Find the indicated roots. Write the answer in polar form.
-Fourth roots of $256 \left( \cos 280 ^ { \circ } + i \sin 280 ^ { \circ } \right)$
A) $4 \left( \cos 70 ^ { \circ } + i \sin 70 ^ { \circ } \right) , 4 \left( \cos 160 ^ { \circ } + i \sin 160 ^ { \circ } \right) , 4 \left( \cos 250 ^ { \circ } + i \sin 250 ^ { \circ } \right) , 4 \left( \cos 340 ^ { \circ } + i \sin 340 ^ { \circ } \right)$
B) $4 \left( \cos 280 ^ { \circ } + i \sin 280 ^ { \circ } \right) , 4 \left( \cos 370 ^ { \circ } + i \sin 370 ^ { \circ } \right) , 4 \left( \cos 460 ^ { \circ } + i \sin 460 ^ { \circ } \right) , 4 \left( \cos 190 ^ { \circ } + i \sin 190 ^ { \circ } \right)$
C) $4 \left( \cos 140 ^ { \circ } + i \sin 140 ^ { \circ } \right) , 4 \left( \cos 230 ^ { \circ } + i \sin 230 ^ { \circ } \right) , 4 \left( \cos 320 ^ { \circ } + i \sin 320 ^ { \circ } \right) , 4 \left( \cos 50 ^ { \circ } + i \sin 50 ^ { \circ } \right)$
D) $4 \left( \cos 70 ^ { \circ } + i \sin 70 ^ { \circ } \right) , 4 \left( \cos 160 ^ { \circ } + i \sin 160 ^ { \circ } \right) , 4 \left( \cos 320 ^ { \circ } + i \sin 320 ^ { \circ } \right) , 4 \left( \cos 50 ^ { \circ } + i \sin 50 ^ { \circ } \right)$

Question

Quizplus · Accepted Answer

To find the fourth roots of $256(\cos 280^\circ + i\sin 280^\circ)$ in polar form, we first take the fourth root of the magnitude, which is $\sqrt[4]{256} = 4$, and then divide the angle by 4 and add $k \cdot 90^\circ$ (where $k = 0, 1, 2, 3$) because there are four roots for a fourth root operation. The initial angle is $280^\circ$, so dividing by 4 gives $70^\circ$. Adding $k \cdot 90^\circ$ for $k = 0, 1, 2, 3$ gives the angles $70^\circ$, $160^\circ$, $250^\circ$, and $340^\circ$. Therefore, the roots are $4(\cos 70^\circ + i\sin 70^\circ)$, $4(\cos 160^\circ + i\sin 160^\circ)$, $4(\cos 250^\circ + i\sin 250^\circ)$, and $4(\cos 340^\circ + i\sin 340^\circ)$.

Find the Indicated Roots 256(cos⁡280∘+isin⁡280∘)256 \left( \cos 280 ^ { \circ } + i \sin 280 ^ { \circ } \right)256(cos280∘+isin280∘)

Find the Indicated Roots $256 \left( \cos 280 ^ { \circ } + i \sin 280 ^ { \circ } \right)$