De Moivre's Theorem shows that to raise a complex number to the nth power, the absolute value is raised to the nth power and the argument is multiplied by n.

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Evaluate \[\left(\frac{\sqrt{2}+i\sqrt{2}}{2}\right)^{64}.\]

Let \(z\) be a complex number such that \[z = \sqrt{2}(\cos 4 ^{\circ} + i \sin 4 ^{\circ}).\] Then \( z^{8} \) can be expressed as \( r( \cos \alpha^{\circ} + i \sin \alpha^{\circ}) \), where \(r\) is a real number and \( 0 \leq \alpha \leq 90\). What is the value of \(r+\alpha ?\)

**Details and assumptions**

\(i\) is the imaginary number such that \(i^2=-1\).

What is the value of \(\left( \cos \frac{\pi}{8} + i \sin \frac{\pi}{8} \right)^{16} \)?

**Details and assumptions**

\(i\) is the imaginary unit, where \(i^2=-1\).

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