This is easily the simplest question in all tutorials I have yet it's the one that I can't do...

Show that

$\displaystyle e^{z_1 + z_2} = e^{z_1} e^{z_2}$ where $\displaystyle z_1$, $\displaystyle z_2$ are complex numbers.

I'm using the binomial theorem but it's going in a huge mess and I'm getting nowhere. Can someone show me how it's done or show me a simpler way, binomial theorems just making me rage.