If $x$ and $y$ are irrational, then $x^y$ is irrational

Question

I thought it was true, however my textbook claims it to be false. I need a counter example but I can't really think of one.

Answer 1

You can easily construct a counterexample.

Let $x=\sqrt{3}$ and $y=\sqrt{2}$ so both $x,y$ are irrational.

Now either $x^y$ is rational in which case we have a counterexample. Or it is irrational and when you raise it to the power of $y$ which is irrational you definitely get something rational

$$ (x^y)^y=3. $$

where both base and exponent were irrational.

Answer 2

Here is a simple example using Euler's Identity:

$$ e^{i\pi} = -1 $$

where $$ x = e$$

and $$ y = {i\pi} $$