Imaginary part of ${\left(\sqrt{3}-\mathrm{i}\right)}^{2016}+{\left(-\sqrt{3}-\mathrm{i}\right)}^{2019}$is

${\left(\sqrt{3}-\mathrm{i}\right)}^{2016}+{\left(-\sqrt{3}-\mathrm{i}\right)}^{2019}$

$\begin{array}{rcl} & & 2^{2016}{\left(\frac{\sqrt{3}}{2}-\frac{\mathrm{i}}{2}\right)}^{2016}-2^{2019} {\left(\frac{\sqrt{3}}{2}+\frac{\mathrm{i}}{2}\right)}^{2019}\end{array}$

$\begin{array}{rcl} & & 2^{2016}{\left(\cos \frac{\mathrm{\pi }}{6}-\mathrm{i} \sin \frac{\mathrm{\pi }}{6}\right)}^{2016}-2^{2019}{\left(\cos \frac{\mathrm{\pi }}{6}+\mathrm{i} \sin \frac{\mathrm{\pi }}{6}\right)}^{2019}\end{array}$

$\begin{array}{rcl} & & 2^{2016}\left(\cos 1008\frac{\mathrm{\pi }}{3}-\mathrm{i} \sin 1008\frac{\mathrm{\pi }}{3}\right)-2^{2018}\left(\cos 2019\frac{\mathrm{\pi }}{6}+\mathrm{i} \sin 2019\frac{\mathrm{\pi }}{6}\right)\end{array}$

$\begin{array}{rcl} & & 2^{2016}\left(\cos 336 \mathrm{\pi }-\mathrm{i} \sin 336 \mathrm{\pi }\right)-2^{2019}\left(\cos \left(336\mathrm{\pi }+\frac{3\mathrm{\pi }}{6}\right)+\mathrm{i} \sin \left(336\mathrm{\pi }+\frac{3\mathrm{\pi }}{6}\right)\right)\end{array}$

$\begin{array}{rcl} & =& 2^{2016} \left(1-\mathrm{i}\left(0\right)\right)-2^{2019} \left(\cos \frac{\mathrm{\pi }}{2}+\mathrm{i} \sin \frac{\mathrm{\pi }}{2}\right)\\ & =& 2^{2016}-2^{2019} \mathrm{i}\end{array}$

Imaginary part $=-2^{2019}$