Problem:
What is the sum of the roots of z12=64 that have a positive real part?
Answer Choices:
A. 2
B. 4
C. 2+23
D. 22+6
E. (1+3)+(1+3)i Solution:
The principal root of the equation z12=64 is
z=64121⋅(cos6π+isin6π)=2⋅(cos6π+isin6π)
The 12 roots lie in the complex plane on the circle of radius 2 centered at the origin. The roots with positive real part make angles of 0,±6π, and ±3π with the positive real axis. When these five numbers are added, the imaginary parts cancel out and the sum is