{"id":695,"date":"2021-09-07T19:56:25","date_gmt":"2021-09-07T19:56:25","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/?post_type=chapter&p=695"},"modified":"2023-08-10T19:20:29","modified_gmt":"2023-08-10T19:20:29","slug":"3-2-3-exponents-on-variables-part-1","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/chapter\/3-2-3-exponents-on-variables-part-1\/","title":{"raw":"3.2.2: Exponential Expressions","rendered":"3.2.2: Exponential Expressions"},"content":{"raw":"
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Learning Outcomes<\/h3>\r\n
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  • Simplify expressions using the Product Property of Exponents<\/li>\r\n \t
  • Simplify expressions using the Power Property of Exponents<\/li>\r\n \t
  • Simplify expressions using the Product to a Power Property of Exponents<\/li>\r\n<\/ul>\r\n<\/div>\r\n
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    Key words<\/h3>\r\n
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    • Product Property of Exponents<\/strong>: to multiply exponential terms with the same base, add the exponents.<\/li>\r\n \t
    • Power\u00a0Property of Exponents<\/strong>: to raise a power to a power, multiply the exponents.<\/li>\r\n \t
    • Product to a Power\u00a0Property of Exponents<\/strong>: to raise a product to a power, raise each factor to that power.<\/li>\r\n<\/ul>\r\n<\/div>\r\n

      The Product Property of Exponents<\/h2>\r\nIn 2.4.1, we derived the properties of exponents for multiplication using integers. This property also works for variables.\r\n\r\nFor example, [latex]\\color{blue}{x^4}\\cdot\\color{green}{x^5}=\\color{blue}{x\\cdot x\\cdot x\\cdot x}\\cdot\\color{green}{x\\cdot x\\cdot x\\cdot x\\cdot x}=x^9[\/latex].\r\n\r\nWe restate the product property here.\r\n
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      The Product Property OF Exponents<\/h3>\r\nFor any real number [latex]x[\/latex] and any rational numbers [latex]m[\/latex]\u00a0<\/i>and [latex]n[\/latex],\u00a0[latex]\\left(x^{m}\\right)\\left(x^{n}\\right) = x^{m+n}[\/latex].\r\n\r\nTo multiply exponential terms with the same base, add the exponents.\r\n\r\n<\/div>\r\n
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      example<\/h3>\r\nSimplify: [latex]{x}^{5}\\cdot {x}^{7}[\/latex]\r\n\r\nSolution\r\n\r\n\r\n\r\n\r\n\r\n
      <\/td>\r\n[latex]{x}^{5}\\cdot {x}^{7}[\/latex]<\/td>\r\n<\/tr>\r\n
      Use the product property, [latex]{a}^{m}\\cdot {a}^{n}={a}^{m+n}[\/latex].<\/td>\r\n[latex]x^{\\color{red}{5+7}}[\/latex]<\/td>\r\n<\/tr>\r\n
      Simplify.<\/td>\r\n[latex]{x}^{12}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n
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      Example<\/h3>\r\nSimplify.\r\n

      [latex](a^{3})(a^{7})[\/latex]<\/p>\r\n\r\n

      Solution<\/h4>\r\nThe base of both exponents is [latex]a[\/latex], so the product rule applies.<\/span>\r\n

      [latex]\\left(a^{3}\\right)\\left(a^{7}\\right)[\/latex]<\/p>\r\nAdd the exponents with a common base.\r\n

      [latex]a^{3+7}[\/latex]<\/p>\r\n\r\n

      Answer<\/h4>\r\n[latex]\\left(a^{3}\\right)\\left(a^{7}\\right) = a^{10}[\/latex]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n
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      try it<\/h3>\r\n