# Can an irrational number raised to an irrational power be rational?

Can an irrational number raised to an irrational power be rational?

If it can be rational, how can one prove it?

There is a classic example here. Consider $$A=√2√2A=\sqrt{2}^\sqrt{2}$$. Then $$AA$$ is either rational or irrational. If it is irrational, then we have $$A√2=√22=2A^\sqrt{2}=\sqrt{2}^2=2$$.