# Lesson Explainer: Roots of Monomials Mathematics

In this explainer, we will learn how to simplify roots of monomials involving single and multiple variables.

### Definition: Square and Cubic Roots of a Number

Taking the square root is the inverse operation of squaring a number. This means that taking the square root of a given number is finding the side of the square whose area is this number.

Taking the cubic root is the inverse operation of cubing a number. This means that taking the cubic root of a given number is finding the side of the cube whose volume is this number.

For instance, taking the cubic root of twenty-seven (written ) is finding the length of the side of the cube whose volume is 27 volume units. We find that this is 3 length units, as illustrated in the diagram below.

We are going now to see important properties of square and cubic roots: first, the product and quotient properties of square and cubic roots, which we illustrate in the case of cubic roots.

### Product Property of Square and Cubic Roots

For any numbers and , the square root of the product is equal to the product of the square root of and the square root of :

For any numbers and , the cubic root of the product is equal to the product of the cubic root of and the cubic root of :

Consider . We are looking for the side of the cube whose volume is . We start with a cube of volume 27.

Then, we take 64 of them.