Answer
The four fourth roots are:
\[
\sqrt{3}+i, \quad-1+i \sqrt{3}, \quad-\sqrt{3}-i, \quad \text { , } \quad 1-i \sqrt{3}
\]
Work Step by Step
\[
\left(\cos \frac{2 \pi}{3}+i \sin \frac{2 \pi}{3}\right)16=z
\]
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 $0=k$ :
\[
\left[\cos \left(\frac{2 \pi}{3 \times 4}+\frac{2 \pi}{4}(0)\right)+i \sin \left(\frac{2 \pi}{3 \times 4}+\frac{2 \pi}{4}(0)\right)\right]16^{\frac{1}{4}}=u_{0}
\]
Simplify:
\[
\begin{aligned}
&\left[\cos \left(\frac{\pi}{6}\right)+i \sin \left(\frac{\pi}{6}\right)\right] \\
&=2\left[\frac{\sqrt{3}}{2}+\frac{1}{2} i\right]2=u_{0}
\end{aligned}
\]
Multiply:
\[
=\sqrt{3}+i
\]
When $k=1:$
\[
w_{1}=16^{\frac{1}{2}}\left[\cos \left(\frac{2 \pi}{3 \times 4}+\frac{2 \pi}{4}(1)\right)+i \sin \left(\frac{2 \pi}{3 \times 4}+\frac{2 \pi}{4}(1)\right)\right]
\]
Simplify:
\[
=2\left[\cos \left(\frac{\pi}{6}+\frac{\pi}{2}\right)+i \sin \left(\frac{\pi}{6}+\frac{\pi}{2}\right)\right]
\]
Unify the denominators and add:
\[
\begin{array}{c}
=2\left[\cos \left(\frac{2 \pi}{3}\right)+i \sin \left(\frac{2 \pi}{3}\right)\right] \\
=2\left[-\frac{1}{2}+\frac{\sqrt{3}}{2} i\right]
\end{array}
\]
Multiply:
\[
=-1+i \sqrt{3}
\]
When $k=2$ :
\[
w_{2}=16^{\frac{1}{4}}\left[\cos \left(\frac{2 \pi}{3 \times 4}+\frac{2 \pi}{4}(2)\right)+i \sin \left(\frac{2 \pi}{3 \times 4}+\frac{2 \pi}{4}(2)\right)\right]
\]
Simplify:
\[
=2\left[\cos \left(\frac{\pi}{6}+\pi\right)+i \sin \left(\frac{\pi}{6}+\pi\right)\right]
\]
Unify the denominators and add:
\[
\begin{array}{c}
=2\left[\cos \left(\frac{7 \pi}{6}\right)+i \sin \left(\frac{7 \pi}{6}\right)\right] \\
=2\left[-\frac{\sqrt{3}}{2}-\frac{1}{2} i\right]
\end{array}
\]
Multiply:
\[
=-\sqrt{3}-i
\]
When $3=k$ :
\[
\left[\cos \left(\frac{2 \pi}{3 \times 4}+\frac{2 \pi}{4}(3)\right)+i \sin \left(\frac{2 \pi}{3 \times 4}+\frac{2 \pi}{4}(3)\right)\right]16^{\frac{1}{4}}=w_{3}
\]
Simplify:
\[
=2\left[\cos \left(\frac{\pi}{6}+\frac{3 \pi}{2}\right)+i \sin \left(\frac{\pi}{6}+\frac{3 \pi}{2}\right)\right]
\]
Unify the denominators and add:
\[
\begin{array}{c}
=2\left[\cos \left(\frac{5 \pi}{3}\right)+i \sin \left(\frac{5 \pi}{3}\right)\right] \\
=2\left[\frac{1}{2}-\frac{\sqrt{3}}{2} i\right]
\end{array}
\]
Multiply:
\[
=[1-i \sqrt{3}]
\]
