Algebra
# De Moivre's Theorem

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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 $i^2=-1$.