Use DeMoivre's Theorem to find the indicated power. $( 1 + i ) ^ { 20 }$
A) $1024$
B) $512 i$
C) $- 1024$
D) $256$
E) $- 256 i$

Question

Quizplus · Accepted Answer

DeMoivre's Theorem states that for a complex number in polar form $ r(\cos \theta + i \sin \theta) $, its power is given by $ r^n (\cos(n\theta) + i \sin(n\theta)) $. The complex number $ 1 + i $ has a modulus $ \sqrt{2} $ and an argument $ \frac{\pi}{4} $. Raising it to the 20th power gives $ (\sqrt{2})^{20} (\cos(5\pi) + i \sin(5\pi)) = 1024(-1) = -1024 $.

Use DeMoivre's Theorem to Find the Indicated Power (1+i)20( 1 + i ) ^ { 20 }(1+i)20

Use DeMoivre's Theorem to Find the Indicated Power $( 1 + i ) ^ { 20 }$