Is it known whether ${\sqrt{2}}^{\sqrt{2}}$ is irrational?

Question

I know the famous proof that uses $x={\sqrt{2}}^{\sqrt{2}}$ to prove that there must exist an irrational to an irrational power that evaluates to a rational. But I don't know if $x$ itself is known to be irrational or rational.

Could someone enlighten me? Thanks!

Answer 1

It's transcendental by the Gelfond-Schneider theorem.