If $X$ and $Y$ are continuous random variables uniformly distributed over $[0,1]$, find $E(X^Y)$.
My first thought was that $E(X^Y) = E(X)^{E(Y)}$, but through simulation I found that this was not the case.
I then tried using a double integral from $0$ to $1$ with respect to $x$ and $y$, but doing so results in a $1/\log$ term which cannot be evaluated at $0$ or $1$.
I'm not sure what to try next