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

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DeMoivre's Theorem states that for a complex number in polar form $ r(\cos \theta + i \sin \theta) $, its nth power is $ r^n (\cos(n\theta) + i \sin(n\theta)) $. First, convert $ 3 + 3\sqrt{3}i $ to polar form: the modulus is $ 6 $ and the argument is $ \pi/3 $. Raising to the 6th power gives $ 6^6 (\cos(2\pi) + i \sin(2\pi)) = 46,656 $.

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