Roots of Complex Number/Examples/6th Roots of -27 i

Example of Roots of Complex Number
The complex $6$th roots of $-27 i$ are given by:
 * $\paren {-27 i}^{1/6} = \set {\sqrt 3 \, \map \cis {45 + 60 k} \degrees}$

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

That is:

Proof

 * Complex 6th Roots of -27 i.png

Let $z^6 = -27 i$.

We have that:
 * $z^6 = 27 \, \map \cis {\dfrac {3 \pi} 2 + 2 k \pi}$

Let $z = r \cis \theta$.

Then: