Find the given power. Write answer in rectangular form.
-$( - \sqrt { 3 } + i ) ^ { 6 }$
A) $- 64$
B) $64 - 64 \sqrt { 3 } \mathrm { i }$
C) $- 64 \sqrt { 3 } + 64 \mathrm { i }$
D) $64 \mathrm { i }$

Question

Quizplus · Accepted Answer

To find $(- \sqrt { 3 } + i ) ^ { 6 }$, we first express the complex number in polar form. The magnitude $r$ is $\sqrt{(-\sqrt{3})^2 + 1^2} = 2$, and the angle $\theta$ (in radians) is $\tan^{-1}(\frac{1}{-\sqrt{3}}) = -\frac{\pi}{6}$ (since the complex number is in the fourth quadrant). Thus, the polar form is $2(\cos(-\frac{\pi}{6}) + i\sin(-\frac{\pi}{6}))$. Using De Moivre's theorem to raise it to the 6th power, we get $2^6(\cos(-\pi) + i\sin(-\pi)) = 64(-1 + 0i) = -64$, which is in rectangular form.

Find the Given Power (−3+i)6( - \sqrt { 3 } + i ) ^ { 6 }(−3​+i)6

Find the Given Power $( - \sqrt { 3 } + i ) ^ { 6 }$