finding powers of a complex number

**robasc** · December 10th 2008, 07:29 PM

I am checking to see if this is correct?

Use Demoivre's Theorem to find the indicated power of the complex number.

$ (\sqrt(3) - i)^9 $

$ 2^9[(cos 9*\pi/6 + i \sin 9*\pi/6)] $

$ 512(0+(-1))= -512 $

**Mathstud28** · December 10th 2008, 07:35 PM

Originally Posted by robasc

I am checking to see if this is correct?

Use Demoivre's Theorem to find the indicated power of the complex number.

$ (\sqrt(3) - i)^9 $

$ 2^9[(cos 9*\pi/6 + i \sin 9*\pi/6)] $

$ 512(0+(-1))= -512 $

$\begin{aligned}\left(\sqrt{3}-i\right)^9&=\left(2\cos\left(\frac{\pi}{6}\right)-2i\sin\left(\frac{\pi}{6}\right)\right)^9\\ &=2^9\left(\cos\left(\frac{-\pi}{6}\right)+i\sin\left(\frac{-\pi}{6}\right)\right)^9\\ &=2^9e^{\frac{-9i\pi}{6}}\\ &=2^9\left(\cos\left(\frac{-9\pi}{6}\right)+i\sin\left(\frac{-9\pi}{6}\right)\right)\\ &=2^9(0--i)\\ &=512i\end{aligned}$

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