Use De Moivre's Theorem to find the indicated power of the complex number. Write your answer in standard form a + bi.
-$( 1 + i ) ^ { 20 }$
A) $- 1024 i$
B) 1024
C) $1024 \mathrm { i }$
D) $- 1024$

Question

Quizplus · Accepted Answer

De Moivre's Theorem states that $(\cos \theta + i \sin \theta)^n = \cos(n\theta) + i\sin(n\theta)$. The complex number $1 + i$ can be written in polar form as $\sqrt{2}(\cos(45^\circ) + i\sin(45^\circ))$. Applying De Moivre's Theorem for $n = 20$, we get $\sqrt{2}^{20}(\cos(900^\circ) + i\sin(900^\circ))$, which simplifies to $1024(\cos(0) + i\sin(0)) = 1024$, hence the answer is $-1024$.

Use De Moivre's Theorem to Find the Indicated Power of the Complex