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One of the values of $\left(\frac{1+i}{\sqrt{2}}\right)^{2 / 3}$ is
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Verified Answer
The correct answer is:
$\frac{1}{2}(\sqrt{3}+i)$
$$
\begin{aligned}
&\left(\frac{1+i}{\sqrt{2}}\right)^{2 / 3}=\left(\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{2}} i\right)^{2 / 3} \\
=&\left(\cos 45^{\circ}+i \sin 45^{\circ}\right)^{2 / 3} \\
=&\left(\cos \frac{2}{3} \times 45^{\circ}+i \sin \frac{2}{3} 45^{\circ}\right) \\
=& \cos 30^{\circ}+i \sin 30^{\circ} \\
=& \frac{\sqrt{3}}{2}+i \times \frac{1}{2}=\frac{1}{2}(\sqrt{3}+i)
\end{aligned}
$$
\begin{aligned}
&\left(\frac{1+i}{\sqrt{2}}\right)^{2 / 3}=\left(\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{2}} i\right)^{2 / 3} \\
=&\left(\cos 45^{\circ}+i \sin 45^{\circ}\right)^{2 / 3} \\
=&\left(\cos \frac{2}{3} \times 45^{\circ}+i \sin \frac{2}{3} 45^{\circ}\right) \\
=& \cos 30^{\circ}+i \sin 30^{\circ} \\
=& \frac{\sqrt{3}}{2}+i \times \frac{1}{2}=\frac{1}{2}(\sqrt{3}+i)
\end{aligned}
$$
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