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I am trying to prove the following: If $a$ and $b$ are non-zero rational numbers, then $a^{b}$ is rational. Any ideas or hints how to prove this?

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It is false. $2^{1/2}=\sqrt 2$ is irrational.

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take $a=2$ and $b=\frac{3}{4}$ we get $2^{3/4}=\sqrt[4]{2^3}$ this is not rational

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