Find the indicated roots. Write the answer in polar form.
-Fifth roots of $7 \left( \cos \frac { \pi } { 3 } + i \sin \frac { \pi } { 3 } \right)$
A) $\sqrt [ 5 ] { 7 } \left( \cos \frac { \pi } { 15 } + \mathrm { i } \sin \frac { \pi } { 15 } \right) , \sqrt [ 5 ] { 7 } \left( \cos \frac { 7 \pi } { 15 } + \mathrm { i } \sin \frac { 7 \pi } { 15 } \right) , \sqrt [ 5 ] { 7 } \left( \cos \frac { 13 \pi } { 15 } + \mathrm { i } \sin \frac { 13 \pi } { 15 } \right)$, $\sqrt [ 5 ] { 7 } \left( \cos \frac { 19 \pi } { 15 } + i \sin \frac { 19 \pi } { 15 } \right) , \sqrt [ 5 ] { 7 } \left( \cos \frac { 5 \pi } { 3 } + i \sin \frac { 5 \pi } { 3 } \right)$
B) $\frac { 7 } { 5 } \left( \cos \frac { \pi } { 15 } + i \sin \frac { \pi } { 15 } \right) , \frac { 7 } { 5 } \left( \cos \frac { 7 \pi } { 15 } + i \sin \frac { 7 \pi } { 15 } \right) , \frac { 7 } { 5 } \left( \cos \frac { 13 \pi } { 15 } + i \sin \frac { 13 \pi } { 15 } \right)$, $\frac { 7 } { 5 } \left( \cos \frac { 19 \pi } { 15 } + i \sin \frac { 19 \pi } { 15 } \right) , \frac { 7 } { 5 } \left( \cos \frac { 5 \pi } { 3 } + i \sin \frac { 5 \pi } { 3 } \right)$
C) $\frac { 7 } { 5 } \left( \cos \frac { \pi } { 15 } + i \sin \frac { \pi } { 15 } \right) , \frac { 7 } { 5 } \left( \cos \frac { 4 \pi } { 15 } + i \sin \frac { 4 \pi } { 15 } \right) , \frac { 7 } { 5 } \left( \cos \frac { 7 \pi } { 15 } + i \sin \frac { 7 \pi } { 15 } \right)$, $\frac { 7 } { 5 } \left( \cos \frac { 2 \pi } { 3 } + i \sin \frac { 2 \pi } { 3 } \right) , \frac { 7 } { 5 } \left( \cos \frac { 13 \pi } { 15 } + i \sin \frac { 13 \pi } { 15 } \right)$
D) $\sqrt [ 5 ] { 7 } \left( \cos \frac { \pi } { 15 } + i \sin \frac { \pi } { 15 } \right) , \sqrt [ 5 ] { 7 } \left( \cos \frac { 4 \pi } { 15 } + i \sin \frac { 4 \pi } { 15 } \right) , \sqrt [ 5 ] { 7 } \left( \cos \frac { 7 \pi } { 15 } + i \sin \frac { 7 \pi } { 15 } \right)$, $\sqrt [ 5 ] { 7 } \left( \cos \frac { 2 \pi } { 3 } + i \sin \frac { 2 \pi } { 3 } \right) , \frac { 7 } { 5 } \left( \cos \frac { 13 \pi } { 15 } + i \sin \frac { 13 \pi } { 15 } \right)$

Question

Quizplus · Accepted Answer

The fifth roots of a complex number in polar form, $r(\cos \theta + i\sin \theta)$, are given by $\sqrt[5]{r}(\cos \frac{\theta + 2k\pi}{5} + i\sin \frac{\theta + 2k\pi}{5})$ for $k = 0, 1, 2, 3, 4$. Here, $r = 7$ and $\theta = \frac{\pi}{3}$, so the roots are correctly represented in choice A with the magnitudes $\sqrt[5]{7}$ and angles $\frac{\pi}{15}$, $\frac{7\pi}{15}$, $\frac{13\pi}{15}$, $\frac{19\pi}{15}$, and $\frac{5\pi}{3}$ after applying the formula for each $k$.

Find the Indicated Roots 7(cos⁡π3+isin⁡π3)7 \left( \cos \frac { \pi } { 3 } + i \sin \frac { \pi } { 3 } \right)7(cos3π​+isin3π​)

Find the Indicated Roots $7 \left( \cos \frac { \pi } { 3 } + i \sin \frac { \pi } { 3 } \right)$