Originally Posted by

**scherz0** Hello everyone,

Could someone pleae check my solution for the following problem?

Thank you!

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1. Let $\displaystyle a $ and $\displaystyle b $ be real numbers with $\displaystyle a > 1 $ and $\displaystyle b > 0 $. If $\displaystyle ab = a^b $ and $\displaystyle \frac{a}{b} = a^{3b} $, determine the value of $\displaystyle a $.

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When the first equation is multiplied by a:

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

After substitution of (3) in the second equation:

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

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

Therefore:

$\displaystyle 2 - b = 3b \Rightarrow b = 1/2 $

After substitution of *b* into (1):

$\displaystyle 0.5a = \sqrt{a}$

$\displaystyle 0.25a^2 - a = 0$

$\displaystyle a(0.25a - 1) = 0 \Longrightarrow a = 0, 4 $