Roots of Complex Number/Examples/Cube Roots of -11-2i

Example of Roots of Complex Number: Corollary
The complex cube roots of $-11 - 2 i$ are given by:
 * $\paren {-11 - 2 i}^{1/3} = \set {1 + 2 i, -\dfrac 1 2 + \sqrt 3 + \paren {-1 - \dfrac {\sqrt 3} 2} i, -\dfrac 1 2 - \sqrt 3 + \paren {-1 + \dfrac {\sqrt 3} 2} i}$

Proof
Let $z^3 = -11 - 2 i = \paren {p + iq}^3$.

Then:

From this we have:

and:

Thus:

Let $w = \dfrac p q$:

Putting $\dfrac p q = \dfrac 1 2$ leads to::
 * $2 p = q$

and hence:

So this gives:


 * $z = \begin {cases} 1 + 2i \\ -\dfrac 1 2 + \sqrt 3 + i \paren {-1 - \dfrac {\sqrt 3} 2} \\ -\dfrac 1 2 - \sqrt 3 + i \paren {-1 + \dfrac {\sqrt 3} 2} \end{cases}$