<s>[INST] Find $z^{10}+\frac{1}{z^{10}}$ given $z^2+z+1=0$ $z$ is a complex number and $z^2+z+1=0$. 
$$z^{10}+\frac{1}{z^{10}}=?$$
For the solution:


*

*the roots of $z^2+z+1$ are: $z_1=-\frac12+\frac{\sqrt3}{2}i$ and $z_2=-\frac12-\frac{\sqrt3}{2}i$

*converting these to their trigonometrical forms, we get: $z_1=\cos\frac{2\pi}{3}+i\sin\frac{2\pi}{3}$ and $z_2=\cos\frac{7\pi}{6}+i\sin\frac{7\pi}{6}$

*How do I proceed?

 [/INST] The solutions to $z^2 + z + 1 = 0$ are the complex $z = 1 \angle (\pm 1/3)$.
So 
$$\begin{align}
z^{10} + z^{-10} & = \left(1\angle (\pm 1/3)\right)^{10} + \left(1\angle (\pm 1/3)\right)^{-10} 
\\& = \left(1\angle (\pm 10/3)\right) + \left(1\angle (\mp 10/3)\right) \
\\ & = (1\angle 10/3) + (1\angle -10/3)
\\ & = (1\angle 1/3) + (1\angle -1/3)
\\ & = -1
\end{align}$$
</s>