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

Using DeMoivre's Theorem, the magnitude is raised to the power, and the angle is multiplied by the power. The magnitude is $(-5)^{10} = 9765625$, and the angle is $2100^\circ$, which is equivalent to $60^\circ$. Thus, the result is $\frac{9765625}{2} - \frac{9765625\sqrt{3}}{2}i$.

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