Proving an irrational to an irrational is rational

[hew]

Homework Statement

prove that it is possible that an irrational number raised to another irrational, can be rational.
you are given root2 to root2 to root2

Homework Equations

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

 

[HallsofIvy]

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?

 

[hew]

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

 

[HallsofIvy]

Either that or get someone to prove it for you!