Fourth Power of Complex Number
Jump to navigation
Jump to search
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: $3.7.20$