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Question

# Use the polar form to find each power. Write the power in rectangular form.$(\sqrt{3}-i)^{3}$

Solution

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Let$z=r(\cos\theta+i\sin\theta)$ be the complex number in polar form, if n is positive then nth power of z i.e $z^n$ is given as

$z^n=r^n(\cos n\theta+i\sin n\theta)$

Here

$z=\sqrt{3}-i=2(\dfrac{\sqrt{3}}{2}-\dfrac{1}{2}i)$

$\cos\theta=\dfrac{\sqrt{3}}{2} \:\:\sin \theta =-\dfrac{1}{2}\:\text{ therefore}\: \: \theta=\dfrac{2\pi}{3}$

\begin{align*}z^3&=(\sqrt{3}-i)^3\\&=2^3(\cos (3..\dfrac{2\pi}{3})+i\sin( 3.\dfrac{2\pi}{3}))\\&=8(1+0i)\\&=8 \end{align*}

Rectangular form for given power of complex number is $z^3=8+0i$

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