel{{3\\text{ factors}}}{{{\\left({x}^{2}\\right)\\cdot \\left({x}^{2}\\right)\\cdot \\left({x}^{2}\\right)}}}\\hfill \\\\ & =& \\hfill \\stackrel{{3\\text{ factors}}}{{{\\left(\\stackrel{{2\\text{ factors}}}{{\\overbrace{x\\cdot x}}}\\right)\\cdot \\left(\\stackrel{{2\\text{ factors}}}{{\\overbrace{x\\cdot x}}}\\right)\\cdot \\left(\\stackrel{{2\\text{ factors}}}{{\\overbrace{x\\cdot x}}}\\right)}}}\\\\ & =& x\\cdot x\\cdot x\\cdot x\\cdot x\\cdot x\\hfill \\\\ & =& {x}^{6}\\hfill \\end{array}[\/latex]<\/div>\n

        The exponent of the answer is the product of the exponents: [latex]{\\left({x}^{2}\\right)}^{3}={x}^{2\\cdot 3}={x}^{6}[\/latex]. In other words, when raising an exponential expression to a power, we write the result with the common base and the product of the exponents.<\/p>\n

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

        Be careful to distinguish between uses of the product rule and the power rule. When using the product rule, different terms with the same bases are raised to exponents. In this case, you add the exponents. When using the power rule, a term in exponential notation is raised to a power. In this case, you multiply the exponents.<\/p>\n\n\n\n\n\n\n\n
        Product Rule<\/th>\nPower Rule<\/th>\n<\/tr>\n<\/thead>\n
        [latex]5^{3}\\cdot5^{4}[\/latex]<\/td>\n=<\/td>\n\u00a0[latex]5^{3+4}[\/latex]<\/td>\n=<\/td>\n[latex]5^{7}[\/latex]<\/td>\nbut<\/td>\n[latex]\\left(5^{3}\\right)^{4}[\/latex]<\/td>\n=<\/td>\n[latex]5^{3\\cdot4}[\/latex]<\/td>\n=<\/td>\n[latex]5^{12}[\/latex]<\/td>\n<\/tr>\n
        [latex]x^{5}\\cdot x^{2}[\/latex]<\/td>\n=<\/td>\n[latex]x^{5+2}[\/latex]<\/td>\n=<\/td>\n[latex]x^{7}[\/latex]<\/td>\nbut<\/td>\n[latex]\\left(x^{5}\\right)^{2}[\/latex]<\/td>\n=<\/td>\n\u00a0[latex]x^{5\\cdot2}[\/latex]<\/td>\n=<\/td>\n[latex]x^{10}[\/latex]<\/td>\n<\/tr>\n
        [latex]\\left(3a\\right)^{7}\\cdot\\left(3a\\right)^{10}[\/latex]<\/td>\n=<\/td>\n[latex]\\left(3a\\right)^{7+1-}[\/latex]<\/td>\n=<\/td>\n[latex]\\left(3a\\right)^{17}[\/latex]<\/td>\nbut<\/td>\n[latex]\\left(\\left(3a\\right)^{7}\\right)^{10}[\/latex]<\/td>\n=<\/td>\n[latex]\\left(3a\\right)^{7\\cdot10}[\/latex]<\/td>\n=<\/td>\n[latex]\\left(3a\\right)^{70}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n
        <\/div>\n
        \n

        A General Note: The Power Rule of Exponents<\/h3>\n

        For any real number [latex]a[\/latex] and positive integers [latex]m[\/latex] and [latex]n[\/latex], the power rule of exponents states that<\/p>\n

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

        Example 3: Using the Power Rule<\/h3>\n

        Write each of the following products with a single base. Do not simplify further.<\/p>\n

          \n
        1. [latex]{\\left({x}^{2}\\right)}^{7}[\/latex]<\/li>\n
        2. [latex]{\\left({\\left(2t\\right)}^{5}\\right)}^{3}[\/latex]<\/li>\n
        3. [latex]{\\left({\\left(-3\\right)}^{5}\\right)}^{11}[\/latex]<\/li>\n<\/ol>\n<\/div>\n
          \n

          Solution<\/h3>\n

          Use the power rule to simplify each expression.<\/p>\n

            \n
          1. [latex]{\\left({x}^{2}\\right)}^{7}={x}^{2\\cdot 7}={x}^{14}[\/latex]<\/li>\n
          2. [latex]{\\left({\\left(2t\\right)}^{5}\\right)}^{3}={\\left(2t\\right)}^{5\\cdot 3}={\\left(2t\\right)}^{15}[\/latex]<\/li>\n
          3. [latex]{\\left({\\left(-3\\right)}^{5}\\right)}^{11}={\\left(-3\\right)}^{5\\cdot 11}={\\left(-3\\right)}^{55}[\/latex]<\/li>\n<\/ol>\n<\/div>\n
            \n

            Try It 3<\/h3>\n

            Write each of the following products with a single base. Do not simplify further.<\/p>\n

              \n
            1. [latex]{\\left({\\left(3y\\right)}^{8}\\right)}^{3}[\/latex]<\/li>\n
            2. [latex]{\\left({t}^{5}\\right)}^{7}[\/latex]<\/li>\n
            3. [latex]{\\left({\\left(-g\\right)}^{4}\\right)}^{4}[\/latex]<\/li>\n<\/ol>\n

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

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