Roots of Complex Number/Examples/5th Roots of -16 + 16 root 3 i

Example of Roots of Complex Number
The complex $5$th roots of $-16 + 16 \sqrt 3 i$ are given by:
 * $\paren {-16 + 16 \sqrt 3 i}^{1/5} = \set {2 \, \map \cis {24 + 72 k} \degrees}$

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

That is:

Proof

 * Complex 5th Roots of -16 + 16 root 3 i.png

Let $z^5 = -16 + 16 \sqrt 3 i$.

We have that:
 * $z^5 = 16 \sqrt 2 \, \map \cis {\dfrac {2 \pi} 3 + 2 k \pi} = 16 \sqrt 2 \, \map \cis {120 \degrees + k \times 360 \degrees}$

Let $z = r \cis \theta$.

Then: