1. ## DeMoivre's theorem

"use DeMoivre's theorem to find the indicated power of the complex number. Write the answer in rectangular form."

(squareroot{3(cos 5pie/6 + isin 5pie/6))^4

Am I doing this correctly?

(sqrt3)^4(cos(4*5pie/6)+isin(4*5pie/6)
9(cos 20pie/6)+isin(20pie/6)
9(-.5)+(-.866i)
=-4.5-7.8i

I'm worried I might have done this wrong due to the step in reducing 20pie/6 I couldn't find anything to reduce it by so just left it in that form.

2. Originally Posted by ConfusedMath
"use DeMoivre's theorem to find the indicated power of the complex number. Write the answer in rectangular form."

(squareroot{3(cos 5pie/6 + isin 5pie/6))^4

Am I doing this correctly?

(sqrt3)^4(cos(4*5pie/6)+isin(4*5pie/6)
9(cos 20pie/6)+isin(20pie/6)
9(-.5)+(-.866i)
=-4.5-7.8i

I'm worried I might have done this wrong due to the step in reducing 20pie/6 I couldn't find anything to reduce it by so just left it in that form.
$\left[\sqrt{3}\left(\cos{\frac{5\pi}{6}} + i\sin{\frac{5\pi}{6}}\right)\right]^4 = (\sqrt{3})^4 \left(\cos{\frac{4\cdot 5\pi}{6}} + i\sin{\frac{4\cdot 5\pi}{6}}\right)$

$= 9\left(\cos{\frac{10\pi}{3}} + i\sin{\frac{10\pi}{3}}\right)$

$= 9\left(\cos{\frac{4\pi}{3}} + i\sin{\frac{4\pi}{3}}\right)$

$= 9\left(-\cos{\frac{\pi}{3}} - i\sin{\frac{\pi}{3}}\right)$

$= 9\left(-\frac{1}{2} - i\frac{\sqrt{3}}{2}\right)$

$= -\frac{9}{2} - \frac{9\sqrt{3}}{2}i$.