a.
![z=-\dfrac{5\sqrt3}2+\dfrac52i=5\left(-\dfrac{\sqrt3}2+\dfrac12i\right)=5e^{i5\pi/6}](https://tex.z-dn.net/?f=z%3D-%5Cdfrac%7B5%5Csqrt3%7D2%2B%5Cdfrac52i%3D5%5Cleft%28-%5Cdfrac%7B%5Csqrt3%7D2%2B%5Cdfrac12i%5Cright%29%3D5e%5E%7Bi5%5Cpi%2F6%7D)
![w=1+\sqrt3\,i=2\left(\dfrac12+\dfrac{\sqrt3}2i\right)=2e^{i\pi/3}](https://tex.z-dn.net/?f=w%3D1%2B%5Csqrt3%5C%2Ci%3D2%5Cleft%28%5Cdfrac12%2B%5Cdfrac%7B%5Csqrt3%7D2i%5Cright%29%3D2e%5E%7Bi%5Cpi%2F3%7D)
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
and
is as simple as multiplying the moduli and adding the arguments:
![zw=5\cdot2e^{i(5\pi/6+\pi/3)}=10e^{i7\pi/6}](https://tex.z-dn.net/?f=zw%3D5%5Ccdot2e%5E%7Bi%285%5Cpi%2F6%2B%5Cpi%2F3%29%7D%3D10e%5E%7Bi7%5Cpi%2F6%7D)
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}](https://tex.z-dn.net/?f=%5Cdfrac%20zw%3D%5Cdfrac52e%5E%7Bi%285%5Cpi%2F6-%5Cpi%2F3%29%7D%3D%5Cdfrac52e%5E%7Bi%5Cpi%2F2%7D)
Explanation:
5.
The initial cost is $65.42
25% of 65.42 is 16.355
Add 16.355 + 65.42 and the total cost is: $81.78
6.
The initial cost is $12.50
3.2% of 12.50 is 0.4
Add 0.4 + 12.50 and the total cost is: $12.90
B...........................
Identity Property of Zero!
I hope this helps xD