{"id":244,"date":"2015-09-18T20:16:13","date_gmt":"2015-09-18T20:16:13","guid":{"rendered":"https:\/\/courses.candelalearning.com\/collegealgebra1xmaster\/?post_type=chapter&p=244"},"modified":"2017-04-03T18:30:23","modified_gmt":"2017-04-03T18:30:23","slug":"using-the-zero-exponent-rule-of-exponents","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/ivytech-collegealgebra\/chapter\/using-the-zero-exponent-rule-of-exponents\/","title":{"raw":"Using the Zero Exponent Rule of Exponents","rendered":"Using the Zero Exponent Rule of Exponents"},"content":{"raw":"Return to the quotient rule. We made the condition that [latex]m>n[\/latex] so that the difference [latex]m-n[\/latex] would never be zero or negative. What would happen if [latex]m=n[\/latex]? In this case, we would use the zero exponent rule of exponents<\/em> to simplify the expression to 1. To see how this is done, let us begin with an example.\r\n

[latex]\\frac{{t}^{8}}{{t}^{8}}=\\frac{\\cancel{t}^{8}}{\\cancel{t}^{8}}=1[\/latex]<\/p>\r\nIf we were to simplify the original expression using the quotient rule, we would have\r\n

[latex]\\frac{{t}^{8}}{{t}^{8}}={t}^{8 - 8}={t}^{0}[\/latex]<\/div>\r\nIf we equate the two answers, the result is [latex]{t}^{0}=1[\/latex]. This is true for any nonzero real number, or any variable representing a real number.\r\n
[latex]{a}^{0}=1[\/latex]<\/div>\r\nThe sole exception is the expression [latex]{0}^{0}[\/latex]. This appears later in more advanced courses, but for now, we will consider the value to be undefined.\r\n
\r\n

A General Note: The Zero Exponent Rule of Exponents<\/h3>\r\nFor any nonzero real number [latex]a[\/latex], the zero exponent rule of exponents states that\r\n
[latex]{a}^{0}=1[\/latex]<\/div>\r\n<\/div>\r\n
\r\n

Example 4: Using the Zero Exponent Rule<\/h3>\r\nSimplify each expression using the zero exponent rule of exponents.\r\n
    \r\n \t
  1. [latex]\\frac{{c}^{3}}{{c}^{3}}[\/latex]<\/li>\r\n \t
  2. [latex]\\frac{-3{x}^{5}}{{x}^{5}}[\/latex]<\/li>\r\n \t
  3. [latex]\\frac{{\\left({j}^{2}k\\right)}^{4}}{\\left({j}^{2}k\\right)\\cdot {\\left({j}^{2}k\\right)}^{3}}[\/latex]<\/li>\r\n \t
  4. [latex]\\frac{5{\\left(r{s}^{2}\\right)}^{2}}{{\\left(r{s}^{2}\\right)}^{2}}[\/latex]<\/li>\r\n<\/ol>\r\n<\/div>\r\n
    \r\n

