Fourth Power of Complex Number

From ProofWiki
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

Retrieved from "https://proofwiki.org/w/index.php?title=Fourth_Power_of_Complex_Number&oldid=632912"