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Math Help - Work out the power

  1. #1
    Senior Member Mukilab's Avatar
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    Work out the power

    64^{-\frac{2}{3}}

    Method not answer please

    No calculator
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  2. #2
    Super Member bigwave's Avatar
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    64^{-\frac{2}{3}}

    \frac{1}{64^{\frac{2}{3}}}

    \frac{1}{\left(64^{\frac{1}{3}}\right)^2}

    = \frac{1}{16}
    Last edited by bigwave; January 7th 2010 at 11:08 AM. Reason: latex
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  3. #3
    Member rowe's Avatar
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    64^{-\frac{2}{3}} = \frac{1}{64^{\frac{2}{3}}}

    Now:

    64^{\frac{2}{3}} = \sqrt[3]{64}^2 = 4^2 = 16

    But we have the reciprocal so the answer is:

    \frac{1}{16}
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  4. #4
    Senior Member Mukilab's Avatar
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    Quote Originally Posted by rowe View Post
    64^{-\frac{2}{3}} = \frac{1}{64^{\frac{2}{3}}}

    Now:

    64^{\frac{2}{3}} = \sqrt[3]{64}^2 = 4^2 = 16

    But we have the reciprocal so the answer is:

    \frac{1}{16}
    Quote Originally Posted by bigwave View Post
    64^{-\frac{2}{3}}

    \frac{1}{64^{\frac{2}{3}}}

    \frac{1}{\left(64^{\frac{1}{3}}\right)^2}

    = \frac{1}{16}
    Thank you but I prefer bigwave's method. Thanks bigwave!
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  5. #5
    Senior Member Mukilab's Avatar
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    Actually. Both are hard without a calculator :/
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  6. #6
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    e^(i*pi)'s Avatar
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    Quote Originally Posted by Mukilab View Post
    Actually. Both are hard without a calculator :/
    The key to this question is knowing/working out that 64 = 4^3

    If you prefer you can use prime factors

    64 = 2 \times 32
    32 = 2 \times 16
    16 = 2 \times 8
    8 = 2 \times 4
    4 = 2 \times 2

    Therefore 64 = 2^6 = 2^{3+3} = 2^3 \cdot 2^3

    (\sqrt[3]{64})^2 = [\sqrt[3]{2^3} \times \sqrt[3]{2^3}]^2

    Cubes and cube roots cancel to leave (2 \times 2)^2 = 16<br />
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  7. #7
    Senior Member Mukilab's Avatar
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    Oh ok. Simplified a complicated looking question ^^
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