Answer
The four fourth roots are:
\[
\begin{array}{l}
\cos \left(\frac{5 \pi}{6}\right)+i \sin \left(\frac{5 \pi}{6}\right) \\
\cos \left(\frac{4 \pi}{3}\right)+i \sin \left(\frac{4 \pi}{3}\right) \\
\cos \left(\frac{\pi}{3}\right)+i \sin \left(\frac{\pi}{3}\right) \\
\cos \left(\frac{11 \pi}{6}\right)+i \sin \left(\frac{11 \pi}{6}\right)
\end{array}
\]
Work Step by Step
\[
\cos \frac{4 \pi}{3}+i \sin \frac{4 \pi}{3}=z
\]
Using the roots theorem:
\[
\left[\cos \left(\frac{\theta}{n}+\frac{360^{\circ}}{n} k\right)+i \sin \left(\frac{\theta}{n}+\frac{360^{\circ}}{n} k\right)\right]r^{\frac{1}{n}}=w_{k}
\]
When $0=k$ :
\[
\cos \left(\frac{4 \pi}{3 \times 4}+\frac{2 \pi}{4}(0)\right)+i \sin \left(\frac{4 \pi}{3 \times 4}+\frac{2 \pi}{4}(0)\right)=u_{0}
\]
Simplify:
\[
=\sqrt{\cos \left(\frac{\pi}{3}\right)+i \sin \left(\frac{\pi}{3}\right)}
\]
When $1=k:$
\[
\cos \left(\frac{4 \pi}{3 \times 4}+\frac{2 \pi}{4}(1)\right)+i \sin \left(\frac{4 \pi}{3 \times 4}+\frac{2 \pi}{4}(1)\right)=w_{1}
\]
Simplify:
\[
=\cos \left(\frac{\pi}{3}+\frac{\pi}{2}\right)+i \sin \left(\frac{\pi}{3}+\frac{\pi}{2}\right)
\]
Unify the denominators and add:
\[
=\cos \left(\frac{5 \pi}{6}\right)+i \sin \left(\frac{5 \pi}{6}\right)
\]
When $2=k$ :
\[
\cos \left(\frac{4 \pi}{3 \times 4}+\frac{2 \pi}{4}(2)\right)+i \sin \left(\frac{4 \pi}{3 \times 4}+\frac{2 \pi}{4}(2)\right)=w_{2}
\]
Simplify:
\[
=\cos \left(\frac{\pi}{3}+\pi\right)+i \sin \left(\frac{\pi}{3}+\pi\right)
\]
Unify the denominators and add:
\[
={\cos \left(\frac{4 \pi}{3}\right)+i \sin \left(\frac{4 \pi}{3}\right)}
\]
When $3=k$ :
\[
\cos \left(\frac{4 \pi}{3 \times 4}+\frac{2 \pi}{4}(3)\right)+i \sin \left(\frac{4 \pi}{3 \times 4}+\frac{2 \pi}{4}(3)\right)=v_{3}
\]
Simplify:
\[
=\cos \left(\frac{\pi}{3}+\frac{3 \pi}{2}\right)+i \sin \left(\frac{\pi}{3}+\frac{3 \pi}{2}\right)
\]
Unify the denominators and add:
\[
={\cos \left(\frac{11 \pi}{6}\right)+i \sin \left(\frac{11 \pi}{6}\right)}
\]
