Use DeMoivre's Theorem to find the indicated power of the following complex number.
$$( 5 + 5 \sqrt { 3 } i ) ^ { 6 }$$
A) $1,953,125$
B) 390,625
C) 78,125
D) $1,000,000$
E) 3,125

Question

Quizplus · Accepted Answer

DeMoivre's Theorem states that $(r(\cos \theta + i\sin \theta))^n = r^n (\cos n\theta + i\sin n\theta)$. First, convert $5 + 5\sqrt{3}i$ to polar form. The modulus $r = \sqrt{5^2 + (5\sqrt{3})^2} = 10$, and the argument $\theta = \tan^{-1}(\frac{5\sqrt{3}}{5}) = \frac{\pi}{3}$. Applying DeMoivre's Theorem, $(10(\cos \frac{\pi}{3} + i\sin \frac{\pi}{3}))^6 = 10^6 (\cos 2\pi + i\sin 2\pi) = 1,000,000$, since $\cos 2\pi = 1$ and $\sin 2\pi = 0$.

Use DeMoivre's Theorem to Find the Indicated Power of the Following