# Complex Arithmetic/Examples/Modulus of ((2 z 2 + z 1 - 5 - 1) (2 z 1 - z 2 + 3 - 1)^-1)^2

## 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.

$\blacksquare$