Explainer: Roots of Monomials

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 27) 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 𝑎 and 𝑏, the square root of the product 𝑎𝑏 is equal to the product of the square root of 𝑎 and the square root of 𝑏: 𝑎×𝑏=𝑎×𝑏.

For any numbers 𝑎 and 𝑏, the cubic root of the product 𝑎𝑏 is equal to the product of the cubic root of 𝑎 and the cubic root of 𝑏: 𝑎×𝑏=𝑎×𝑏.

Consider 64×27. We are looking for the side of the cube whose volume is 64×27. We start with a cube of volume 27.

Then, we take 64 of them.