Question

If $i=\sqrt{-1}$,then $-1-9{(-\frac{1}{2}+\frac{i\sqrt{3}}{2})}^{55}+7{(\frac{-1}{2}+\frac{i\sqrt{3}}{2})}^{86}$ is equal to

• A. $9i\sqrt{3}$
• B. $-8i\sqrt{3}$
• C. $8i\sqrt{3}$
• D. $7i\sqrt{3}$

Answer 1

Answer: (B)

$-8i\sqrt{3}$

$-1-9{(-\frac{1}{2}+\frac{i\sqrt{3}}{2})}^{55}+7{(-\frac{1}{2}+\frac{i\sqrt{3}}{2})}^{86}=-1-9{\omega }^{55}+7{\omega }^{86}=-1-9\omega +7{\omega }^2$ $=7({\omega }^2+\omega +1)-8-16\omega =7\left(0\right)-8[1+2\omega ]$ $=-8[1+2\left(\frac{-1+i\sqrt{3}}{2}\right)]$ $=-8\left[i\sqrt{3}\right]=-8i\sqrt{3}$