Find the indicated roots. Write the answer in a + bi form.
-Fourth roots of $- 16$
A) $2 + i , 2 - i , - 2 + i , - 2 - i$
B) $8 \sqrt { 2 } + 8 \sqrt { 2 } i , 8 \sqrt { 2 } - 8 \sqrt { 2 } i , - 8 \sqrt { 2 } + 8 \sqrt { 2 } i , - 8 \sqrt { 2 } - 8 \sqrt { 2 } i$
C) $\sqrt { 2 } + \sqrt { 2 } i , \sqrt { 2 } - \sqrt { 2 } i , - \sqrt { 2 } + \sqrt { 2 } i , - \sqrt { 2 } - \sqrt { 2 } i$
D) $1 + i , 1 - i , - 1 + i , - 1 - i$

Question

Quizplus · Accepted Answer

The fourth roots of -16 can be found by converting -16 into polar form and then applying De Moivre's Theorem. -16 can be represented as $16(\cos(\pi) + i\sin(\pi))$. The fourth roots are found by dividing the angle by 4 and taking the fourth root of the radius, resulting in roots of the form $\sqrt{2}(\cos(\frac{\pi + 2k\pi}{4}) + i\sin(\frac{\pi + 2k\pi}{4}))$ for $k = 0, 1, 2, 3$, which simplifies to the options in C.

Find the Indicated Roots −16- 16−16 A) 2+i,2−i,−2+i,−2−i2 + i , 2 - i , - 2 + i , - 2 - i2+i,2−i,−2+i,−2−i

Find the Indicated Roots $- 16$ A) $2 + i , 2 - i , - 2 + i , - 2 - i$