Question

There are six different sixth roots of 64. That is, there are six complex numbers that solve x^6-64.

Find them.

Please describe the process if possible. Thanks!

Answer 1

There are 6 6th roots of 64 To find them, first write 64 in polar form:
64 = 64(cos0 + isin0)
If z = r(cosθ + isinθ) is a 6th root of 64, then z6 = 64 By DeMoivre's Theorem, this gives us:
r6[cos(6θ) + isin(6θ)] = 64[cos0 + isin0]

So, r6 = 64 and 6θ = 0° + k(360°)
r = 2 and θ = k(360°)/6 = k(60°), k = 0, 1, 2 ,...
If z is a 6th root of 64, then z = 2[cos(k(60°)) + isin(k(60°))], where k = 0, 1, 2, ...
1st 6th root (set k = 0): 2[cos0° + isin0°] = 2
2nd 6th root (set k = 1): 2[cos60° + isin60°] = 1 + √(3) i
3rd 6th root (set k = 2): 2[cos120° + isin120°] = -1 +√(3) i
4th 6th root (set k = 3): 2[cos180° + isin180°] = -2
5th 6th root (set k = 4): 2[cos240° + isin240°] = -1 - √(3) i
6th 6th root (set k = 5): 2[cos300° +isin300°] = 1 - √(3) i