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Thread: determine the value of a

  1. #1
    rcs
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    determine the value of a

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  2. #2
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    $\displaystyle ab = a^b$ and $\displaystyle \frac{a}{b} = a^{3b}$.

    Rearranging the second equation gives $\displaystyle b = \frac{a}{a^{3b}} = a^{1 - 3b}$.

    Substituting into the first gives

    $\displaystyle a(a^{1 - 3b}) = a^b$

    $\displaystyle a^{2 - 3b} = a^b$

    $\displaystyle 2 - 3b = b$

    $\displaystyle 2 = 4b$

    $\displaystyle b = \frac{1}{2}$.


    Back substituting gives

    $\displaystyle b = a^{1 - 3b}$

    $\displaystyle \frac{1}{2} = a^{1 - \frac{3}{2}}$

    $\displaystyle \frac{1}{2} = a^{-\frac{1}{2}}$

    $\displaystyle 2 = a^{\frac{1}{2}}$

    $\displaystyle a = 4$.
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