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How to prove that the power $b^n$ is an irrational number, where $b$ is an integer greater than $1$ and $n$ is an irrational number?

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This is not true. A counterexample is $b = 2$, $n = \log_2 3$, $b^n = 3$. –  Rahul Nov 14 '10 at 2:14

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up vote 3 down vote accepted

The Gelfond-Schneider theorem shows that if you assume that $n$ is algebraic, then $b^n$ is indeed irrational, in fact transcendental! You can even take $b$ to be algebraic.

So we can ask whether there's an easy proof that $ b^n $ is irrational for b integral and n an algebraic irrational.

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