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

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@Pink Elephants : Perhaps this should be an answer. –  Patrick Da Silva Jul 22 '12 at 8:50
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If it were so easy, it wouldn't have been on the list of Hilbert's problems, would it? –  J. M. Jul 22 '12 at 8:58
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@J.M.: As far as I understand it the Hilber's problem is to decide wheter it is trascendental, not to decide whether it is irrational. –  Marco Jul 22 '12 at 11:57
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up vote 6 down vote accepted

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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