Lesson Explainer: Simplifying Monomials: Quotient Rule Mathematics • 9th Grade

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, letโ€™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. Letโ€™s say we want to simplify the quotient of 12๐‘ฅ๏Šฉ and 4๐‘ฅ๏Šจ. We simplify this quotient by recalling that powers are defined by repeated multiplication. So, ๐‘ฅ=๐‘ฅร—๐‘ฅร—๐‘ฅ๏Šฉ and ๐‘ฅ=๐‘ฅร—๐‘ฅ๏Šจ. We can use this to rewrite the quotient: 12๐‘ฅ4๐‘ฅ=12ร—๐‘ฅร—๐‘ฅร—๐‘ฅ4ร—๐‘ฅร—๐‘ฅ.๏Šฉ๏Šจ

At this point, it is worth noting that ๐‘ฅ 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 ๐‘ฅ if ๐‘ฅ is 0. In cases like this, we can say that ๐‘ฅ is nonzero since the quotient 12๐‘ฅ4๐‘ฅ๏Šฉ๏Šจ would be undefined otherwise.

Thus, when ๐‘ฅโ‰ 0, we can cancel the shared factors of ๐‘ฅ. We can use commutativity and associativity of multiplication (noting that division is equivalent to multiplication by the inverse) to rewrite the expression as follows: 12ร—๐‘ฅร—๐‘ฅร—๐‘ฅ4ร—๐‘ฅร—๐‘ฅ=๏€ผ124๏ˆร—๐‘ฅร—๐‘ฅร—๐‘ฅ๐‘ฅร—๐‘ฅ=3ร—๐‘ฅร—๐‘ฅร—๐‘ฅ๐‘ฅร—๐‘ฅ1=3๐‘ฅ.

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: 12ร—๐‘ฅร—๐‘ฅร—๐‘ฅ4ร—๐‘ฅร—๐‘ฅ=312ร—๐‘ฅร—๐‘ฅร—๐‘ฅ14ร—๐‘ฅร—๐‘ฅ=3๐‘ฅ

We can generalize this process to the quotient of any two monomials. To do this, letโ€™s start with the quotient of the monomials ๐‘ฅ๏‰ and ๐‘ฅ๏Š for nonnegative integers ๐‘š and ๐‘›, with ๐‘šโ‰ฅ๐‘›. We write the products out in full as follows.

Since ๐‘šโ‰ฅ๐‘›, the noncanceled factors of ๐‘ฅ will be in the numerator. We have the following.

Therefore, we have that ๐‘ฅ๐‘ฅ=๐‘ฅ๏‰๏Š๏‰๏Šฑ๏Š; this is known as the quotient rule for exponents. It is worth noting that this is the same as the quotient rule for exponents ๐‘๐‘=๐‘๏‰๏Š๏‰๏Šฑ๏Š, with ๐‘ being a nonzero number; however, instead of ๐‘, we have an unknown base ๐‘ฅ.

Rule: Quotient Rule for Exponents

For any integers ๐‘š and ๐‘› with ๐‘šโ‰ฅ๐‘› and any nonzero value of ๐‘ฅ, we have ๐‘ฅ๐‘ฅ=๐‘ฅ.๏‰๏Š๏‰๏Šฑ๏Š

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 ๐‘ฅ๐‘ฅ๏Šญ๏Šฌ.

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 ๐‘š and ๐‘› and any nonzero value of ๐‘ฅ, we have ๐‘ฅ๐‘ฅ=๐‘ฅ.๏‰๏Š๏‰๏Šฑ๏Š

In this case, we have that ๐‘š=7 and ๐‘›=6. We substitute these values into the quotient rule for exponents and evaluate to get ๐‘ฅ๐‘ฅ=๐‘ฅ=๐‘ฅ.๏Šญ๏Šฌ๏Šญ๏Šฑ๏Šฌ๏Šง

Finally, we recall that raising a number to the first power leaves it unchanged. Hence, ๐‘ฅ๐‘ฅ=๐‘ฅ.๏Šญ๏Šฌ

