Question

Find all the seventh roots of (3 4i).

Answer 1

To find the 7th roots of (3 + 4i), first we have to convert (3 + 4i) to polar form (r cis theta), and then set up an equation where we let z^7 = r cis (theta + k2 pi)

(where of course r is |3 + 4i| and theta is the argument (3 + 4i), or the angle between the real axis and the complex number. I take it you've done at least some of this before, so I'll move on)

We then have
z = ( r cis (theta + k2 pi))^(1/7)
= (r^(1/7)) cis ((theta/7) + (k2 pi/7)) According to De Moiré's Theorem

Sub in k = 0, 1, 2, 3, 4, 5, 6 to get all of the roots in polar form. You can then convert them back to the other form, but it's very likely to be messy.