Roots of Complex Number/Examples/Square Roots of 2 root 3 - 2i

Example of Roots of Complex Number

The complex square roots of $2 \sqrt 3 - 2 i$ are given by:

$\paren {2 \sqrt 3 - 2 i}^{1/2} = \set {2 \cis 165 \degrees, 2 \cis 345 \degrees}$

Let $z^2 = 2 \sqrt 3 - 2 i$.

We have that:

$z^2 = 4 \paren {\dfrac {\sqrt 3} 2 - \dfrac 1 2 i}$

and it is seen (indirectly) from Cube Roots of Unity that:

$\dfrac {\sqrt 3} 2 - \dfrac 1 2 i = \cis \dfrac {11 \pi} 6$

Hence

$\ds z^2$

$=$

$\ds 4 \cis \dfrac {11 \pi} 6$

$\ds \leadsto \ \ $

$\ds z$

$=$

$\ds \sqrt 4 \, \paren {\cis \dfrac {11 \pi} {12} + k \pi}$

where $k = 0, 1$

$\ds $

$=$

$\ds 2 \cis \dfrac {11 \pi} {12} \text { and } 2 \cis \dfrac {23 \pi} {12}$

$\ds $

$=$

$\ds 2 \cis 165 \degrees \text { and } 2 \cis 345 \degrees$

$\blacksquare$

1981: Murray R. Spiegel: Theory and Problems of Complex Variables (SI ed.) ... (previous) ... (next): $1$: Complex Numbers: Supplementary Problems: Roots of Complex Numbers: $95 \ \text{(a)}$