Question

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

• A. $-2i\sqrt{3}$
• B. a real number
• C. purely imagimary
• D. a complex number

Answer 1

The correct option is C purely imagimary

The value of ${\left(\frac{1+\sqrt{3}i}{1-i\sqrt{3}}\right)}^{10}-{\left(\frac{1-\sqrt{3}i}{1+i\sqrt{3}}\right)}^{10}$ is
${\left(\frac{2e^{i\frac{\pi }{3}}}{2e^{-i\frac{\pi }{3}}}\right)}^{10}-{\left(\frac{2e^{-i\frac{\pi }{3}}}{2e^{i\frac{\pi }{3}}}\right)}^{10}$
$e^{i\left(\frac{20\pi }{3}\right)}-e^{i\left(\frac{-20\pi }{3}\right)}$
$=2i\sin \left(\frac{20\pi }{3}\right)$
$=2i\sin (6\pi +\frac{2\pi }{3})$
$=2i\sin \left(\frac{2\pi }{3}\right)$
$=\sqrt{3}i$