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I would like to plot the function $x(x^2-1)^\frac{1}{3}$. When I do it in a statistical computing package, I dont get anything between $-1$ and $1$. However, when I plot this on google, I get a complete plot and when plotting in wolfram alpha, an even different plot. I sense there is something going on with complex valued roots, but am not really sure what is going on. Can anyone tell me? Thanks!

user136503
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  • Hi thanks, do you know why the google graph and one that appears in my graphing calculator looks like this: https://www.google.com/#q=x(x%5E2-1)%5E(1/3) ? Which is different it seems than the wolframalpha graph. – user136503 Feb 17 '16 at 05:41
  • Hi, I see that, however if you notice google's graph, it has a positive graph from -1 to 0 while in wolfram it is below the y-axis. Are they plotting two different things? – user136503 Feb 17 '16 at 05:57
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    They are plotting different things. In the Wolfram plot you're looking at just the real part of a complex-valued number. Put those parts together and you'll realize that the combined value is really entirely off the real Cartesian plane. – Daniel R. Collins Feb 17 '16 at 06:00

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Note that when $x$ is between $-1$ and $1$, that is, $|x| < 1$, then $x^2 < 1$, and so the expression under the rational exponent, $x^2 - 1$, is negative.

The problem is that there are several contradictory definitions for the principal root of negative numbers. A real-valued definition produces a negative real number from this exponent $\frac{1}{3}$; for example, in this context, $(-1)^\frac{1}{3} = -1$. But a complex-valued definition instead produces a complex number; in this context, $(-1)^\frac{1}{3} = \frac{1}{2} + \frac{i \sqrt{3}}{2}$. And so in this latter case, points in this range cannot be graphed on the real Cartesian coordinate system.

Google and your graphing calculator are using the real-valued definition. Wolfram Alpha and your statistical package use the complex-valued definition by default. This clash of definitions may be responsible for a surprisingly large proportion of the questions here on Math SE. If you're asked to produce such a graph, be sure to know exactly what context and what definitions you're working under.

See also: What are the Laws of Rational Exponents?

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