Fun With Roots

I was recently investigating the graph of the cubic root function.  Someone I know that is taking a business calculus class believed their professor have wrongly plotted the graph of the cubic root function.  Their calculator did not define the cubic root function for negative numbers, and this professor had drawn the function as defined for those numbers.  As it turns out the answer is interesting, and involves the very important Fundamental Theorem of Algebra by Gauss.
A consequence of the fundamental theorem is that all nth roots have n answers.  So there are 2 possible answers for the square, or 2nd, root; \(\sqrt{4}\) namely 2, and -2, since \(2^2 = 4\) and \((-2)^2 = 4\).  In the case of the cubic, or 3rd, root there would be three possible answers.
\begin{array}{lcl}
\sqrt[3]{-8} & = & -2 \\
 & = & i\sqrt{3} + 1 \\
 & = & i\sqrt{3} - 1
\end{array}
What is striking is that
\begin{array}{lcl}
|-2| & = & 2 \\
|i\sqrt{3} + 1| & = \\
|i\sqrt{3} - 1| & =
\end{array}
And so the roots form a circle of radius 2 around the origin of the complex plane.  The same is true of the nth roots of any given x; they will all have the same absolute value.  This demonstrates a very interesting symmetry among the nth natural number roots of any given x on the complex plane.  There will be n of them, and they will be the same distance from the origin.

It will be noted that one of the roots lies on the real axis above.  The same will be true for any odd numbered root of a real--there will always be at least one real root.  So one of the cube roots of x is real for any real x, and the function is defined for that x.  But it is also true that there are complex roots too.  As it turns out, both he and the professor were right.  Sadly the calculator didn't specify there is more to roots than meets the eye.