Is $2^\sqrt{2}$ is rational or irrational?

Question

There are many popular problems on irrational power rational. A very interesting read is this blog. And there are many specific examples where rational power irrational is rational (e.g. $2^{\log_2(3)}$). But what about $2^{\sqrt{2}}$? Is it rational or irrational?

Answer 1

According to the Gelfond-Schneider theorem, it is transcendental (and, in particular, irrational).