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Evaluating N[(-1)^(1/3)] returns 0.5 + 0.866i in my v12.0 notebook. Of course, the equation x^3 = -1 has three distinct solutions for x. Is there a global (or local) option that will force Mathematica to output the real solution in favor of the complex ones? In this case, simply -1?

Further to this, if my notebook defaults to evaluating (-1)^(1/3) to a complex number, can I assume that Mathematica is doing the same when plotting complicated expressions with cube roots? Namely that some areas of the plot may appear imaginary when in fact there is a real solution available?

Thanks

subzero366
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  • Have you seen Surd[]? – J. M.'s missing motivation Feb 27 '21 at 20:21
  • Hi J. M.

    Surd[(-1),3] would give the desired result here, but for a complicated expression where some terms contain a cube root, it would be very cumbersome to have to manually call Surd[ ] for the terms which happen to be cube roots.

     – subzero366 Feb 27 '21 at 20:50
  • Can you give an example where you think making this replacement is difficult? That said, the principal branch of $\sqrt[3]{z}$ is generally agreed on, but you can always premultiply such radicals with Exp[2 I π/3] if need be. – J. M.'s missing motivation Feb 27 '21 at 21:00
  • Perhaps more detail is required. I am examining a function that has an additive term like:

    $f(x)=(- x^3 + 5 x + (-2 + \sqrt{-27 x^2 + 10 x^3 - 48 x + 115}))^{1/3}$

    The question is, for what values of x is this function real? If one simply plots the function, it will appear to be real on the interval (0,1.9). The term under the square root however, is real on (0,1.9) AND when $x>3$. Suppose $x = 4$, then the function reduces to:

    $f(4)=(\sqrt{131}-46)^{1/3}$

     – subzero366 Feb 28 '21 at 17:55
  • This clearly CAN be real and evaluate to $f(4)\sim 3.2$, if one chooses the correct branch, but N[(-46 + Sqrt[131])^(1/3)] does not give the real value, nor does the plot, which assumes the function is imaginary when $x>3$.

    As I am examining many plots visually that contain this term, it is important to get Mathematica to display the real branch (when $x>3$) in my plots.

     – subzero366 Feb 28 '21 at 17:55
  • Perhaps you can refocus on asking for a way to convert a complicated expression containing Power[w, 1/3] into Surd[w, 3], if so. – J. M.'s missing motivation Feb 28 '21 at 17:59

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