Find the given power. Write answer in rectangular form.
-$( 2 - 2 \mathrm { i } ) ^ { 5 }$
A) $- \sqrt { 2 } + \sqrt { 2 } i$
B) $- 128 + 128 \mathrm { i }$
C) $- 64 + 64 i$
D) $- 64 \sqrt { 2 } + 64 \sqrt { 2 } i$

Question

Quizplus · Accepted Answer

To find $(2 - 2i)^5$ in rectangular form, first express $2 - 2i$ in polar form. The magnitude $r$ is $\sqrt{2^2 + (-2)^2} = \sqrt{8} = 2\sqrt{2}$, and the angle $\theta$ is $\tan^{-1}\left(\frac{-2}{2}\right) = \tan^{-1}(-1) = -\frac{\pi}{4}$ (or $7\pi/4$ for a positive angle). In polar form, $2 - 2i = 2\sqrt{2}(\cos(-\pi/4) + i\sin(-\pi/4))$. Using De Moivre's Theorem to raise to the 5th power: $(2\sqrt{2})^5(\cos(5(-\pi/4)) + i\sin(5(-\pi/4))) = 32\sqrt{2}(\cos(-5\pi/4) + i\sin(-5\pi/4))$. Since $\cos(-5\pi/4) = \cos(\pi/4) = \frac{1}{\sqrt{2}}$ and $\sin(-5\pi/4) = \sin(\pi/4) = \frac{1}{\sqrt{2}}$, the result is $32\sqrt{2}(\frac{1}{\sqrt{2}} + i\frac{1}{\sqrt{2}}) = 32(1 + i) = -128 + 128i$.

Find the Given Power (2−2i)5( 2 - 2 \mathrm { i } ) ^ { 5 }(2−2i)5

Find the Given Power $( 2 - 2 \mathrm { i } ) ^ { 5 }$