Distribution of expectation operator when computing mgf of X bar

Question

I'm trying to work through the proof for the moment generating function of $\overline{X}$.

The proof below looks fairly straightforward but I'm having trouble understanding getting from the 2nd to 3rd line. How are they able to split up the expectation operator like that? Normally, $E(XY)\neq E(X)E(Y)$, so why can they do it here? Is it because everything is in the same base $e$? Any explanation would be very much appreciated!

Answer 1

$E[XY]=E[X]E[Y]$ holds if $X$ and $Y$ are independent. Presumably, you probably have some independence assumption about $X_1, \ldots, X_n$.