{"id":250,"date":"2015-09-18T20:18:49","date_gmt":"2015-09-18T20:18:49","guid":{"rendered":"https:\/\/courses.candelalearning.com\/collegealgebra1xmaster\/?post_type=chapter&p=250"},"modified":"2015-11-02T20:01:59","modified_gmt":"2015-11-02T20:01:59","slug":"simplifying-exponential-expressions","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/ivytech-collegealgebra\/chapter\/simplifying-exponential-expressions\/","title":{"raw":"Simplifying Exponential Expressions","rendered":"Simplifying Exponential Expressions"},"content":{"raw":"Recall that to simplify an expression means to rewrite it by combing terms or exponents; in other words, to write the expression more simply with fewer terms. The rules for exponents may be combined to simplify expressions.\r\n
\r\n

Example 9: Simplifying Exponential Expressions<\/h3>\r\nSimplify each expression and write the answer with positive exponents only.\r\n
    \r\n\t
  1. [latex]{\\left(6{m}^{2}{n}^{-1}\\right)}^{3}[\/latex]<\/li>\r\n\t
  2. [latex]{17}^{5}\\cdot {17}^{-4}\\cdot {17}^{-3}[\/latex]<\/li>\r\n\t
  3. [latex]{\\left(\\frac{{u}^{-1}v}{{v}^{-1}}\\right)}^{2}[\/latex]<\/li>\r\n\t
  4. [latex]\\left(-2{a}^{3}{b}^{-1}\\right)\\left(5{a}^{-2}{b}^{2}\\right)[\/latex]<\/li>\r\n\t
  5. [latex]{\\left({x}^{2}\\sqrt{2}\\right)}^{4}{\\left({x}^{2}\\sqrt{2}\\right)}^{-4}[\/latex]<\/li>\r\n\t
  6. [latex]\\frac{{\\left(3{w}^{2}\\right)}^{5}}{{\\left(6{w}^{-2}\\right)}^{2}}[\/latex]<\/li>\r\n<\/ol>\r\n<\/div>\r\n
    \r\n

