How to simplify the product of two $\exp$ functions

Question

It's been a while since I did any of this. I have the following product: $\exp(-j2 \pi u|k|x) \cdot \exp(-j2 \pi v |k|x)$. This seems like it is something that can be simplified, but how? Note, that is not convolution, it is simple multiplication. Thanks!

Answer 1

Since $e^x\cdot e^y=e^{x+y}$, we get $$ \exp(-j2 \pi u|k|x) \cdot \exp(-j2 \pi v |k|x)=\exp(-j2 \pi (u+v)|k|x) $$

Answer 2

$$\exp(-j2 \pi u|k|x) \cdot \exp(-j2 \pi v |k|x)=\exp\big(-j2\pi u|k|x-j2\pi v|k|x\big) $$

$$=\exp(-j2\pi (u+v)|k|x).$$

Exponentials obey $a^ba^c=a^{b+c}$ (though there can be branching issues for complex $a$).