[tex]\dfrac{6\sqrt 3~ \text{cis} \left( \dfrac{7 \pi} 6 \right)}{3 \sqrt 5 ~\text{cis} \left( \dfrac{\pi}{3}\right)}\\\\\\=\dfrac{2\sqrt 3 \cdot e^{i\tfrac{7\pi}{6}}}{\sqrt 5\cdot e^{i \tfrac{\pi}{3}}}\\\\\\=\dfrac{2 \sqrt 3}{\sqrt 5} \cdot \left(e^i \right)^{\tfrac{7\pi}{6} - \tfrac{\pi }{3}}~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~;\left[\text{cis}~ \theta = e^{i \theta}\right]\\\\\\=\dfrac{2 \sqrt{3}}{\sqrt 5} \cdot \left(e^i\right)^{\tfrac{5\pi}6}\\\\\\=\dfrac{2\sqrt{3}}{\sqrt 5}\cdot e^{i\tfrac{5 \pi}6}\\\\\\[/tex]
[tex]=\dfrac{2\sqrt{3}}{\sqrt 5} \left[ \cos \left(\dfrac{5\pi}{6}\right) + i \sin \left(\dfrac{5\pi}{6}\right) \right]~~~~~~~~~~~~~~~~~~;\left[e^{i\theta} = \cos \theta + i \sin \theta} \right]\\\\\\=\dfrac{2\sqrt{3}}{\sqrt 5}\left[ \cos \left(2 \times \dfrac{\pi}2 -\dfrac{\pi}{6} \right) + i\sin \left(2 \times \dfrac{\pi}2 -\dfrac{\pi}{6} \right) \right]\\\\\\=\dfrac{2\sqrt{3}}{\sqrt 5} \left[ -\cos \left(\dfrac{\pi}{6}\right) + i \sin \left(\dfrac{\pi}{6}\right) \right]\\\\\\[/tex]
[tex]=\dfrac{2\sqrt{3}}{\sqrt 5}\left( -\dfrac {\sqrt{3}}{2} + i \cdot \dfrac 12 \right)\\\\\\=-\dfrac{3}{\sqrt 5} + i \dfrac{\sqrt 3}{\sqrt 5}[/tex]
