This is a classic (very old).
If you've seen it, please don't post the answer.
An irrational number raised to an irrational power is always irrational.
Prove it or provide a counterexample.
The "classic" solution goes like this:
Theorem: . can be rational.
Proof
There are only two possibilities:
. . (1) is rational.
. . (2) is irrational.
If (1) is rational, the theroem is verified.
If (2) is irrational, consider: .
. . an irrational number raised to an irrational power.
Then we have: . . . . a rational number.
Either way, an irrational raised to an irrational power can be rational.
~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~
I've always found this proof amusing.
Note that we still don't know if is rational or irrational
. . but it doesn't matter . . .