Roots of Complex Number/Examples/Cube Roots of 2 + 2 root 3 i

Example of Roots of Complex Number
The complex cube roots of $2 + 2 \sqrt 3 i$ are given by:
 * $\paren {2 + 2 \sqrt 3 i}^{1/3} = \set {\sqrt [3] 4 \, \map \cis {20 + 120 k} \degrees}$

for $k = 0, 1, 2$.

That is:

Proof

 * Complex Cube Roots of 2 + 2 root 3 i.png

Let $z^3 = 2 + 2 \sqrt 3 i$.

We have that:
 * $z^3 = 4 \, \map \cis {\dfrac \pi 3 + 2 k \pi} = 4 \, \map \cis {60 \degrees + k \times 360 \degrees}$

Let $z = r \cis \theta$.

Then: