The numerical value of  $\tan \frac{\mathrm{\pi }}{3}+2\tan \frac{2\mathrm{\pi }}{3}+4\tan \frac{4\mathrm{\pi }}{3}+8\tan \frac{8\mathrm{\pi }}{3}$ is equal to

$\begin{array}{c}\tan \frac{\mathrm{\pi }}{3}+2\tan \frac{2\mathrm{\pi }}{3}+4\tan \frac{4\mathrm{\pi }}{3}+8\tan \frac{8\mathrm{\pi }}{3}\\ =\tan \frac{\mathrm{\pi }}{3}+2\tan \left(\mathrm{\pi }-\frac{\mathrm{\pi }}{3}\right)+4\tan \left(\mathrm{\pi }+\frac{\mathrm{\pi }}{3}\right)+8\tan \left(3\mathrm{\pi }-\frac{\mathrm{\pi }}{3}\right)\\ =\tan \frac{\mathrm{\pi }}{3}-2\tan \frac{\mathrm{\pi }}{3}+4\tan \frac{\mathrm{\pi }}{3}-8\tan \frac{\mathrm{\pi }}{3}\\ =-5\tan \frac{\mathrm{\pi }}{3}=-5\sqrt{3}\end{array}$