<s>[INST] Where is the mistake in the equality of roots? I have this statement:

If $f(x) =  \frac{\sqrt[3]{x}}{\sqrt{2}}, g(x) =
> \frac{\sqrt[6]{8x^2}}{2}$, with $x <  0$ is $f(x) = g(x)$ ?

My attempt was:
$(1)$ $f(x) =  \frac{\sqrt[3]{x}}{\sqrt{2}} = \frac{\sqrt{2}\sqrt[3]{x}}{2}$
$(2)$ $g(x) = \frac{\sqrt[6]{8x^2}}{2}$ using the property of $\sqrt[n]{k^m} = k^{\frac{m}{n}}$ and since $x^2 \geq 0$, thus:
$(3)$ $g(x) = \frac{\sqrt[6]{2^3}\sqrt[6]{x^2}}{2} = \frac{2^{\frac{3}{6}}\cdot (x^2)^{\frac{1}{6}}}{2} = \frac{\sqrt{2}\sqrt[3]{x}}{2}$, that is equal to $f(x)$ from the step $(1)$
But according to the guide, this is false. So, what is wrong?
 [/INST] with $x<0$, $\sqrt[3]x<0$, but $\sqrt[6]{x^2}>0$
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