Complex Arithmetic/Examples/(Conjugate of z 3)^4

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Example of Complex Arithmetic

Let $z^3 = -\dfrac 1 2 + \dfrac {\sqrt 3} 2 i$.

Then:

$\paren {\overline {z_3} }^4 = -\dfrac 1 2 - \dfrac {\sqrt 3} 2 i$


Proof

\(\ds \paren {\overline {z_3} }^4\) \(=\) \(\ds \paren {-\dfrac 1 2 - \dfrac {\sqrt 3} 2 i}^4\) Definition of Complex Conjugate
\(\ds \) \(=\) \(\ds \paren {-\dfrac 1 2 - \dfrac {\sqrt 3} 2 i}^3 \times \paren {-\dfrac 1 2 - \dfrac {\sqrt 3} 2 i}\)


But from Cube Roots of Unity:

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


Hence the result.

$\blacksquare$


Sources