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Math Help - evaluation of exponential expression

  1. #1
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    evaluation of exponential expression

    without a calculator, how do i determine that :
    32^3/5
    = 8
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  2. #2
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    Quote Originally Posted by dymand68 View Post
    without a calculator, how do i determine that :
    32^3/5
    = 8
    (32^{1/5})^3 and you're expected to know what 32^{1/5} is.
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  3. #3
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    Quote Originally Posted by dymand68 View Post
    without a calculator, how do i determine that :
    32^3/5
    = 8

    Definitions: 32^{3\slash 5} = \left(32^{1\slash 5}\right)^3=\left(\sqrt[5]{32}\right)^3 . Now just check that 2^5 = 32 so...

    Tonio
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  4. #4
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    the problem is:
    evaluate:
    32^-3/5
    =1/32^3/5
    =???
    1/8 ?
    thus my original question: without a calculator, how do i Know that
    32^3/5
    = 8
    ...okay, find the fifth root of 32 and cube it, gotcha
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  5. #5
    Member integral's Avatar
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    If you want to be fancy

    This does not require a calculator at all!

    32^{3/5}=n
    (take the log of both sides)

    \textrm{log}_{32}32^{\frac{3}{5}}=\textrm{log}_{32  }8

    \textrm{log}_a(a)^x=x\therefore

    \frac{3}{5}=\textrm{log}_{32}8
    log base change formula. (changes and base to two base 10 logs):
    \textrm{log}_ab=\frac{\textrm{log}(b)}{\textrm{log  }(a)} \therefore

    \frac{3}{5}=\frac{\textrm{log}8}{\textrm{log}32}

    \textrm{log}(a)^x=x\textrm{log}a
    and
    \textrm{log}(ab)=\textrm{log}(a)+\textrm{log}(b)

    \frac{3}{5}=\frac{3\textrm{log}2}{\textrm{log}4+\t  extrm{log}8}

    \frac{3}{5}=\frac{3\textrm{log}2}{2\textrm{log}2+3  \textrm{log}2}

    \frac{3}{5}=\frac{3\textrm{log}2}{5\textrm{log}2}

    \frac{3}{5}=\frac{3}{5}

    \therefore 32^{3/5}=8


    ((This is my 100th post! Whooo! ))
    Last edited by integral; April 1st 2010 at 11:35 PM.
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