Glossary Checklist FAQ

Entire and Mixed Radicals


Radicals can be expressed in two forms:

  • Entire Radical – a radical expression with the entire value sitting under the radical sign. For example, \(\sqrt{226}\), \(\sqrt[{3}]{45}\), or \(\sqrt[{5}]{108}\) are all entire radicals.
  • Mixed Radical – a radical expression, usually written in simplest form, that has a coefficient in front of the radical sign and a value under the radical sign. For example, \(2\sqrt{5}\), \(3\sqrt[{3}]{4}\), or \(8\sqrt[{5}]{2}\) are all mixed radicals.

A simplified radical is one where no perfect \(n^{th}\) root factors (perfect squares, perfect cubes, etc.) can be removed from the radicand and placed as a coefficient of the radical. A radical expression is also in simplest form when there are no radicals in the denominator. Lesson 3.1 focuses on removing perfect \(n^{th}\) root factors from the radicand, and Lesson 3.2 continues with removing radicals from the denominator.

There are two methods used to change an entire radical into a mixed radical. (In other words, removing perfect \(n^{th}\) root factors from the radicand) The first uses prime factorization. The second method uses perfect \(n^{th}\) root factors. Both methods will be demonstrated in the following examples.