# Thread: finding powers of a complex number

1. ## finding powers of a complex number

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
$

2. 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}