Irrationality of some specific numbers

Question

Is $(\sqrt 2)^\sqrt3+(\sqrt 5)^\sqrt{7}$ an irrational number? is there any result about rationality\irrationality of numbers of the form $a^b+c^d$ where $a,b,c$ and $d$ are well-known irrational numbers (like $\sqrt{2}, \pi$ and...)

Answer 1

While the transcendentality (and even the irrationality) of $\sqrt 2 ^\sqrt3 + \sqrt 5 ^\sqrt7$ is still unknown, we can prove that $\sqrt 2 ^\sqrt3$ and $\sqrt 5 ^\sqrt7$ are transcendental using the Gelfond-Schneider theorem. The theorem states that:

If $a$ and $b$ are algebraic numbers with $a ≠ 0,1$ and $b$ non-rational, then any value of $a^b$ is a transcendental number.

This not only gives rise to trivial results such that $2^{\sqrt 2}$ is irrational, but to some more interesting ones as well. For instance, by taking $a=-1$ and $b=-i$, where $i=\sqrt {-1}$, we can obtain the following result:

$$ (-1)^{-i} = \left( e^{i \pi} \right)^{-i} = e^\pi, $$

showing that $e^\pi$ is transcendental.