    Solution<\/h3>\r\nUse the zero exponent and other rules to simplify each expression.\r\n
      \r\n \t
    1. [latex]\\begin{array}\\text{ }\\frac{c^{3}}{c^{3}} \\hfill& =c^{3-3} \\\\ \\hfill& =c^{0} \\\\ \\hfill& =1\\end{array}[\/latex]<\/li>\r\n \t
    2. [latex]\\begin{array}{ccc}\\hfill \\frac{-3{x}^{5}}{{x}^{5}}& =& -3\\cdot \\frac{{x}^{5}}{{x}^{5}}\\hfill \\\\ & =& -3\\cdot {x}^{5 - 5}\\hfill \\\\ & =& -3\\cdot {x}^{0}\\hfill \\\\ & =& -3\\cdot 1\\hfill \\\\ & =& -3\\hfill \\end{array}[\/latex]<\/li>\r\n \t
    3. [latex]\\begin{array}{cccc}\\hfill \\frac{{\\left({j}^{2}k\\right)}^{4}}{\\left({j}^{2}k\\right)\\cdot {\\left({j}^{2}k\\right)}^{3}}& =& \\frac{{\\left({j}^{2}k\\right)}^{4}}{{\\left({j}^{2}k\\right)}^{1+3}}\\hfill & \\text{Use the product rule in the denominator}.\\hfill \\\\ & =& \\frac{{\\left({j}^{2}k\\right)}^{4}}{{\\left({j}^{2}k\\right)}^{4}}\\hfill & \\text{Simplify}.\\hfill \\\\ & =& {\\left({j}^{2}k\\right)}^{4 - 4}\\hfill & \\text{Use the quotient rule}.\\hfill \\\\ & =& {\\left({j}^{2}k\\right)}^{0}\\hfill & \\text{Simplify}.\\hfill \\\\ & =& 1& \\end{array}[\/latex]<\/li>\r\n \t
    4. [latex]\\begin{array}{cccc}\\hfill \\frac{5{\\left(r{s}^{2}\\right)}^{2}}{{\\left(r{s}^{2}\\right)}^{2}}& =& 5{\\left(r{s}^{2}\\right)}^{2 - 2}\\hfill & \\text{Use the quotient rule}.\\hfill \\\\ & =& 5{\\left(r{s}^{2}\\right)}^{0}\\hfill & \\text{Simplify}.\\hfill \\\\ & =& 5\\cdot 1\\hfill & \\text{Use the zero exponent rule}.\\hfill \\\\ & =& 5\\hfill & \\text{Simplify}.\\hfill \\end{array}[\/latex]<\/li>\r\n<\/ol>\r\n<\/div>\r\n
      \r\n

      Try It 4<\/h3>\r\nSimplify each expression using the zero exponent rule of exponents.\r\n

      a. [latex]\\frac{{t}^{7}}{{t}^{7}}[\/latex]\r\nb. [latex]\\frac{{\\left(d{e}^{2}\\right)}^{11}}{2{\\left(d{e}^{2}\\right)}^{11}}[\/latex]\r\nc. [latex]\\frac{{w}^{4}\\cdot {w}^{2}}{{w}^{6}}[\/latex]\r\nd. [latex]\\frac{{t}^{3}\\cdot {t}^{4}}{{t}^{2}\\cdot {t}^{5}}[\/latex]<\/p>\r\nSolution<\/a>\r\n\r\n<\/div>","rendered":"

      Return to the quotient rule. We made the condition that [latex]m>n[\/latex] so that the difference [latex]m-n[\/latex] would never be zero or negative. What would happen if [latex]m=n[\/latex]? In this case, we would use the zero exponent rule of exponents<\/em> to simplify the expression to 1. To see how this is done, let us begin with an example.<\/p>\n

      [latex]\\frac{{t}^{8}}{{t}^{8}}=\\frac{\\cancel{t}^{8}}{\\cancel{t}^{8}}=1[\/latex]<\/p>\n

      If we were to simplify the original expression using the quotient rule, we would have<\/p>\n

      [latex]\\frac{{t}^{8}}{{t}^{8}}={t}^{8 - 8}={t}^{0}[\/latex]<\/div>\n

      If we equate the two answers, the result is [latex]{t}^{0}=1[\/latex]. This is true for any nonzero real number, or any variable representing a real number.<\/p>\n

      [latex]{a}^{0}=1[\/latex]<\/div>\n

      The sole exception is the expression [latex]{0}^{0}[\/latex]. This appears later in more advanced courses, but for now, we will consider the value to be undefined.<\/p>\n

      \n

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

      For any nonzero real number [latex]a[\/latex], the zero exponent rule of exponents states that<\/p>\n

