Question

The Euler form of $\frac{ 2 + 6\sqrt{3}i}{5 + i\sqrt{3}}$ is

• A. $2\cdot e^{i \frac{\pi}{6}}$
• B. $e^{i\pi /3}$
• C. $e^{-2\pi/3}$
• D. $2e^{i \frac{\pi}{3}}$

Answer 1

Answer: (D)

Let $z = \frac{2 + 6\sqrt{3}i}{5 + i\sqrt{3}}$
$ = \frac{(2 + 6\sqrt{3}i)( 5 - i\sqrt{3})}{28}$
$ = 2\left(\frac{ 1 + i\sqrt{3}}{2}\right) = 2\left(cos \frac{\pi}{3} + i\, sin \frac{\pi}{3}\right) = 2e^{i\pi/3}$