Power of Complex Modulus equals Complex Modulus of Power/Examples/(2-i)^6

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 $4 \ \text{(ii)}$