Mathematics | VITEEE 2018 | Discussion Page

One of the values of $\left(\frac{1+i}{\sqrt{2}}\right)^{\frac{2}{3}}$ is

A) $\left(\frac{1}{2}\times\sqrt{3}+i\right)$

B) -i

C) i

D) $-\sqrt{3}+i$

Option A

$\left(\frac{1+i}{\sqrt{2}}\right)^{2/3}$ = $\left(\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{2}i}\right)^{2/3}$

= (cos 45^º + i sin45^º)^2/3 = (cos 2/3 × 45° + i sin 2/3 × 45°)

= (cos 30° + i sin 30°)

= $\frac{\sqrt{3}}{2} + i \times\frac{1}{2}$ = 1/2 × (√3 +i)

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