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 ๐‘ฅ 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 ๐‘™, we must have ๐‘™ร—4๐‘ฅ=24๐‘ฅ.๏Šจ

We note that ๐‘ฅโ‰ 0 since the width and area of the rectangle are nonzero; so, we can divide both sides of the equation by 4๐‘ฅ to get ๐‘™ร—4๐‘ฅ4๐‘ฅ=24๐‘ฅ4๐‘ฅ.๏Šจ

On the left-hand side of the equation, we can cancel the shared factors of 4๐‘ฅ to get ๐‘™=24๐‘ฅ4๐‘ฅ.๏Šจ

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 ๐‘š and ๐‘› and any nonzero value of ๐‘ฅ, we have ๐‘ฅ๐‘ฅ=๐‘ฅ.๏‰๏Š๏‰๏Šฑ๏Š

Since ๐‘ฅ=๐‘ฅ๏Šง, we can apply this rule with ๐‘š=2 and ๐‘›=1 as follows: 24๐‘ฅ4๐‘ฅ=๏€ผ244๏ˆร—๏€พ๐‘ฅ๐‘ฅ๏Š=6๐‘ฅ=6๐‘ฅ=6๐‘ฅ.๏Šจ๏Šจ๏Šง๏Šจ๏Šฑ๏Šง๏Šง

Hence, the length of the rectangle is given by 6๐‘ฅ.

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 10๐‘ฅ๐‘ฆ๏Šจ๏Šซ and 4๐‘ฅ๐‘ฆ๏Šจ, we can rewrite the quotient as 10๐‘ฅ๐‘ฆ4๐‘ฅ๐‘ฆ=10ร—๐‘ฅร—๐‘ฆ4ร—๐‘ฅร—๐‘ฆ=๏€ผ104๏ˆร—๏€พ๐‘ฅ๐‘ฅ๏Šร—๏€พ๐‘ฆ๐‘ฆ๏Š.๏Šจ๏Šซ๏Šจ๏Šจ๏Šซ๏Šง๏Šจ๏Šจ๏Šง๏Šซ๏Šจ

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 ๐‘ฅ and ๐‘ฆ are nonzero. We get ๏€ผ104๏ˆร—๏€พ๐‘ฅ๐‘ฅ๏Šร—๏€พ๐‘ฆ๐‘ฆ๏Š=52ร—๏€น๐‘ฅ๏…ร—๏€น๐‘ฆ๏…=52๐‘ฅ๐‘ฆ.๏Šจ๏Šง๏Šซ๏Šจ๏Šจ๏Šฑ๏Šง๏Šซ๏Šฑ๏Šจ๏Šฉ

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 โˆ’12๐‘ฅ๐‘ฆ6๐‘ฅ๐‘ฆ๏Šฌ๏Šซ๏Šช.

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: โˆ’12๐‘ฅ๐‘ฆ6๐‘ฅ๐‘ฆ=๏€ผโˆ’126๏ˆร—๏€พ๐‘ฅ๐‘ฅ๏Šร—๏€พ๐‘ฆ๐‘ฆ๏Š.๏Šฌ๏Šซ๏Šช๏Šฌ๏Šช๏Šซ

We can then simplify the final two quotients by using the quotient rule for exponents, which states that for any integers ๐‘š and ๐‘› and any nonzero value of ๐‘ฅ, we have ๐‘ฅ๐‘ฅ=๐‘ฅ.๏‰๏Š๏‰๏Šฑ๏Š

We assume that ๐‘ฅ and ๐‘ฆ are nonzero, and we note that ๐‘ฆ=๐‘ฆ๏Šง. Thus, ๏€ผโˆ’126๏ˆร—๏€พ๐‘ฅ๐‘ฅ๏Šร—๏€พ๐‘ฆ๐‘ฆ๏Š=(โˆ’2)ร—๏€พ๐‘ฅ๐‘ฅ๏Šร—๏€พ๐‘ฆ๐‘ฆ๏Š=(โˆ’2)๏€น๐‘ฅ๏…๏€น๐‘ฆ๏…=โˆ’2๐‘ฅ๐‘ฆ.๏Šฌ๏Šช๏Šซ๏Šฌ๏Šช๏Šซ๏Šง๏Šฌ๏Šฑ๏Šช๏Šซ๏Šฑ๏Šง๏Šจ๏Šช

