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De Moivre’s Theorem

If the polar form of a complex number is $x=r(cos\theta +i\cdot sin\theta)$ , then for positive integers n. $z^{n}=[r(cos\theta +i\cdot sin\theta)]^{n}$ or $r^{n}(cos(n\theta)+i\cdot sin(n\theta))$ .

Example: Find $(4_4i)^{6}$

express in polar form

$r=\sqrt{a^{2}+b^{2}}$

$r=\sqrt{4^{2}+4^{2}}$

$r=\sqrt{16+16}$

$r=\sqrt{32}$

$r=4\sqrt{2}$

$\theta=tan^{-1}\frac{b}{a}$

$\theta=tan^{-1}\frac{4}{4}$

$\theta=\frac{\pi}{4}$

$[4\sqrt{2}(cos\frac{\pi}{4}+i\cdot sin\frac{\pi}{4})]^{6}$

$(4\sqrt{2})^{6}[cos(6\frac{\pi}{4})+i\cdot sin(6\frac{\pi}{4})]$

$32768(cos\frac{3\pi}{2}+i\cdot sin\frac{3\pi}{2})$

$32768(0-1i)$

$-32768i$