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Wolfram Alpha doesn't give $-2$ for $(-8)^{1/3}$, and it absolutely fails to draw $f(x)=x^{1/3}$ - does anyone know why? Am I missing something very 'deep' Wolfram Alpha is trying to teach me?

Here's what the graph should look like: enter image description here

And here's what Wolfram Alpha draws: enter image description here

Eric Wofsey
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Mark
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    Yes. You are missing the fact that cubic root multi-valued (3 values) and W|A gives you the principle value while the other program uses a simpler convention and spits out a non-complex value. Try plotting $x^{1/2}$ instead in both to understand the difference deeper. – A.S. Mar 11 '16 at 08:29
  • You can get the plot you expect by entering y=Abs(x)^(1/3)*Signum(x) into W/A. – LouisB Mar 11 '16 at 08:50
  • You can enter cbrt(x) for the cube root function in Wolfram|Alpha, which (as implemented in that particular piece of software) is different from $x^{1/3}$. – Mark McClure Mar 11 '16 at 12:56

3 Answers3

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What you are seeing is that every non-zero number has three distinct "cuberoots". In the first picture you've posted, the software very nicely graphed the real cuberoots. The problem is that Wolfram|Alpha is drawing a different branch of the cuberoot than you are used to, hence the labels of real and imaginary parts on the graph.

Arkady
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The principal value of the cube root is given by Mathematica or Wolfram Alpha. If you really just want the real value, input with Sign[x] Abs[x]^(1/3)

Ng Chung Tak
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What you are probably thinking about is that for solving the equation $$z^k = a$$

we often have many solutions (for $z$) in the complex plane. More to the point, for a function one wants to assign one out-value for each in-value. Choosing a particular way to assign function values is called picking a branch for a function. For some functions there exist widely used conventions how to pick these branches and they get called principal branch. Integer roots and logarithms are famous examples of functions which have a principal branch.

mathreadler
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