Find Roots of Complex Numbers in Polar Form
-The complex fourth roots of $- 16$ (rectangular form)
A) $\sqrt { 2 } + \sqrt { 2 } i , \sqrt { 2 } - \sqrt { 2 } i , - \sqrt { 2 } + \sqrt { 2 } i , - \sqrt { 2 } - \sqrt { 2 } i$
B) $1 + i , 1 - i , - 1 + i , - 1 - i$
C) $2 + i , 2 - i , - 2 + i , - 2 - i$
D) $8 \sqrt { 2 } + 8 \sqrt { 2 } i , 8 \sqrt { 2 } - 8 \sqrt { 2 } i , - 8 \sqrt { 2 } + 8 \sqrt { 2 } i , - 8 \sqrt { 2 } - 8 \sqrt { 2 } i$

Question

Quizplus · Accepted Answer

The complex number $-16$ in polar form is $16(\cos(\pi) + i\sin(\pi))$. Its fourth roots can be found using De Moivre's theorem, which gives roots of the form $2(\cos(\frac{\pi + 2k\pi}{4}) + i\sin(\frac{\pi + 2k\pi}{4}))$ for $k = 0, 1, 2, 3$. Calculating these gives the roots $\sqrt{2} + \sqrt{2}i$, $\sqrt{2} - \sqrt{2}i$, $-\sqrt{2} + \sqrt{2}i$, and $-\sqrt{2} - \sqrt{2}i$, matching option A.

Find Roots of Complex Numbers in Polar Form -The Complex −16- 16−16

Find Roots of Complex Numbers in Polar Form
-The Complex $- 16$