What kind of number is $2^\sqrt{2}$

Question

What kind of number is $2^\sqrt{2}$ ? It is real and obviously is not integer nor rational. Is it real algebraic number?

Answer 1

According to the Gelfond–Schneider theorem, since $2$ and $\sqrt2$ are both algebraic with $a\ne0,1$ and $\sqrt2$ irrational, it follows that $2^{\sqrt2}$ is irrational and transcendental.

Your exact number is also known as the Gelfond-Schneider constant, and it has its own Wikipedia full of such information.