Let X and Y are independent random variables. Then
$$M_{X+Y}(t)=M_{X}(t)M_{Y}(t)$$
Proof
\begin{eqnarray*}M_{X+Y}(t)&=&E[\mathrm{e}^{(X+Y)t}]\\&=&E[\mathrm{e}^{Xt}\mathrm{e}^{Yt}]\\&=&E[\mathrm{e}^{Xt}]E[\mathrm{e}^{Yt}]\ \ \ (\text{ ∵ X and Y are independent})\\&=&M_{X}(t)M_{Y}(t)\end{eqnarray*}