Fourth Power of Complex Number

Theorem

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

Then its fourth power is given by:

$z^4 = a^4 - 6 a^2 b^2 b^4 i \paren {4 a^3 b - 4 a b^3}$

Proof

\(\ds z^4\)

\(=\)

\(\ds \paren {a i b}^4\)

by hypothesis

\(\ds \)

\(=\)

\(\ds a^4 4 a^3 \paren {i b} 6 a^2 \paren {i b}^2 4 a \paren {i b}^3 \paren {i b}^4\)

Fourth Power of Sum

\(\ds \)

\(=\)

\(\ds a^4 4 i a^3 b 6 i^2 a^2 b^2 4 i^3 a b^3 i^4 b^4\)

Complex Multiplication is Commutative

\(\ds \)

\(=\)

\(\ds a^4 4 i a^3 b - 6 a^2 b^2 - 4 i a b^3 b^4\)

Definition of Imaginary Unit

\(\ds \)

\(=\)

\(\ds a^4 - 6 a^2 b^2 b^4 i \paren {4 a^3 b - 4 a b^3}\)

gathering like terms

$\blacksquare$

Sources

• 1964: Milton Abramowitz and Irene A. Stegun: Handbook of Mathematical Functions ... (previous) ... (next): $3$: Elementary Analytic Methods: $3.7$ Complex Numbers and Functions: Powers: $3.7.20$