Question

State true or false
${\left(\frac{\sqrt{3}+i}{2}\right)}^6+{\left(\frac{\sqrt{3}-i}{2}\right)}^6=-2$

• A. True
• B. False

Answer 1

The correct option is A True

${\left(\frac{\sqrt{3}+i}{2}\right)}^6=(e^{i\pi /6}{)}^6=e^{i\pi }$ and

${\left(\frac{\sqrt{3}-i}{2}\right)}^6=(e^{i5\pi /6}{)}^6=e^{5\pi }$

Hence,

${\left(\frac{\sqrt{3}+i}{2}\right)}^6+{\left(\frac{\sqrt{3}-i}{2}\right)}^6=e^{i\pi }+e^{i5\pi }=cos\pi +isin\pi +cos5\pi +isin5\pi =-1+0-1+0=-2$