Question

If $i=\sqrt{-1},$ then $4+5{(-\frac{1}{2}+\frac{i\sqrt{3}}{2})}^{334}+3{(-\frac{1}{2}+\frac{i\sqrt{3}}{2})}^{365}$ 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 $i\sqrt{3}$

We know that $\omega =-\frac{1}{2}+\frac{\sqrt{3}}{2}i$

Thus, $4+5{(-\frac{1}{2}+\frac{i\sqrt{3}}{2})}^{334}+3{(-\frac{1}{2}+\frac{i\sqrt{3}}{2})}^{365}$

$=4+5{\omega }^{334}+3{\omega }^{365}=4+5{\left({\omega }^3\right)}^{111}\omega +3{\left({\omega }^3\right)}^{121}{\omega }^2$

$=4+5{\left(1\right)}^{111}\omega +3{\left(1\right)}^{121}{\omega }^2=4+5\omega +3{\omega }^2$

$=1+2\omega +3(1+\omega +{\omega }^2)=1+2\omega +3\left(0\right)$

$=1+2\omega =1-1+\sqrt{3}i=\sqrt{3}i$