Question

The value of ${\left(\frac{1+\sqrt{3}i}{1-\sqrt{3}i}\right)}^{10}$ is equal to:

• A. $1-i\sqrt{3}$
• B. $\frac{-1+i\sqrt{3}}{2}$
• C. $-1-i\sqrt{3}$
• D. $1+i\sqrt{3}$

Answer 1

Answer: (B)

$\frac{-1+i\sqrt{3}}{2}$

$(\frac{1+\sqrt{3}i}{1-\sqrt{3}i}{)}^{10}$
$=(\frac{2e^{i\frac{\pi }{3}}}{2e^{i\frac{-\pi }{3}}}{)}^{10}$
$=(e^{i\frac{2\pi }{3}}{)}^{10}$
$=e^{i\frac{20\pi }{3}}$
$=e^{i(6\pi +\frac{2\pi }{3})}$
$=cos(6\pi +\frac{2\pi }{3})+isin(6\pi +\frac{2\pi }{3})$
$=cos\left(\frac{2\pi }{3}\right)+isin\left(\frac{2\pi }{3}\right)$
$=\frac{-1+i\sqrt{3}}{2}$