Answer:
[tex]z^{4} = \cos \frac{8\pi}{3}+i\,\sin \frac{8\pi}{3}[/tex]
Step-by-step explanation:
We can determine the power of a complex number by the De Moivre's Theorem, which states that for all [tex]z = r\cdot (\cos \theta + i\,\sin\theta)[/tex], where [tex]a, b \in \mathbb{R}[/tex], the power of the complex number is:
[tex]z^{n} = r^{n}\cdot (\cos n\theta + i\,\sin n\theta)[/tex] (1)
Where:
[tex]r[/tex] - Magnitude of the complex number, dimensionless.
[tex]\theta[/tex] - Direction of the complex number.
If we know that [tex]r = 1[/tex], [tex]n = 4[/tex] and [tex]\theta = \frac{2\pi}{3}[/tex], then the fourth power of the complex number is:
[tex]z^{4} = 1^{4}\cdot \left[\cos\left(\frac{8\pi}{3} \right)+i\,\sin\left(\frac{8\pi}{3}\right)\right][/tex]
[tex]z^{4} = \cos \frac{8\pi}{3}+i\,\sin \frac{8\pi}{3}[/tex]
