Question

The value of $4+5{(-\frac{1}{2}+i\frac{\sqrt{3}}{2})}^{334}+3{(-\frac{1}{2}-i\frac{\sqrt{3}}{2})}^{335}$ is

• A. $1-i\sqrt{3}$
• B. $-1+i\sqrt{3}$
• C. $4\sqrt{3}i$
• D. $-i\sqrt{3}$

Answer 1

Answer: (C)

$4\sqrt{3}i$

$4+5{(-\frac{1}{2}+i\frac{\sqrt{3}}{2})}^{334}+3{(-\frac{1}{2}-i\frac{\sqrt{3}}{2})}^{335}$
$=4+5{\omega }^{334}+3{\omega }^{670}[{\omega }^2=-\frac{1}{2}-\frac{i\sqrt{3}}{2}]$
$=4+5\omega +3\omega =4+9\omega$
$=4+8(-\frac{1}{2}+\frac{i\sqrt{3}}{2})=4\sqrt{3}i$
Hence choice (c) is correct.