    Solution<\/h3>\r\n
      \r\n\t
    1. [latex]\\begin{array}{cccc}\\hfill {\\left(6{m}^{2}{n}^{-1}\\right)}^{3}& =& {\\left(6\\right)}^{3}{\\left({m}^{2}\\right)}^{3}{\\left({n}^{-1}\\right)}^{3}\\hfill & \\text{The power of a product rule}\\hfill \\\\ & =& {6}^{3}{m}^{2\\cdot 3}{n}^{-1\\cdot 3}\\hfill & \\text{The power rule}\\hfill \\\\ & =& \\text{ }216{m}^{6}{n}^{-3}\\hfill & \\text{Simplify}.\\hfill \\\\ & =& \\frac{216{m}^{6}}{{n}^{3}}\\hfill & \\text{The negative exponent rule}\\hfill \\end{array}[\/latex]<\/li>\r\n\t
    2. [latex]\\begin{array}{cccc}\\hfill {17}^{5}\\cdot {17}^{-4}\\cdot {17}^{-3}& =& {17}^{5 - 4-3}\\hfill & \\text{The product rule}\\hfill \\\\ & =& {17}^{-2}\\hfill & \\text{Simplify}.\\hfill \\\\ & =& \\frac{1}{{17}^{2}}\\text{ or }\\frac{1}{289}\\hfill & \\text{The negative exponent rule}\\hfill \\end{array}[\/latex]<\/li>\r\n\t
    3. [latex]\\begin{array}{cccc}\\hfill {\\left(\\frac{{u}^{-1}v}{{v}^{-1}}\\right)}^{2}& =& \\frac{{\\left({u}^{-1}v\\right)}^{2}}{{\\left({v}^{-1}\\right)}^{2}}\\hfill & \\text{The power of a quotient rule}\\hfill \\\\ & =& \\frac{{u}^{-2}{v}^{2}}{{v}^{-2}}\\hfill & \\text{The power of a product rule}\\hfill \\\\ & =& {u}^{-2}{v}^{2-\\left(-2\\right)}& \\text{The quotient rule}\\hfill \\\\ & =& {u}^{-2}{v}^{4}\\hfill & \\text{Simplify}.\\hfill \\\\ & =& \\frac{{v}^{4}}{{u}^{2}}\\hfill & \\text{The negative exponent rule}\\hfill \\end{array}[\/latex]<\/li>\r\n\t
    4. [latex]\\begin{array}{cccc}\\hfill \\left(-2{a}^{3}{b}^{-1}\\right)\\left(5{a}^{-2}{b}^{2}\\right)& =& -2\\cdot 5\\cdot {a}^{3}\\cdot {a}^{-2}\\cdot {b}^{-1}\\cdot {b}^{2}\\hfill & \\text{Commutative and associative laws of multiplication}\\hfill \\\\ & =& -10\\cdot {a}^{3 - 2}\\cdot {b}^{-1+2}\\hfill & \\text{The product rule}\\hfill \\\\ & =& -10ab\\hfill & \\text{Simplify}.\\hfill \\end{array}[\/latex]<\/li>\r\n\t
    5. [latex]\\begin{array}{cccc}\\hfill {\\left({x}^{2}\\sqrt{2}\\right)}^{4}{\\left({x}^{2}\\sqrt{2}\\right)}^{-4}& =& {\\left({x}^{2}\\sqrt{2}\\right)}^{4 - 4}\\hfill & \\text{The product rule}\\hfill \\\\ & =& \\text{ }{\\left({x}^{2}\\sqrt{2}\\right)}^{0}\\hfill & \\text{Simplify}.\\hfill \\\\ & =& 1\\hfill & \\text{The zero exponent rule}\\hfill \\end{array}[\/latex]<\/li>\r\n\t
    6. [latex]\\begin{array}{cccc}\\hfill \\frac{{\\left(3{w}^{2}\\right)}^{5}}{{\\left(6{w}^{-2}\\right)}^{2}}& =& \\frac{{\\left(3\\right)}^{5}\\cdot {\\left({w}^{2}\\right)}^{5}}{{\\left(6\\right)}^{2}\\cdot {\\left({w}^{-2}\\right)}^{2}}\\hfill & \\text{The power of a product rule}\\hfill \\\\ & =& \\frac{{3}^{5}{w}^{2\\cdot 5}}{{6}^{2}{w}^{-2\\cdot 2}}\\hfill & \\text{The power rule}\\hfill \\\\ & =& \\frac{243{w}^{10}}{36{w}^{-4}}\\hfill & \\text{Simplify}.\\hfill \\\\ & =& \\frac{27{w}^{10-\\left(-4\\right)}}{4}\\hfill & \\text{The quotient rule and reduce fraction}\\hfill \\\\ & =& \\frac{27{w}^{14}}{4}\\hfill & \\text{Simplify}.\\hfill \\end{array}[\/latex]<\/li>\r\n<\/ol>\r\n<\/div>\r\n
      \r\n

      Try It 9<\/h3>\r\nSimplify each expression and write the answer with positive exponents only.\r\n

      a. [latex]{\\left(2u{v}^{-2}\\right)}^{-3}[\/latex]\r\nb. [latex]{x}^{8}\\cdot {x}^{-12}\\cdot x[\/latex]\r\nc. [latex]{\\left(\\frac{{e}^{2}{f}^{-3}}{{f}^{-1}}\\right)}^{2}[\/latex]\r\nd. [latex]\\left(9{r}^{-5}{s}^{3}\\right)\\left(3{r}^{6}{s}^{-4}\\right)[\/latex]\r\ne. [latex]{\\left(\\frac{4}{9}t{w}^{-2}\\right)}^{-3}{\\left(\\frac{4}{9}t{w}^{-2}\\right)}^{3}[\/latex]\r\nf. [latex]\\frac{{\\left(2{h}^{2}k\\right)}^{4}}{{\\left(7{h}^{-1}{k}^{2}\\right)}^{2}}[\/latex]<\/p>\r\nSolution<\/a>\r\n\r\n<\/div>","rendered":"

