Why is :$\displaystyle {e}^\sqrt{2}$ is known to be transcedental number but ${\sqrt{2}}^ {e}$ is not known?

Question

I'm confused why $\displaystyle {e}^\sqrt{2}$ is known to be transcendental number but in the same time ${\sqrt{2}}^ {e}$ is not even known , why we can't deduce any thing from $\displaystyle {e}^\sqrt{2}$ to know more about irrationality of ${\sqrt{2}}^ {e}$ ?

Answer 1

There is no reason to expect $a^b$ to share properties with $b^a$, in general. For example, consider $({1\over 2})^{-1}$ versus $(-1)^{1\over 2}$.

(In fact, "swapping the exponent" isn't even well defined! E.g. $1^2=1^3$ but $2^1\not=3^1$.)