Definition:Root (Analysis)

Let $$x \in \mathbb{R}$$ be a real number such that $$x > 0$$.

Let $$m \in \mathbb{Z}$$ be an integer such that $$m \ne 0$$.

Then there always exists a unique $y \in \mathbb{R}: y > 0$ such that $$y^m = x$$.

This $$y$$ is called the $$m$$th root of $$x$$, and is denoted $$y = \sqrt [m] x$$.

When $$m = 2$$, we write $$y = \sqrt x$$ and call $$y$$ the square root of $$x$$.

When $$m = 2$$, we write $$y = \sqrt [3] x$$ and call $$y$$ the cube root of $$x$$.

The $$m$$th root of $$x$$ can also be written, using the power notation, as $$x^{1/m}$$.

Note the special case where $$x = 0$$: $$\sqrt [m] 0 = 0$$.

Root of a Polynomial
Let $$R$$ be a commutative ring with unity whose zero is $$0_R$$ and whose unity is $$1_R$$.

Let $$f \left({x}\right)$$ be a polynomial in $$x$$, where $$x \in R$$.

Then the values of $$x$$ for which $$f \left({x}\right) = 0$$ are known as the roots of the polynomial $$f$$.

Root of a Function
Let $$f: K \to K$$ be a function on a field $$K$$.

Let $$x \in K$$.

Then the values of $$x$$ for which $$f \left({x}\right) = 0$$ are known as the roots of the function $$f$$.

This is simply a generalization of the case where $$f$$ is a polynomial.

The field $$K$$ is usually the set of real numbers $$\mathbb{R}$$ or complex numbers $$\mathbb{C}$$.