Complex Modulus of Quotient of Complex Numbers/Examples/(1+2i)^12 (1-2i)^-9
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Example of Complex Modulus of Quotient of Complex Numbers
- $\cmod {\dfrac {\paren {1 + 2 i}^{12} } {\paren {1 - 2 i}^9} } = 5 \sqrt 5$
Proof
| \(\ds \cmod {\dfrac {\paren {1 + 2 i}^{12} } {\paren {1 - 2 i}^9} }\) | \(=\) | \(\ds \dfrac {\cmod {\paren {1 + 2 i}^{12} } } {\cmod {\paren {1 - 2 i}^9} }\) | Complex Modulus of Quotient of Complex Numbers | |||||||||||
| \(\ds \) | \(=\) | \(\ds \dfrac {\cmod {1 + 2 i}^{12} } {\cmod {1 - 2 i}^9}\) | Power of Complex Modulus equals Complex Modulus of Power | |||||||||||
| \(\ds \) | \(=\) | \(\ds \dfrac {\sqrt{1^2 + 2^2}^{12} } {\sqrt{1^2 + 2^2}^9}\) | Definition of Complex Modulus | |||||||||||
| \(\ds \) | \(=\) | \(\ds \sqrt{1^2 + 2^2}^3\) | simplification | |||||||||||
| \(\ds \) | \(=\) | \(\ds \sqrt 5^3\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 5 \sqrt 5\) |
$\blacksquare$
Sources
- 1960: Walter Ledermann: Complex Numbers ... (previous) ... (next): $\S 1$. Algebraic Theory of Complex Numbers: Exercise $5$