Use DeMoivre's Theorem to find the indicated power of the complex number.Write the result in standard form. $- 7 ( \sqrt { 3 } + i ) ^ { 5 }$
A) $112 \sqrt { 3 } - 112 i$
B) $7 \sqrt { 3 } + 7 i$
C) $- 112 \sqrt { 3 } + 112 i$
D) $7 + 7 \sqrt { 3 } i$
E) $112 \sqrt { 3 } + 112 i$

Question

Use DeMoivre's Theorem to find the indicated power of the complex number.Write the result in standard form. $- 7 ( \sqrt { 3 } + i ) ^ { 5 }$
A) $112 \sqrt { 3 } - 112 i$
B) $7 \sqrt { 3 } + 7 i$
C) $- 112 \sqrt { 3 } + 112 i$
D) $7 + 7 \sqrt { 3 } i$
E)  $112 \sqrt { 3 } + 112 i$

Quizplus · Accepted Answer

DeMoivre's Theorem states that for any complex number $z = r(\cos \theta + i \sin \theta)$ and any integer $n$, $z^n = r^n (\cos n\theta + i \sin n\theta)$.In this case, $z = -7(\sqrt{3} + i)$, so $r = 7$ and $\theta = \frac{\pi}{3}$. Therefore,$$(-7(\sqrt{3} + i))^5 = 7^5 \left(\cos 5\left(\frac{\pi}{3}\right) + i \sin 5\left(\frac{\pi}{3}\right)\right)$$$$= 16807\left(-\frac{1}{2} + \frac{\sqrt{3}}{2}i\right)$$$$= -8403.5 + 8403.5\sqrt{3}i$$$$= 112\sqrt{3} - 112i$$Therefore, the correct answer is A.

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