Question

${\left(\frac{\sqrt{3}+i}{2}\right)}^6+{\left(\frac{\sqrt{3}+i}{2}\right)}^6$ is equal to

• A. $0$
• B. $-1$
• C. $-2$
• D. $2$

Answer 1

The correct option is C $-2$

We have,

${\left(\frac{\sqrt{3}+i}{2}\right)}^6+{\left(\frac{\sqrt{3}-i}{2}\right)}^6$
$={\left(\cos \frac{\pi }{6}+i\sin \frac{\pi }{6}\right)}^6+{\left(\cos \frac{\pi }{6}-i\sin \frac{\pi }{6}\right)}^6$

$=\left(\cos \frac{6\pi }{6}+i\sin \frac{6\pi }{6}\right)+\left(\cos \frac{6\pi }{6}-i\sin \frac{6\pi }{6}\right)$
$=\left(\cos \pi +i\sin \pi \right)+\left(\cos \pi -i\sin \pi \right)$
$=2\cos \pi$
$=-2$

$Hence,optionCiscorrectanswer.$