      Recall that to simplify an expression means to rewrite it by combing terms or exponents; in other words, to write the expression more simply with fewer terms. The rules for exponents may be combined to simplify expressions.<\/p>\n

      \n

      Example 9: Simplifying Exponential Expressions<\/h3>\n

      Simplify each expression and write the answer with positive exponents only.<\/p>\n

        \n
      1. [latex]{\\left(6{m}^{2}{n}^{-1}\\right)}^{3}[\/latex]<\/li>\n
      2. [latex]{17}^{5}\\cdot {17}^{-4}\\cdot {17}^{-3}[\/latex]<\/li>\n
      3. [latex]{\\left(\\frac{{u}^{-1}v}{{v}^{-1}}\\right)}^{2}[\/latex]<\/li>\n
      4. [latex]\\left(-2{a}^{3}{b}^{-1}\\right)\\left(5{a}^{-2}{b}^{2}\\right)[\/latex]<\/li>\n
      5. [latex]{\\left({x}^{2}\\sqrt{2}\\right)}^{4}{\\left({x}^{2}\\sqrt{2}\\right)}^{-4}[\/latex]<\/li>\n
      6. [latex]\\frac{{\\left(3{w}^{2}\\right)}^{5}}{{\\left(6{w}^{-2}\\right)}^{2}}[\/latex]<\/li>\n<\/ol>\n<\/div>\n
        \n

