I roll a fair, 6-sided dice twice. Let $X$ and $Y$ be the outcomes of my first and second rolls. I subsequently find the moment generating function of $S = X + Y$ by finding $E(e^{t(X+Y)})$.
Why does $E(e^{t(X + Y)}) = E(e^{tX})E(e^{tY})$? I can see how $E(e^{t(X + Y)}) = E(e^{tX}e^{tY})$, but why can we just separate this expression into the product of two expected values?