Question

The value of $z = \left( \frac{\sqrt{3}}{2} + \frac{i}{2} \right)^{5} + \left( \frac{\sqrt{3}}{2} - \frac{i}{2} \right)^{5}$ is.

• A. 16
• B. – 16
• C. 32
• D. – 32

Answer 1

Answer: (C)

32

Answer 2

$z_{1}^{2} + z_{2}^{2} + z_{3}^{2} = z_{1}z_{2} + z_{2}z_{3} + z_{3}z_{1}$ and $- \frac{1}{2} + \frac{1}{2}i\sqrt{3}$

$\Rightarrow ( 1 + i ) ^ { 8 } + ( 1 - i ) ^ { 8 } = ( 2 i ) ^ { 4 } + ( - 2 i ) ^ { 4 }$ $\omega^{1000} = \omega^{999}\omega = (\omega^{3})^{333}\omega = \omega = - \frac{1}{2} + \frac{\sqrt{3}}{2}i$

$p < 0$