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

Question

Compare these two functions: plot $\sqrt[3]{x}$ 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 $\sqrt[3]{x}$.

Is there any reason why the different approach? In the "input interpretation" it displays both as $\sqrt[3]{x}$ and aren't they in fact supposed to mean the same? Isn't $\sqrt[x]{y}$ defined as $y^{1/x}$ ?

Answer 1

It is taking $\sqrt[3]x$ as the inverse of $x^3$, while $x^{\frac13}$ is define through exponential (aproximating the values with Taylor maybe) for the graph

Answer 2

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.