I asked a similar question in a different thread but that one got answered and I thought this would be the natural extension.
Is $\pi^{b}$ transcendental for any algebraic $b$? Is this a known result? If this is too broad maybe something like $\pi^{\sqrt{2}}$. I feel like if $\pi^{2}$ is transcendental and if there is any justice in the world so should $\pi^{\sqrt{2}}$.
Thanks.
According to Wikipedia, most sums, products, powers, etc., of $\pi$ have unknown status; in particular, it isn't known whether $\pi^{\sqrt{2}}$ is transcendental. However, your suggestion that this should follow from the transcendental nature of $\pi^2$ (together with the existence of justice in the world) is certainly false. Consider $x=2^{\sqrt{2}/2}$. It provides a counterexample, in that $x^2=2^{\sqrt{2}}$ is transcendental (the Gelfond-Schneider constant), while $x^{\sqrt{2}}=2$ (two) is not.