Question

Let z= -5 sqrt 3/2 + 5/2i and w=1 + sqrt 3i

Answer 1

a.

$z=-\dfrac{5\sqrt3}2+\dfrac52i=5\left(-\dfrac{\sqrt3}2+\dfrac12i\right)=5e^{i5\pi/6}$

$w=1+\sqrt3\,i=2\left(\dfrac12+\dfrac{\sqrt3}2i\right)=2e^{i\pi/3}$

b. Not exactly sure how DeMoivre's theorem is relevant, since it has to do with taking powers of complex numbers... At any rate, multiplying $z$ and $w$ is as simple as multiplying the moduli and adding the arguments:

$zw=5\cdot2e^{i(5\pi/6+\pi/3)}=10e^{i7\pi/6}$

c. Similar to (b), except now you divide the moduli and subtract the arguments:

$\dfrac zw=\dfrac52e^{i(5\pi/6-\pi/3)}=\dfrac52e^{i\pi/2}$