Answer:

Step-by-step explanation:
We can determine the power of a complex number by the De Moivre's Theorem, which states that for all
, where
, the power of the complex number is:
(1)
Where:
- Magnitude of the complex number, dimensionless.
- Direction of the complex number.
If we know that
,
and
, then the fourth power of the complex number is:
![z^{4} = 1^{4}\cdot \left[\cos\left(\frac{8\pi}{3} \right)+i\,\sin\left(\frac{8\pi}{3}\right)\right]](https://tex.z-dn.net/?f=z%5E%7B4%7D%20%3D%201%5E%7B4%7D%5Ccdot%20%5Cleft%5B%5Ccos%5Cleft%28%5Cfrac%7B8%5Cpi%7D%7B3%7D%20%5Cright%29%2Bi%5C%2C%5Csin%5Cleft%28%5Cfrac%7B8%5Cpi%7D%7B3%7D%5Cright%29%5Cright%5D)
