This might be naive. Is $x^x$ a rational number for $x=\sqrt{2}^\sqrt{2}$ ?
I remember reading somewhere a long time ago that such $x^x$ is a rational number, as an example of issues with non-constructive math and excluded middle. (I think the question was whether there exists irrational $x$, such that $x^x$ is rational.) But I didn't give it much thought or figured out why. I dismissed the issue as I mistakenly thought exponentiation is associative, and assumed $x^x = \sqrt{2}^2=2$.
Can someone help show constructively whether this is a rational number?
Related:
Can an irrational number raised to an irrational power be rational?