Good morning; I'm having doubts about a perhaps simple question involving the cubic root of a function. Say I do have the following $$f(x) = \sqrt[3]{\dfrac{x^3}{2-x^2}}$$
Its domain is clearly $x\neq\pm\sqrt{2}$
Yet I'm having perplexities about the way to write a cubic root. For example, using W. Mathematica to get a plot for the function, I immediately say that there is a huge difference in writing the function with the $\texttt{CubeRoot}$ function, which would define the function I wrote above, and the following way: $$\left(\dfrac{x^3}{2-x^2}\right)^{1/3}$$
In this case the domain would be $x\in(-\infty, -\sqrt{2}) \cup [0, +\sqrt{2})$, rather different.
So my question is: why and how are there difference between
$$\sqrt[3]{x} ~~~~~~~ \text{and} ~~~~~~~ x^{1/3}$$
There is no difference between $x^{3}$ and $\sqrt[3]{x}$. By definition, $x^{1/n} = \sqrt[n]{x}$. I plotted the two functions on Desmos's graphing calculator (one using a cube root and one using $\frac{1}{3}$ as the exponent) and they showed to be exactly the same. I am not sure why it doesn't work on W. Mathematica.