Use DeMoivre's Theorem to find the indicated power of the complex number.Write the result in standard form. $\left[ - 5 \left( \cos \left( 210 ^ { \circ } \right) + i \sin \left( 210 ^ { \circ } \right) \right) \right] ^ { 10 }$
A) $- \frac { 9765625 } { 2 } + \frac { 9765625 \sqrt { 3 } } { 2 } i$
B) $\frac { 9765625 } { 2 } + \frac { 9765625 \sqrt { 3 } } { 2 } i$
C) $\frac { 9765625 } { 2 } - \frac { 9765625 \sqrt { 3 } } { 2 } i$
D) $\frac { 9765625 \sqrt { 3 } } { 2 } - \frac { 9765625 } { 2 } i$
E) $- \frac { 9765625 \sqrt { 3 } } { 2 } + \frac { 9765625 } { 2 } i$

Question

Quizplus · Accepted Answer

DeMoivre's Theorem states that $(r(\cos \theta + i \sin \theta))^n = r^n (\cos(n\theta) + i \sin(n\theta))$. Here, $r = 5$, $\theta = 210^\circ$, and $n = 10$. So, $(-5)^{10} = 9765625$ and $10 \times 210^\circ = 2100^\circ$. Reducing $2100^\circ$ modulo $360^\circ$ gives $2100^\circ \equiv 60^\circ$. Thus, $\cos(60^\circ) = \frac{1}{2}$ and $\sin(60^\circ) = \frac{\sqrt{3}}{2}$. Therefore, the result is $\frac{9765625}{2} - \frac{9765625\sqrt{3}}{2}i$.

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