algebra

Lemma: If $x + \dfrac1x = 2\cos \alpha$, then $x^n + \dfrac1{x^n} = 2\cos n\alpha$

If you want to see the proof of this, go here. I have done the proof in a previous answer before.

Now, if we consider $\alpha = \dfrac{2\pi}3$ and $n = 99$, using the formula directly gives

$x^{99} + \dfrac1{x^{99}} = 2\cos\left(\dfrac{2\pi}3\cdot 99\right) = \boxed{2}$