<s>[INST] Why If $x^2=-\frac{1}{3}$ then $x=\pm\frac{i\sqrt{3}}{3}$ If $x^2=-\frac{1}{3}$ then $x=\pm\frac{i\sqrt{3}}{3}$ according to my text book.
I understand that $x=\pm\sqrt{-\frac{1}{3}}$ but do not fully grasp how to express this as $\pm\frac{i\sqrt{3}}{3}$
I understand that $i$ is $\sqrt{-1}$
As far as I can get with my understanding starting from:
$$x=\pm\sqrt{-\frac{1}{3}}$$
$$x=\pm\frac{\sqrt{-1}}{\sqrt{-3}}$$
$$x=\pm\frac{i}{\sqrt{3}i}$$
$$x=\pm\sqrt{3}$$
Where did I go wrong and how can I arrive at $x=\pm\frac{i\sqrt{3}}{3}$
[Edit]
From the comments I now know that $-\frac{1}{3}$ = $\frac{-1}{3}$
Taking that back into my working I still arrive at a different answer:
$$x=\pm\sqrt{-\frac{1}{3}}$$
$$x=\pm\sqrt{\frac{-1}{3}}$$
$$x=\pm\frac{\sqrt{-1}}{3}$$
$$x=\pm\frac{i}{3}$$
How can I get to $x=\pm\frac{i\sqrt{3}}{3}$
 [/INST] After your edit; compare your simplification with
\begin{align}
x&=\pm\sqrt{-\frac{1}{3}}\\
x&=\pm\frac{\sqrt{-1}}{\sqrt{3}}\\
x&=\pm\frac{i}{\sqrt{3}}\\
x&=\pm i \frac{\sqrt{3}}{{3}}
\end{align}
The last step is achieved since
\begin{align}
\sqrt{3} \sqrt{3} = 3
\end{align}
Thus
$$\frac{1}{\sqrt{3}} = \frac{\sqrt{3}}{3}$$
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