Roots of Complex Numbers

DeMoivre's Theorem:

Think about reversing the process:  Raise    to the nth power. The answer is
     r cis (q + 360^o k) = r cis q, which is the original number for which we're trying to find the nth root.

Example:

Find three complex roots of 8.

8 = 8 + 0i = 8 cis 0^o

³\/(8 cis 0 ^o) = 2 cis (0^o + 360^o(0)) = 2 cis 0^o = 2(1 + 0) = 2
                                        3

³\/(8 cis 0 ^o) = 2 cis (0 ^o + 360^o(1)) = 2 cis 120^o = 2(-1/2 + i\/3 /2) = -1 + i\/3
                                      3

³\/(8 cis 9 ^o) = 2 cis (0 ^o + 360^o(2)) = 2 cis 240^o = 2(-1/2 - i\/3 /2) = -1 - i\/3
                                        3

Basically, dividing the angles by 3 divides the unit circle into 3 equal parts.  Notice,

(q ^o + 360^o k) = q ^o   +  360^o  k   =   q ^o + 120^o k
        3                 3             3

The first angle in the answer is the original angle divided by the root number.  The second angle in the answer is 360^o divided by the root number.

Problems