Discussion
The modulus of the complex number \(\left(2\sqrt{2} + i2\sqrt{2}\right)^{2}\) is equal to
Answer: E
The modulus is 16.
The modulus of a power is the power of the modulus, \(|w^{2}| = |w|^{2}\). Here \(w = 2\sqrt{2} + 2\sqrt{2}\,i\):
\[ \begin{aligned} |w| &= \sqrt{(2\sqrt{2})^{2} + (2\sqrt{2})^{2}} = \sqrt{8 + 8} = 4 \\ |w^{2}| &= 4^{2} = 16 \end{aligned} \]
4 is \(|w|\) without squaring; 64 is \(|w|^{3}\); 8 and 32 come from adding the parts instead of squaring them.
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