$z$ is a complex number and $z^2+z+1=0$. 

$$z^{10}+\frac{1}{z^{10}}=?$$

**For the solution:**

1. the roots of $z^2+z+1$ are: $z\_1=-\frac12+\frac{\sqrt3}{2}i$ and $z\_2=-\frac12-\frac{\sqrt3}{2}i$
2. converting these to their trigonometrical forms, we get: $z\_1=\cos\frac{2\pi}{3}+i\sin\frac{2\pi}{3}$ and $z\_2=\cos\frac{7\pi}{6}+i\sin\frac{7\pi}{6}$
3. How do I proceed?