Roots of Complex Number/Examples/Square Roots of 4 root 2 + 4 root 2 i

Example of Roots of Complex Number

The complex square roots of $4 \sqrt 2 + 4 \sqrt 2 i$ are given by:

$\paren {4 \sqrt 2 + 4 \sqrt 2 i}^{1/2} = \set {\sqrt 8 \cis 22.5 \degrees, \sqrt 8 \cis 202.5 \degrees}$

Let $z^2 = 4 \sqrt 2 + 4 \sqrt 2 i$.

We have that:

$z^2 = 8 \paren {\dfrac {\sqrt 2} 2 + \dfrac {\sqrt 2} 2 i}$

and it is seen that:

$\dfrac {\sqrt 2} 2 + \dfrac {\sqrt 2} 2 i = \cis \dfrac \pi 4$

Hence

$\ds z^2$

$=$

$\ds 8 \cis \dfrac \pi 4$

$\ds \leadsto \ \ $

$\ds z$

$=$

$\ds \sqrt 8 \, \paren {\cis \dfrac \pi 8 + k \pi}$

where $k = 0, 1$

$\ds $

$=$

$\ds \sqrt 8 \cis \dfrac \pi 8 \text { and } \sqrt 8 \cis \dfrac {9 \pi} 8$

$\ds $

$=$

$\ds \sqrt 8 \cis 22.5 \degrees \text { and } \sqrt 8 \cis 202.5 \degrees$

$\blacksquare$

1981: Murray R. Spiegel: Theory and Problems of Complex Variables (SI ed.) ... (previous) ... (next): $1$: Complex Numbers: Supplementary Problems: Roots of Complex Numbers: $96 \ \text{(b)}$