Question

If $i=\sqrt{-1}$,then $4+5{(-\frac{1}{2}+\frac{i\sqrt{3}}{2})}^{331}+3{(-\frac{1}{2}-\frac{i\sqrt{3}}{2})}^{367}$ is equal to

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

Answer 1

The correct option is C $i\sqrt{3}$

$4+5{(-\frac{1}{2}+\frac{i\sqrt{3}}{2})}^{331}+3{(-\frac{1}{2}-\frac{i\sqrt{3}}{2})}^{367}$ $=4+5{\omega }^{331}+3({\omega }^2{)}^{367}=4+5{\omega }^{331}+3{\omega }^{734}$
$4+5\omega +3{\omega }^2=3(1+\omega +{\omega }^2)+(1+2\omega )=0+1+2\frac{(-1+i\sqrt{3})}{2}=i\sqrt{3}$