Deciding whether $2^{\sqrt2}$ is irrational/transcendental [duplicate]
Is $2^\sqrt{2}$ irrational? Is it transcendental?
Solution 1:
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.