Use DeMoivre's Theorem to find the indicated power of the following complex number.
$$( 3 + 3 \sqrt { 3 } i ) ^ { 6 }$$
A) 19,683
B) 6,561
C) 2,187
D) 46,656
E) 243

Question

Quizplus · Accepted Answer

DeMoivre's Theorem states that for a complex number in polar form $ r(\cos \theta + i \sin \theta) $, its $ n $-th power is $ r^n (\cos(n\theta) + i \sin(n\theta)) $. The given complex number $ 3 + 3\sqrt{3}i $ can be converted to polar form with $ r = 6 $ and $ \theta = \frac{\pi}{3} $. Raising it to the 6th power gives $ 6^6 = 46,656 $.

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