Power of Complex Modulus equals Complex Modulus of Power/Examples/(2-i)^6
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Example of Power of Complex Modulus equals Complex Modulus of Power
- $\cmod {\paren {2 - i}^6} = 125$
Proof
\(\ds \cmod {2 - i}\) | \(=\) | \(\ds \sqrt {2^2 + \paren {-1}^2}\) | Definition of Complex Modulus | |||||||||||
\(\ds \) | \(=\) | \(\ds \sqrt 5\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \cmod {\left({2 - i}\right)^6}\) | \(=\) | \(\ds \paren {\sqrt 5}^6\) | Power of Complex Modulus equals Complex Modulus of Power | ||||||||||
\(\ds \) | \(=\) | \(\ds 5^3\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 125\) |
$\blacksquare$
Sources
- 1960: Walter Ledermann: Complex Numbers ... (previous) ... (next): $\S 1$. Algebraic Theory of Complex Numbers: Exercise $4 \ \text{(ii)}$