Is $\sqrt{3}^\sqrt{5}$ rational or irrational?

Question

Is $\sqrt{3}^\sqrt{5}$ rational or irrational?

One way is to let $x$=$\sqrt{3}^\sqrt{5}$ and then calculate $antilog \ (log (\sqrt 3) \times \sqrt(5))$ which gives irrational number.

But is there a way to check it without calculators..

Answer 1

This is a direct application of the Gelfond–Schneider theorem. Interestingly, I don't think it's easier to prove that this number is irrational than to prove it is transcendental!