Think the complex plane. Note that $1$, $2$, and $\sqrt{3}$ are the three sides of a right triangle. This implies that $|x|= 1$. Furthermore, this also suggests that $x^4$ and $x^7$ are perpendicular to each other. In another word, $x^3$ represents $90^{\circ}$, or $$x=\cos 30^{\circ} + i \sin 30^{\circ}=\boxed{\frac{\sqrt{3}}{2}+\frac{1}{2}i}$$