# What is the definition of a radical number in math?

Sep 19, 2015

A normal radical is a root of a polynomial of the form ${x}^{n} - a = 0$

If $n = 2$ then we call $x$ a square root of $a$

If $n = 3$ then we call $x$ a cube root of $a$

#### Explanation:

Normal radicals are otherwise known as $n$th roots.

If $a \ge 0$ then ${x}^{n} - a = 0$ will have a positive Real root known as the principal $n$th root, written $\sqrt[n]{a}$.

If $n$ is even, then $- \sqrt[n]{a}$ will also be an $n$th root of $a$.

If a polynomial is of degree $\le 4$ then its zeros can be found and expressed using just normal radicals: square roots and cube roots. (Note that fourth roots are just square roots of square roots).

If a polynomial is of degree $5$ - a quintic, then its roots may not be expressible in terms of normal radicals.

To get beyond this limitation, the Bring radical is a root of the polynomial equation ${x}^{5} + x + a = 0$

It is possible to reduce any quintic equation to a form (Bring-Jerrard normal form) that only has terms in ${x}^{5}$, $x$ and a constant term, and hence to express its roots in terms of a Bring radical.