Taking 4th roots is harder than multiplying. So let's look at multiplying complex numbers first.
We're just following the basic rules first, nothing fancy. Remember i^2 = -1:
firsts lasts inners outers
(cos(x) + i sin(x)) * (cos(y) + i sin(y)) = cos(x)*cos(y) - sin(x)*sin(y) + i (sin(x)*cos(y) + sin(y)*cos(x))
You might recognize that the Real terms are the identity for cos(x+y) and the Imaginary terms are the identity for sin(x+y).
This means that if you multiply two complex numbers, if they're both on the unit circle, then you can just add the angles together. This is pretty amazing. So to take roots of complex numbers on the unit circle, you can just divide the angle. If I started with A*(cos(x) + i sin(x)) and B*(cos(y) + i sin(y))) I hope you can see that the result would just be A*B*(cis (x + y)). The angles add and the magnitudes multiply.
So your angle was 160 and your magnitude was 81. The 4th root of 81 is 3 and 160/4 = 40.
So one solution to your problem is 3(cos(40) + i sin(40)).
What are the other 3 solutions? Well, many angles can refer to the same angle:
160 = 160+360 = 160+360+360 = ...
We found one 4th root angle: 160/4 = 40. So let's try this with 3 other angles:
(360+160)/4 = 130
(360+360+160)/4 = 220
(360+360+360+160)/4 = 310
So our 4 possible angles are 40, 40+90, 40+90+90, and 40+90+90+90. Our magnitude is 3.
I hope this helps.
