Question about the answer to a root of a complex number. I am supposed to solve for the roots of the complex number $(2(\cos(\pi/3)+i\sin(\pi/3))^{\frac{1}{2}}$.
The answer I get is that the roots are $\sqrt2e^{i\frac{\pi}{6}}$ and $\sqrt2e^{i\frac{7\pi}{6}}$. However, the answer key says that the roots are $\frac{\sqrt3+i}{\sqrt2}$and $-\frac{\sqrt3+i}{\sqrt2}$. Why is this?
Doesn't $(2(\cos(\pi/3)+i\sin(\pi/3))^{\frac{1}{2}}=1+\sqrt(3)i$, where did the answer key get (sqrt(3)+i) from?
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You've lost a $2$ in divisor. Roots are
$\sqrt{2}e^{i\pi/6}$ and $\sqrt{2}e^{i7\pi/6}=\sqrt{2}e^{-i5\pi/6}$
Which are
$$\pm\frac{\sqrt{3}+i}{\sqrt{2}}$$
