Is there supposed to be a difference between $x^{1/3}$ and $\sqrt{x}$ ? (Wolfram Alpha shows different results)

Compare these two functions: plot $$\sqrt{x}$$ and plot $$x^{1/3}$$

I understand how roots are ambiguous, and Wolfram Alpha apparently takes the principle root with the $$x^{1/3}$$ case and the real root with $$\sqrt{x}$$.

Is there any reason why the different approach? In the "input interpretation" it displays both as $$\sqrt{x}$$ and aren't they in fact supposed to mean the same? Isn't $$\sqrt[x]{y}$$ defined as $$y^{1/x}$$ ?

• I feel it as an example of the necessity of a standard of notation in maths, much like as IUPAC in Chemistry and Physics. – ajotatxe Jan 4 at 12:42

It is taking $$\sqrtx$$ as the inverse of $$x^3$$, while $$x^{\frac13}$$ is define through exponential (aproximating the values with Taylor maybe) for the graph
As far as Wolfram Alpha is concerned, $$x^{\frac13}$$ is the principal cube root of $$x$$. Since that's not a real number when $$x<0$$, you can't see it in the graph.