Let $z={\left(\frac{\sqrt{3}}{2}+\frac{i}{2}\right)}^5+{\left(\frac{\sqrt{3}}{2}-\frac{i}{2}\right)}^5$ .If $R\left(z\right)$ and $I\left(z\right)$ respectively denote the real and imaginary parts of z, then

We have $z={\left(\frac{\sqrt{3}}{2}+\frac{i}{2}\right)}^5+{\left(\frac{\sqrt{3}}{2}-\frac{i}{2}\right)}^3$

$\begin{array}{l}={\left(\cos \frac{\pi }{6}+i\sin \frac{\pi }{6}\right)}^5+{\left(\cos \frac{\pi }{6}-i\sin \frac{\pi }{6}\right)}^5\\ =\left(\cos \frac{5\pi }{6}+i\sin \frac{5\pi }{6}\right)+\left(\cos \frac{5\pi }{6}-i\sin \frac{5\pi }{6}\right)=2\cos \frac{5\pi }{6}<0\end{array}$

Here, $I\left(z\right)>0$ and $R\left(z\right)>0$.