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## De Moivre's Theorem

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.

# Raising to a Power - Basic



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 $$i^2=-1$$.

If \begin{align} x & =\cos \frac{\pi}{14}+i\sin \frac{\pi}{14},\\ y &= \cos \frac{9}{14}\pi+i\sin \frac{9}{14}\pi, \end{align} what is the value of $$x^{5} y^{15}$$?

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