## Roots in Mathematica

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Plots of functions containing rational exponents such as $Latex formula$ is the source of much commotion. In this short report, I will highlight how Mathematica defines these roots and why it makes sense mathematically. Next, I will show how to resolve the issue using the `Surd`

### Roots

We know that every number (real or complex) has $Latex formula$ distinct roots. Therefore, $Latex formula$ has three roots, only one of which is real. As a matter of fact, more generally, if $Latex formula$, then $Latex formula$ has $Latex formula$ $Latex formula$ roots, only one of which is real. Let’s consider the simplest case. Every positive real number has 2 square roots. It turns out both of these roots are real, one is positive and one is negative. For example $Latex formula$ has two square roots, $Latex formula$ and $Latex formula$. We identify $Latex formula$ as the principal square root. Why? In algebra texts, the the principal square root is defined to be the positive root. However, this is very misleading and is the root of the problem we are addressing in this report. The principal square root is defined to be the complex root with the smallest argument. $Latex formula$ therefore, the arguments of its roots are $Latex formula$ and $Latex formula$. We identify $Latex formula$ as the principal square root because it has the smallest argument; not because it is positive!

Generalizing this argument, let $Latex formula$, then $Latex formula$ and therefore the arguments of its roots are $Latex formula$ for $Latex formula$. The smallest of these is of course the root corresponding to $Latex formula$ in which case we have $Latex formula$ which means that $Latex formula$ is real (and positive).
Now suppose that $Latex formula$, then $Latex formula$ and therefore, the arguments of its roots are $Latex formula$ for $Latex formula$. As before, the root with the smallest argument corresponds to $Latex formula$, namely $Latex formula$. Finally, since $Latex formula$ for $Latex formula$ we know that $Latex formula$. In other words, the principal $Latex formula$ root of every negative real number is complex, even when $Latex formula$ is odd!

Going back to Mathematica, $Latex formula$ is defined as the principal cube root of $Latex formula$. Per our discussion above, if $Latex formula$ then $Latex formula$ and therefore wouldn’t (and shouldn’t) appear in the real plot.

### Surds

Notwithstanding the mathematics of roots, Mathematica has a built-in function called `Surd`. This function gives the real roots of real numbers (if they exist) even if these real roots are not the principal roots. For example `Surd[-1,3]` gives the real cube root of $Latex formula$. While`(-1)^(1/3)` is left unevaluated (see figure 1).

Note that $Latex formula$ and therefore the principal cube root is $Latex formula$. We can get a numerical result by entering `(-1.)^(1/3)` as in figure 2 to confirm our analysis.