Complex Arithmetic/Examples/(z 3 - conj z 3)^5

Example of Complex Arithmetic

Let $z_3 = \sqrt 3 - 2 i$.

Then:

$\paren {z_3 - \overline {z_3} }^5 = -1024 i$

$\ds \paren {z_3 - \overline {z_3} }^5$

$=$

$\ds \paren {\paren {\sqrt 3 - 2 i} - \overline {\paren {\sqrt 3 - 2 i} } }^5$

$\ds $

$=$

$\ds \paren {\paren {\sqrt 3 - 2 i} - \paren {\sqrt 3 + 2 i} }^5$

$\ds $

$=$

$\ds \paren {-4 i}^5$

$\ds $

$=$

$\ds \paren {-1}^5 \times 4^5 \times i^5$

$\ds $

$=$

$\ds \paren {-1} \times 1024 \times i$

$\ds $

$=$

$\ds -1024 i$

$\blacksquare$

1981: Murray R. Spiegel: Theory and Problems of Complex Variables (SI ed.) ... (previous) ... (next): $1$: Complex Numbers: Supplementary Problems: Fundamental Operations with Complex Numbers: $54 \ \text {(c)}$