        Solution<\/h3>\n
          \n
        1. [latex]\\begin{array}{cccc}\\hfill {\\left(6{m}^{2}{n}^{-1}\\right)}^{3}& =& {\\left(6\\right)}^{3}{\\left({m}^{2}\\right)}^{3}{\\left({n}^{-1}\\right)}^{3}\\hfill & \\text{The power of a product rule}\\hfill \\\\ & =& {6}^{3}{m}^{2\\cdot 3}{n}^{-1\\cdot 3}\\hfill & \\text{The power rule}\\hfill \\\\ & =& \\text{ }216{m}^{6}{n}^{-3}\\hfill & \\text{Simplify}.\\hfill \\\\ & =& \\frac{216{m}^{6}}{{n}^{3}}\\hfill & \\text{The negative exponent rule}\\hfill \\end{array}[\/latex]<\/li>\n
        2. [latex]\\begin{array}{cccc}\\hfill {17}^{5}\\cdot {17}^{-4}\\cdot {17}^{-3}& =& {17}^{5 - 4-3}\\hfill & \\text{The product rule}\\hfill \\\\ & =& {17}^{-2}\\hfill & \\text{Simplify}.\\hfill \\\\ & =& \\frac{1}{{17}^{2}}\\text{ or }\\frac{1}{289}\\hfill & \\text{The negative exponent rule}\\hfill \\end{array}[\/latex]<\/li>\n
        3. [latex]\\begin{array}{cccc}\\hfill {\\left(\\frac{{u}^{-1}v}{{v}^{-1}}\\right)}^{2}& =& \\frac{{\\left({u}^{-1}v\\right)}^{2}}{{\\left({v}^{-1}\\right)}^{2}}\\hfill & \\text{The power of a quotient rule}\\hfill \\\\ & =& \\frac{{u}^{-2}{v}^{2}}{{v}^{-2}}\\hfill & \\text{The power of a product rule}\\hfill \\\\ & =& {u}^{-2}{v}^{2-\\left(-2\\right)}& \\text{The quotient rule}\\hfill \\\\ & =& {u}^{-2}{v}^{4}\\hfill & \\text{Simplify}.\\hfill \\\\ & =& \\frac{{v}^{4}}{{u}^{2}}\\hfill & \\text{The negative exponent rule}\\hfill \\end{array}[\/latex]<\/li>\n
        4. [latex]\\begin{array}{cccc}\\hfill \\left(-2{a}^{3}{b}^{-1}\\right)\\left(5{a}^{-2}{b}^{2}\\right)& =& -2\\cdot 5\\cdot {a}^{3}\\cdot {a}^{-2}\\cdot {b}^{-1}\\cdot {b}^{2}\\hfill & \\text{Commutative and associative laws of multiplication}\\hfill \\\\ & =& -10\\cdot {a}^{3 - 2}\\cdot {b}^{-1+2}\\hfill & \\text{The product rule}\\hfill \\\\ & =& -10ab\\hfill & \\text{Simplify}.\\hfill \\end{array}[\/latex]<\/li>\n
        5. [latex]\\begin{array}{cccc}\\hfill {\\left({x}^{2}\\sqrt{2}\\right)}^{4}{\\left({x}^{2}\\sqrt{2}\\right)}^{-4}& =& {\\left({x}^{2}\\sqrt{2}\\right)}^{4 - 4}\\hfill & \\text{The product rule}\\hfill \\\\ & =& \\text{ }{\\left({x}^{2}\\sqrt{2}\\right)}^{0}\\hfill & \\text{Simplify}.\\hfill \\\\ & =& 1\\hfill & \\text{The zero exponent rule}\\hfill \\end{array}[\/latex]<\/li>\n
        6. [latex]\\begin{array}{cccc}\\hfill \\frac{{\\left(3{w}^{2}\\right)}^{5}}{{\\left(6{w}^{-2}\\right)}^{2}}& =& \\frac{{\\left(3\\right)}^{5}\\cdot {\\left({w}^{2}\\right)}^{5}}{{\\left(6\\right)}^{2}\\cdot {\\left({w}^{-2}\\right)}^{2}}\\hfill & \\text{The power of a product rule}\\hfill \\\\ & =& \\frac{{3}^{5}{w}^{2\\cdot 5}}{{6}^{2}{w}^{-2\\cdot 2}}\\hfill & \\text{The power rule}\\hfill \\\\ & =& \\frac{243{w}^{10}}{36{w}^{-4}}\\hfill & \\text{Simplify}.\\hfill \\\\ & =& \\frac{27{w}^{10-\\left(-4\\right)}}{4}\\hfill & \\text{The quotient rule and reduce fraction}\\hfill \\\\ & =& \\frac{27{w}^{14}}{4}\\hfill & \\text{Simplify}.\\hfill \\end{array}[\/latex]<\/li>\n<\/ol>\n<\/div>\n
          \n

          Try It 9<\/h3>\n

          Simplify each expression and write the answer with positive exponents only.<\/p>\n

          a. [latex]{\\left(2u{v}^{-2}\\right)}^{-3}[\/latex]
          \nb. [latex]{x}^{8}\\cdot {x}^{-12}\\cdot x[\/latex]
          \nc. [latex]{\\left(\\frac{{e}^{2}{f}^{-3}}{{f}^{-1}}\\right)}^{2}[\/latex]
          \nd. [latex]\\left(9{r}^{-5}{s}^{3}\\right)\\left(3{r}^{6}{s}^{-4}\\right)[\/latex]
          \ne. [latex]{\\left(\\frac{4}{9}t{w}^{-2}\\right)}^{-3}{\\left(\\frac{4}{9}t{w}^{-2}\\right)}^{3}[\/latex]
          \nf. [latex]\\frac{{\\left(2{h}^{2}k\\right)}^{4}}{{\\left(7{h}^{-1}{k}^{2}\\right)}^{2}}[\/latex]<\/p>\n

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

          \n\t\t\t

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