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

In this explainer, we want to focus on simplifying the quotient of monomial terms. To do this, let’s start by recalling the definition of a monomial.

### Definition: Monomial

A monomial is a single algebraic term where every variable is raised to a nonnegative integer power.

To see how we can simplify the quotient of monomials, we can start with an example. Let’s say we want to simplify the quotient of and . We simplify this quotient by recalling that powers are defined by repeated multiplication. So, and . We can use this to rewrite the quotient:

At this point, it is worth noting that represents a number and we need to be careful since we know we cannot divide by 0 and, in the same way, we cannot cancel the shared factors of if is 0. In cases like this, we can say that is nonzero since the quotient would be undefined otherwise.

Thus, when , we can cancel the shared factors of . We can use commutativity and associativity of multiplication (noting that division is equivalent to multiplication by the inverse) to rewrite the expression as follows:

It is worth noting that the formality of using commutativity and associativity to rewrite the expression is often skipped so that we can just cancel the shared factors in the expanded expression:

We can generalize this process to the quotient of any two monomials. To do this, let’s start with the quotient of the monomials and for nonnegative integers and , with . We write the products out in full as follows.

Since , the noncanceled factors of will be in the numerator. We have the following.

Therefore, we have that ; this is known as the quotient rule for exponents. It is worth noting that this is the same as the quotient rule for exponents , with being a nonzero number; however, instead of , we have an unknown base .

### Rule: Quotient Rule for Exponents

For any integers and with and any nonzero value of , we have

We can apply this rule to find the quotient of any two monomials, as we will see in our first example.

### Example 1: Simplifying a Quotient of Monomials

Simplify .

### Answer

Since we are asked to simplify the quotient of two monomials, we can start by recalling the quotient rule for exponents, which states that for any integers and and any nonzero value of , we have

In this case, we have that and . We substitute these values into the quotient rule for exponents and evaluate to get

Finally, we recall that raising a number to the first power leaves it unchanged. Hence,

It is worth noting that we can also solve this problem from first principles by writing the products out in full. We have the following.

We can then cancel the 6 shared factors of in the numerator and denominator to get the following.

In our next example, we will use the quotients of monomials to find an expression for the missing side length in a rectangle.

### Example 2: Solving a 2D Geometric Problem by Dividing Monomials

Find the missing side length of the following rectangle.

### Answer

We start by recalling that the area of a rectangle is given by its length multiplied by its width. If we call the length of this rectangle , we must have

We note that since the width and area of the rectangle are nonzero; so, we can divide both sides of the equation by to get

On the left-hand side of the equation, we can cancel the shared factors of to get

On the right-hand side of the equation, we are taking the quotient of monomials, and we can simplify this by using the quotient rule. This states that for any integers and and any nonzero value of , we have

Since , we can apply this rule with and as follows:

Hence, the length of the rectangle is given by .

Thus far, we have only dealt with the quotients of single-variable monomials. However, we can apply the exact same process to find the quotient of any monomials. For example, if we want to simplify the quotient of and , we can rewrite the quotient as

We now need to apply the quotient rule for exponents on the final two factors, where we note that this is only valid if both and are nonzero. We get

We can apply the same process with any number of variables by using the properties of the multiplication of rational numbers to separate the division into factors with the same base.

In our next example, we will see an example of applying this process to simplify the quotient of two monomials in multiple variables.

### Example 3: Simplifying a Quotient of Monomials including More Than One Variable

Simplify .

### Answer

We note that we are asked to simplify the quotient of two monomials. We recall that we can simplify the quotient of two monomials by first rearranging the quotient to separate the factors with the same base:

We can then simplify the final two quotients by using the quotient rule for exponents, which states that for any integers and and any nonzero value of , we have

We assume that and are nonzero, and we note that . Thus,

Hence,

In our next example, we will apply this process to simplify the quotient of two monomials in multiple variables in a word problem.

### Example 4: Solving Word Problems by Simplifying a Quotient of Monomials including More Than One Variable

What is the length of a rectangle whose area is
cm^{2}
and whose width is
cm?

### Answer

We start by recalling that the area of a rectangle is given by its length multiplied by its width. If we call the length of this rectangle , we must have

We can divide both sides of the equation by to get

Assuming that both and are nonzero, we can cancel the shared factor of in the numerator and denominator on the left-hand side of the equation to get

We now note that the right-hand side of the equation is the quotient of two monomials. We recall that we can simplify the quotient of two monomials by first rearranging the quotient to separate the factors with the same base:

We can then simplify the final two quotients by using the quotient rule for exponents, which states that for any integers and and any nonzero value of , we have

We assume that and are nonzero, and we note that and . Thus,

Any nonzero number raised to the power of 0 is 1; thus,

Since the lengths are given in centimetres, we can say that the length of the rectangle is cm.

In our final example, we will use this process to simplify the product and quotient of monomials.

### Example 5: Simplifying an Algebraic Expression of Monomial Multiplication and the Quotient Rule

Simplify .

### Answer

We first note that the numerator of this expression is the product of two monomials. We can simplify the product of monomials by recalling that the product rule for exponents states that for any rational number and nonnegative integers and , we have

This means that we can simplify the numerator of the expression by adding the exponents of the variables. We have

Substituting this into the original expression gives

We now note that this is the quotient of two monomials. We recall that we can simplify the quotient of two monomials by first rearranging the quotient to separate the factors with the same base:

We can then simplify the final two quotients by using the quotient rule for exponents, which states that for any integers and and any nonzero value of , we have

We assume that and are nonzero and then apply this rule to get

Let’s finish by recapping some of the important points from this explainer.

### Key Points

- A monomial is a single algebraic term where every variable is raised to a nonnegative integer power.
- The quotient rule for exponents states that for any integers and such that and any nonzero value of , we have