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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 $y^{1/x}$ ?

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

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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.

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