Roots of Complex Number/Examples/6th Roots of 64

Example of Roots of Complex Number
The complex $6$th roots of $64$ are given by:
 * $\paren {64}^{1/6} = \set {2 \, \map \cis {60 k} \degrees}$

for $k = 0, 1, 2, 3, 4, 5$.

That is:

Proof

 * Complex 6th Roots of 64.png

Let $z^6 = 64$.

We have that:
 * $z^6 = 64 \, \map \cis {0 + 2 k \pi}$

Let $z = r \cis \theta$.

Then:

When $k = 3$ we have:


 * $z_4 = 2 \cis 180 \degrees = - 2$