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Math Help - Finding the value of...

  1. #1
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    Finding the value of...

    Hi,

    \mathrm{If\ a\ and\ b\ are\ positive\ integers\ and\ (a^\frac12 \cdot b^\frac13)^6\ =\ 432\ what\ is\ the\ value\ of\ ab?}

    (a^\frac12 \cdot b^\frac13)^6\ =\ 432

    \underbrace{a^3\ \cdot\ b^2\ =\ 432}_\mathrm{Stuck\ here}

    Perhaps I went about it incorrectly...
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  2. #2
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    432 = 2(216) = (2)(2)(108) = 2(2)(2)(54) = 2(2)(2)(2)(27) = (4)^2(3)^3.
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  3. #3
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    So ab = 12. I figured this out by removing the exponents from both sides, knowing that the bases on both sides would become equal; seeing that this is math, magic isn't allowed.

    So I ask, what are the intermediary steps between:
    a^3 \cdot b^2 = (4)^2(3)^3 that math allows?
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  4. #4
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    Quote Originally Posted by Hellbent View Post
    So I ask, what are the intermediary steps between:
    a^3 \cdot b^2 = (4)^2(3)^3 that math allows?
    What you did is correct! Magic isn't allowed, but intuition is. We compare the bases: if (a)^3(b)^2 = (4)^2(3)^3,
    then a = 3, and b = 4. It's purely deductive. (It's the same way we compare the coefficients of polynomials).
    Last edited by TheCoffeeMachine; December 30th 2010 at 02:11 PM.
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  5. #5
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    Quote Originally Posted by Hellbent View Post
    Hi,

    \mathrm{If\ a\ and\ b\ are\ positive\ integers\ and\ (a^\frac12 \cdot b^\frac13)^6\ =\ 432\ what\ is\ the\ value\ of\ ab?}

    (a^\frac12 \cdot b^\frac13)^6\ =\ 432

    \underbrace{a^3\ \cdot\ b^2\ =\ 432}_\mathrm{Stuck\ here}

    Perhaps I went about it incorrectly...
    Since a and b are positive integers,

    a^3b^2=432\Rightarrow\ a^3b^3=432b\Rightarrow\ (ab)^3=432b

    which is a multiple of 432 and ab is also an integer.

    Multiples of 432 are 432(2)=864,\;\;432(3)=1296,\;\;432(4)=1728,....

    7^3=343,\;\;8^3=512\ \ne\ 432k

    9^3=729,\;\;10^3=1000,\;\;11^3=1331,\;\;12^3=1728

    We have a hit.

    ab=12


    Alternatively,

    \displaystyle\ a^2b^2=\frac{432}{a}\Rightarrow\ (ab)^2=\frac{432}{a}

    \displaystyle\frac{432}{2}=216,\;\;\frac{432}{3}=1  44,......

    12^2=144\Rightarrow\ ab=12
    Last edited by Archie Meade; January 1st 2011 at 04:55 PM.
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