Find the given power. Write answer in rectangular form.
-$$\left( - \frac { 1 } { 2 } - \frac { \sqrt { 3 } } { 2 } i \right) ^ { 10 }$$
A) $\frac { 1 } { 2 } - \frac { \sqrt { 3 } } { 2 } \mathrm { i }$
B) $- \frac { 1 } { 2 } + \frac { \sqrt { 3 } } { 2 } \mathrm { i }$
C) $\frac { 1 } { 2 } + \frac { \sqrt { 3 } } { 2 } \mathrm { i }$
D) $- \frac { 1 } { 2 } - \frac { \sqrt { 3 } } { 2 } \mathrm { i }$

Question

Quizplus · Accepted Answer

The given complex number is in polar form, $r(\cos \theta + i\sin \theta)$, where $r = 1$ and $\theta = \frac{5\pi}{3}$ or $300^\circ$. Raising it to the 10th power using De Moivre's theorem, we get $r^{10}(\cos 10\theta + i\sin 10\theta)$. Since $10\theta = 3000^\circ$ or $8\pi + \frac{4\pi}{3}$, which is equivalent to $\frac{4\pi}{3}$ or $240^\circ$, the resulting angle corresponds to the complex number $-\frac{1}{2} - \frac{\sqrt{3}}{2}i$, which matches option D.

Find the Given Power (−12−32i)10\left( - \frac { 1 } { 2 } - \frac { \sqrt { 3 } } { 2 } i \right) ^ { 10 }(−21​−23​​i)10

Find the Given Power $\left( - \frac { 1 } { 2 } - \frac { \sqrt { 3 } } { 2 } i \right) ^ { 10 }$