Answer
The two products are equal.
$z_{1}z_{2}=-5-5i\sqrt {3}$
Work Step by Step
$z_{1}z_{2}=(-5)(1+i\sqrt 3)=-5-5i\sqrt 3$
$z_{1}$ in trigonometric form is
$-5=5(\cos\pi+i\sin\pi)$
$z_{2}$ in trigonometric form is
$1+i\sqrt 3=2(\cos \frac{\pi}{3}+i\sin\frac{\pi}{3})$
Applying the formula
$(r_{1}\,cis\,\theta_{1})(r_{2}\,cis\,\theta_{2})=r_{1}r_{2}\,cis\,(\theta_{1}+\theta_{2})$, we get
$z_{1}z_{2}=[5(\cos\pi+i\sin\pi)][2(\cos \frac{\pi}{3}+i\sin\frac{\pi}{3})]$
$=5\cdot2[\cos(\pi+\frac{\pi}{3})+i\sin(\pi+\frac{\pi}{3})]$
$=10(\cos \frac{4\pi}{3}+i\sin \frac{4\pi}{3})$
In standard form, our result is
$=10(-\frac{1}{2}+i\cdot-\frac{\sqrt 3}{2})=-5-5i\sqrt 3$
