# Wolfram Alpha wrong answers on $(-8)^{1/3}$ and more? [duplicate]

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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: And here's what Wolfram Alpha draws: ## marked as duplicate by Mark McClure, Kamil Jarosz, user228113, user223391, Eric WofseyMar 11 '16 at 21:13

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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 8:29
• You can get the plot you expect by entering y=Abs(x)^(1/3)*Signum(x) into W/A. – LouisB Mar 11 '16 at 8:50
• 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 Answers

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.

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)

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.