algebra precalculus - Proving $|z_1z_2|=|z_1||z_2|$ using exponential form of a Complex Number

Problem:

Prove $$|z_1z_2|=|z_1||z_2|$$ where $z_1,z_2$ are Complex Numbers.

I tried to solve this using the exponential form of a Complex Number.

Assuming $z_1=r_1e^{i\theta_1}$ and $z_2=r_2e^{i\theta_2},$
I got $$|z_1z_2|=|r_1e^{i\theta_1}\times r_2e^{i\theta_2}|= |r_1 r_2e^{i(\theta_1+\theta_2)}|$$
Unfortunately I cannot think of how to proceed further. Any help would be greatly appreciated! Many thanks in anticipation!

Answer

$$|re^{i\theta}|=|r|$$

So

$$|z_{1}|=|r_{1}| $$

$$|z_{2}|=|r_{2}|$$

$$|z_{1}z_{2}|=|r_{1}r_{2}|$$

$r_{1},r_{2}$ are real numbers and so $|r_{1}r_{2}|=|r_{1}||r_{2}|=|z_{1}||z_{2}|$

$$|z_{1}z_{2}|=|z_{1}||z_{2}|$$