      [latex]{a}^{0}=1[\/latex]<\/div>\n<\/div>\n
      \n

      Example 4: Using the Zero Exponent Rule<\/h3>\n

      Simplify each expression using the zero exponent rule of exponents.<\/p>\n

        \n
      1. [latex]\\frac{{c}^{3}}{{c}^{3}}[\/latex]<\/li>\n
      2. [latex]\\frac{-3{x}^{5}}{{x}^{5}}[\/latex]<\/li>\n
      3. [latex]\\frac{{\\left({j}^{2}k\\right)}^{4}}{\\left({j}^{2}k\\right)\\cdot {\\left({j}^{2}k\\right)}^{3}}[\/latex]<\/li>\n
      4. [latex]\\frac{5{\\left(r{s}^{2}\\right)}^{2}}{{\\left(r{s}^{2}\\right)}^{2}}[\/latex]<\/li>\n<\/ol>\n<\/div>\n
        \n

        Solution<\/h3>\n

        Use the zero exponent and other rules to simplify each expression.<\/p>\n

          \n
        1. [latex]\\begin{array}\\text{ }\\frac{c^{3}}{c^{3}} \\hfill& =c^{3-3} \\\\ \\hfill& =c^{0} \\\\ \\hfill& =1\\end{array}[\/latex]<\/li>\n
        2. [latex]\\begin{array}{ccc}\\hfill \\frac{-3{x}^{5}}{{x}^{5}}& =& -3\\cdot \\frac{{x}^{5}}{{x}^{5}}\\hfill \\\\ & =& -3\\cdot {x}^{5 - 5}\\hfill \\\\ & =& -3\\cdot {x}^{0}\\hfill \\\\ & =& -3\\cdot 1\\hfill \\\\ & =& -3\\hfill \\end{array}[\/latex]<\/li>\n
        3. [latex]\\begin{array}{cccc}\\hfill \\frac{{\\left({j}^{2}k\\right)}^{4}}{\\left({j}^{2}k\\right)\\cdot {\\left({j}^{2}k\\right)}^{3}}& =& \\frac{{\\left({j}^{2}k\\right)}^{4}}{{\\left({j}^{2}k\\right)}^{1+3}}\\hfill & \\text{Use the product rule in the denominator}.\\hfill \\\\ & =& \\frac{{\\left({j}^{2}k\\right)}^{4}}{{\\left({j}^{2}k\\right)}^{4}}\\hfill & \\text{Simplify}.\\hfill \\\\ & =& {\\left({j}^{2}k\\right)}^{4 - 4}\\hfill & \\text{Use the quotient rule}.\\hfill \\\\ & =& {\\left({j}^{2}k\\right)}^{0}\\hfill & \\text{Simplify}.\\hfill \\\\ & =& 1& \\end{array}[\/latex]<\/li>\n
        4. [latex]\\begin{array}{cccc}\\hfill \\frac{5{\\left(r{s}^{2}\\right)}^{2}}{{\\left(r{s}^{2}\\right)}^{2}}& =& 5{\\left(r{s}^{2}\\right)}^{2 - 2}\\hfill & \\text{Use the quotient rule}.\\hfill \\\\ & =& 5{\\left(r{s}^{2}\\right)}^{0}\\hfill & \\text{Simplify}.\\hfill \\\\ & =& 5\\cdot 1\\hfill & \\text{Use the zero exponent rule}.\\hfill \\\\ & =& 5\\hfill & \\text{Simplify}.\\hfill \\end{array}[\/latex]<\/li>\n<\/ol>\n<\/div>\n
          \n

          Try It 4<\/h3>\n

          Simplify each expression using the zero exponent rule of exponents.<\/p>\n

          a. [latex]\\frac{{t}^{7}}{{t}^{7}}[\/latex]
          \nb. [latex]\\frac{{\\left(d{e}^{2}\\right)}^{11}}{2{\\left(d{e}^{2}\\right)}^{11}}[\/latex]
          \nc. [latex]\\frac{{w}^{4}\\cdot {w}^{2}}{{w}^{6}}[\/latex]
          \nd. [latex]\\frac{{t}^{3}\\cdot {t}^{4}}{{t}^{2}\\cdot {t}^{5}}[\/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
          \n\t\t\t\t\t\t
          \n\t\t\t\t\t\t\t
          CC licensed content, Specific attribution<\/div>