# Proving an irrational to an irrational is rational

1. Oct 18, 2008

### hew

1. The problem statement, all variables and given/known data
prove that it is possible that an irrational number raised to another irrational, can be rational.
you are given root2 to root2 to root2

2. Relevant equations

3. The attempt at a solution
i have shown that root2 to root2 to root2 is rational, but would appreciate a hint on showing root2 to root2 is irrational

2. Oct 18, 2008

### HallsofIvy

Staff Emeritus
Suppose $\sqrt{2}^\sqrt{2}$ is rational. Then you are done!

If it is not rational, then
$$\left(\sqrt{2}^\sqrt{2}\right)^\sqrt{2}$$
is again an "irrational to an irrational power".

Now, what is
$$\left(\sqrt{2}^\sqrt{2}\right)^\sqrt{2}$$?

Do you see how, even though we don't know whether $\sqrt{2}^\sqrt{2}$ is rational or irrational, either way we have an irrational number to an irrational power that is rational?

Last edited: Oct 20, 2008
3. Oct 20, 2008

### hew

wow, i thought you somehow had to prove it. Thanks

4. Oct 20, 2008

### HallsofIvy

Staff Emeritus
Either that or get someone to prove it for you!