TITLE: Square root of a natural number to square root of another natural number
QUESTION [7 upvotes]: Is there such integers $x,y$ which they're not perfect squares and they're not equal, such that:
$\sqrt{x}^\sqrt{y}$ is actually an integer?  Or rational number?

REPLY [10 votes]: By the Gelfond-Schneider Theorem, the number is always transcendental. In particular, it cannot be rational.

REPLY [9 votes]: By Gelfond-Schneider theorem, you can show that $\sqrt{x}^\sqrt{y}$ is irrational.