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

Source Work

1981: Murray R. Spiegel: Theory and Problems of Complex Variables (SI ed.)

Chapter $1$: Complex Numbers

Supplementary Problems: Fundamental Operations with Complex Numbers: $54 \ \text {(c)}$

Mistake

If ... $z_3 = \sqrt 3 - 2 i$, evaluate ...:

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

Ans. $1024 i$

Correction

This result actually works out to $-1024 i$:

\(\ds \paren {z_3 - \overline {z_3} }^5\)

\(=\)

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

\(\ds \)

\(=\)

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

\(\ds \)

\(=\)

\(\ds \paren {-4 i}^5\)

\(\ds \)

\(=\)

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

\(\ds \)

\(=\)

\(\ds \paren {-1} \times 1024 \times i\)

\(\ds \)

\(=\)

\(\ds -1024 i\)

$\blacksquare$

Sources

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