Roots of Complex Number/Corollary/Examples

Examples of Roots of Complex Number: Corollary

Complex Cube Roots

Let $z \in \C$ be a complex number.

Let $z \ne 0$.

Let $w$ be one of the (complex) cube roots of $z$.

Then the complete set of (complex) cube roots of $z$ is:

$\set {w, w \omega, w \omega^2}$

where:

$\omega = e^{2 \pi / 3} = -\dfrac 1 2 + \dfrac {i \sqrt 3} 2$

Fourth Roots of $2 - 2 i$

The complex $4$th roots of $2 - 2 i$ are given by:

$\paren {2 - 2 i}^{1/4} = \set {b, bi, -b, -bi}$

where:

$b = \sqrt [8] 8 \paren {\cos \dfrac \pi {16} + i \sin \dfrac \pi {16} }$