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.
We find that the length of the side of this cube of volume is four times the side of the cube of volume 27, that is, three. So the length of the side is
The four comes from the fact that we have 64 cubes of volume 27, and so we have four of them in each dimension. And 3 is the side of the cube of volume 27, because there are then 3 unit cubes in each dimension. So we have
Quotient Property of Square and Cubic Roots
For any numbers and , the square root of the quotient is equal to the quotient of the square root of and the square root of :
For any numbers and , the cubic root of the quotient is equal to the quotient of the cubic root of and the cubic root of :
To illustrate the quotient property, let’s take . This number is the length of the side of the cube of volume . Let’s start with a cube of volume 125.
We need to divide this cube in 27 to have a cube of volume . This is done by dividing each side into three parts.
The length of the blue cube is then one-third of the length of the largest cube of side 5, that is, . So, .
Now, when we are dealing with monomials, and so with variables and not numbers, the same rules apply of course. Let’s see what happens to a variable when it is raised to a power. For instance, if we take the variable and raise it to the third power, we write this as , and it just means . If we now raise this to another power, let’s say the fifth power, we have .
We see that it is .
This can be written in a general form as
Inversely, this means that when we are taking the square or the cubic root of a variable raised to a given power, we are going to divide its exponent by two or three:
Finally, when we are dealing with negative numbers as well, we need to remember that a squared number is always positive, whatever its sign. This is because mutliplying two negative numbers together gives a positive number. It follows that the square root of a negative number simply does not exist. In other words, does not represent any number. And, by definition, the square root of a positive number is positive.
Raising a negative number to the third power gives a negative number, and so taking the cubic root of a negative number gives a negative number.
Example 1: Simplifying a Cubic Root
We are looking for the number that when raised to the third power is equal to . We know that it is .
We can check our answer by raising to the third power:
The answer is .
Example 2: Cubic Root of a Negative Monomial
First, we see that there is a negative sign in front of the monomial, which means that our answer will have a negative sign as well.
Then, we have a constant times a variable raised to the thirtieth power. We can, therefore, split our cubic root in the product of two cubic roots using the product property of cubic roots:
We can express the first part as a fraction to make the calculation simpler and write the cubic root of a quotient as a quotient of cubic roots using the quotient property of cubic roots:
For the second part, , we need to divide the exponent in by 3, so we get
We finally find
Example 3: Finding the Side Length of a Cube Given Its Volume
A cube-shaped container has a capacity of . Determine the length of one of its sides.
In this question, we are given the capacity of a cube-shaped container. This means that, here, we know the volume of a cube, and we are looking for its side. This is given by the cubic root of the volume, that is, .
We can split the cubic root in the product of three cubic roots using the product property of cubic roots:
We can find by trials and errors or by realizing that a unit of 2 can be obtained only with a number having 8 as unit. And evaluating indeed gives 512.
For the other two cubic roots of a variable raised to a given power, we need to divide their exponents by three, that is, and .
We finally find
Hence, the side of the cube-shaped container is .
Example 4: Finding the Square Root of a Cubic Root
This calculation looks more complicated than it really is. We are going to carry it out in two steps, and you will see that it is not that difficult.
So, we have the square root of a cubic root. Therefore, we need to calculate first the cubic root, and then we will be able to work out the square root of this number.
So, let’s start with . It can be split in the product of two cubic roots using the product property of cubic roots:
We now need to calculate :
Hence, we find that .