MathNet 2023-part-03 — 191 problems (EN–VI) Question 153

Consider the complex number $z=\frac{1}{2}-\frac{\sqrt{3}}{4}+\frac{1}{4} i$. Show that the value of the ratio $\sqrt{3} \cdot \frac{\operatorname{Re}\left(z^{2024}\right)}{\operatorname{Im}\left(z^{2024}\right)}$ is a rational number.