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
- 1981: Murray R. Spiegel: Theory and Problems of Complex Variables (SI ed.) ... (previous) ... (next): $1$: Complex Numbers: Solved Problems: Fundamental Operations with Complex Numbers: $2 \ \text{(c)}$