## Definitions

### General

The n-th root of a number x, is a number r so that r^{n} = x. In modern notation n-th root is written as \(\sqrt[n]{x}\) or x^{1/n} . Typically root degree n is an integer larger or equal to 2. For n=2 the result is called square root while for n=3 cube root. Any non-zero number has n distinct roots of n degree, either real or complex ones. If n is even, there are two real n-th roots of a real x, only if x is positive. These two roots are opposites and the positive one is called principal root. If x is negative and n is even, all roots are complex numbers. On the other hand, if n is odd, there is always a real n-th root for any real x (positive or negative). The n-th roots of 0 are all 0.

### Properties

The derivative of the principal n-th root function is:

\[ \left(\sqrt[n] x\right)' = \frac{1}{n}x^{\frac{1-n}{n}} = \frac{1}{n\sqrt[n]{x^{n-1}}} \]

The integral of the n-th root function is given by:

\[ \int \sqrt[n] x\, \mathrm{d}x = \frac{n}{n+1} x^{\frac{n+1}{n}} +C \]