How can I prove that an irrational number can be raised to another one and the result is rational?

Question

I don't know how to do this I'm just curious how can I.

Answer 1

Let $x$ and $y$ be 2 real numbers. Choose $y = \sqrt{2}$ and $x = \sqrt{3}^{\sqrt{2}}$

Now there are two scenarios possible :-

1. $x$ is rational, in which case your proposition is proved, as x is a number which equals an irrational number raised to the power of another irrational number.

2. $x$ is irrational, in which case we have $x^y = 3$ being rational.