Wolfram Alpha gives wrongs answer for an equation

Question

I have experienced an issue while evaluating $$(x)^\frac{1}{5}-x = 0$$ The values that the program gives are 0 and 1, however, there is an additional solution, which is -1. In addition, the corrispondent graph is wrong because it excludes values of x for negative numbers. I have also noticed that when I try to evaluate: $$(x)^\frac{1}{5}=-1$$ Wolfram Alpha gives no solutions, instead the solution should be -1 (over the reals). I believe that there is a general problem in the program that comes up when trying to evaluate $$(x)^\frac{1}{n} , x<0, n = 2n+1, n = {1,2,3..}$$

Answer 1

Mind the text right under the equation:

Assuming the principal root  |  Use the real‐valued root instead

Click the real-valued root and you get the $3$ real roots $\,\{-1,0,1\}\,$, and the graph over the entire $\mathbb{R}$.

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[ EDIT ]  To clarify what happens under the default Wolfram Alpha interpretation of "assuming the principal root", the equation is taken to be in complex numbers, and $x^{\frac{1}{5}}$ is assumed to be the principal value of the $5^{th}$-root complex power function. With the usual choices for the branch cut along the negative real axis and the principal root being the one with the minimum argument, the principal value of $(-1)^{\frac{1}{5}}$ is $w = \text{cis}\;\frac{\pi}{5}$. This value does not satisfy the equality $w - w^5 = 0$ and therefore WA does not return it as a root of $x^{\frac{1}{5}} - x=0$ under the "principal root" interpretation.

For related discussion and insights see for example What is the principal cubic root of −8?.