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This question already has an answer here:

Is $2^\sqrt{2}$ irrational? Is it transcendental?

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marked as duplicate by Najib Idrissi, Inactive - avoiding CoC, Tom-Tom, Claude Leibovici, hardmath Sep 16 '15 at 12:32

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According to Gel'fond's theorem, if $\alpha$ and $\beta$ are algebraic numbers (which $2$ and $\sqrt 2$ are) and $\beta$ is irrational, then $\alpha^\beta$ is transcendental, except in the trivial cases when $\alpha$ is 0 or 1.

Wikipedia's article about the constant $2^{\sqrt 2}$ says that it was first proved to be transcendental in 1930, by Kuzmin.

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