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

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

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

Then:

From this we have:

and:

Thus:

Let $w = \dfrac p q$:

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

and hence:

So this gives:


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