Fifth Power of Complex Number

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Theorem

Let $z = a + i b$ be a complex number.

Then its fifth power is given by:

$z^5 = a^5 - 10 a^3 b^2 + 5 a b^4 + i \paren {5 a^4 b - 10 a^2 b^3 + b^5}$


Proof

\(\ds z^5\) \(=\) \(\ds \paren {a + i b}^5\) by hypothesis
\(\ds \) \(=\) \(\ds a^5 + 5 a^4 \paren {i b} + 10 a^3 \paren {i b}^2 + 10 a^2 \paren {i b}^3 + 5 a \paren {i b}^4 + \paren {i b}^5\) Fifth Power of Sum
\(\ds \) \(=\) \(\ds a^5 + 5 i a^4 b + 10 i^2 a^3 b^2 + 10 i^3 a^2 b^3 + 5 i^4 a b^4 + i^5 b^5\) Complex Multiplication is Commutative
\(\ds \) \(=\) \(\ds a^5 + 5 i a^4 b - 10 a^3 b^2 - 10 i a^2 b^3 + 5 a b^4 + i b^5\) Definition of Imaginary Unit
\(\ds \) \(=\) \(\ds a^5 - 10 a^3 b^2 + 5 a b^4 + i\paren {5 a^4 b - 10 a^2 b^3 + b^5}\) gathering like terms

$\blacksquare$


Sources

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