Question

What is the definition of a radical number in math?

Answer 1

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 >= 0` then `x^n - a = 0` will have a positive Real root known as the principal `n`th root, written `root(n)(a)`.

If `n` is even, then `-root(n)(a)` will also be an `n`th root of `a`.

If a polynomial is of degree `<= 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.