elementary number theory - How can I prove $sqrt{sqrt2}$ to be irrational?

How can I prove $\sqrt{\sqrt2}$ to be irrational?

I know that $\sqrt2$ is an irrational number, it can be proved by contradiction, but I'm not sure how to prove that $\sqrt{\sqrt2} = \sqrt[4]{2}$ is irrational as well.

Answer

Suppose $x= \sqrt{ \sqrt 2}$ was rational, then so is its square $x^2=\sqrt 2$ which you have shown is irrational. Contradiction!