a.
[tex]z=-\dfrac{5\sqrt3}2+\dfrac52i=5\left(-\dfrac{\sqrt3}2+\dfrac12i\right)=5e^{i5\pi/6}[/tex]
[tex]w=1+\sqrt3\,i=2\left(\dfrac12+\dfrac{\sqrt3}2i\right)=2e^{i\pi/3}[/tex]
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 [tex]z[/tex] and [tex]w[/tex] is as simple as multiplying the moduli and adding the arguments:
[tex]zw=5\cdot2e^{i(5\pi/6+\pi/3)}=10e^{i7\pi/6}[/tex]
c. Similar to (b), except now you divide the moduli and subtract the arguments:
[tex]\dfrac zw=\dfrac52e^{i(5\pi/6-\pi/3)}=\dfrac52e^{i\pi/2}[/tex]