Hence, โˆ’12๐‘ฅ๐‘ฆ6๐‘ฅ๐‘ฆ=โˆ’2๐‘ฅ๐‘ฆ.๏Šฌ๏Šซ๏Šช๏Šจ๏Šช

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 10๐‘ฅ๐‘ฆ๏Šฌ cm2 and whose width is 5๐‘ฅ๐‘ฆ 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 ๐‘™, we must have ๐‘™ร—5๐‘ฅ๐‘ฆ=10๐‘ฅ๐‘ฆ.๏Šฌ

We can divide both sides of the equation by 5๐‘ฅ๐‘ฆ to get ๐‘™ร—5๐‘ฅ๐‘ฆ5๐‘ฅ๐‘ฆ=10๐‘ฅ๐‘ฆ5๐‘ฅ๐‘ฆ.๏Šฌ

Assuming that both ๐‘ฅ and ๐‘ฆ are nonzero, we can cancel the shared factor of 5๐‘ฅ๐‘ฆ in the numerator and denominator on the left-hand side of the equation to get ๐‘™=10๐‘ฅ๐‘ฆ5๐‘ฅ๐‘ฆ.๏Šฌ

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: 10๐‘ฅ๐‘ฆ5๐‘ฅ๐‘ฆ=๏€ผ105๏ˆร—๏€พ๐‘ฅ๐‘ฅ๏Šร—๏€ฝ๐‘ฆ๐‘ฆ๏‰.๏Šฌ๏Šฌ

We can then simplify the final two quotients by using the quotient rule for exponents, which states that for any integers ๐‘š and ๐‘› and any nonzero value of ๐‘ฅ, we have ๐‘ฅ๐‘ฅ=๐‘ฅ.๏‰๏Š๏‰๏Šฑ๏Š

We assume that ๐‘ฅ and ๐‘ฆ are nonzero, and we note that ๐‘ฅ=๐‘ฅ๏Šง and ๐‘ฆ=๐‘ฆ๏Šง. Thus, ๏€ผ105๏ˆร—๏€พ๐‘ฅ๐‘ฅ๏Šร—๏€ฝ๐‘ฆ๐‘ฆ๏‰=2ร—๏€พ๐‘ฅ๐‘ฅ๏Šร—๏€พ๐‘ฆ๐‘ฆ๏Š=2๏€น๐‘ฅ๏…๏€น๐‘ฆ๏…=2๐‘ฅ๐‘ฆ.๏Šฌ๏Šฌ๏Šง๏Šง๏Šง๏Šฌ๏Šฑ๏Šง๏Šง๏Šฑ๏Šง๏Šซ๏Šฆ

Any nonzero number raised to the power of 0 is 1; thus, ๐‘™=2๐‘ฅ.๏Šซ

Since the lengths are given in centimetres, we can say that the length of the rectangle is 2๐‘ฅ๏Šซ 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 ๐‘ฅ๐‘ฆร—๐‘ฅ๐‘ฆ๐‘ฅ๐‘ฆ๏Šช๏Šช๏Šจ๏Šช๏Šช๏Šฉ.

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 ๐‘ฅ and nonnegative integers ๐‘š and ๐‘›, we have ๐‘ฅร—๐‘ฅ=๐‘ฅ.๏‰๏Š๏‰๏Šฐ๏Š

This means that we can simplify the numerator of the expression by adding the exponents of the variables. We have ๐‘ฅ๐‘ฆร—๐‘ฅ๐‘ฆ=๏€น๐‘ฅร—๐‘ฅ๏…ร—๏€น๐‘ฆร—๐‘ฆ๏…=๐‘ฅร—๐‘ฆ=๐‘ฅ๐‘ฆ.๏Šช๏Šช๏Šจ๏Šช๏Šช๏Šจ๏Šช๏Šช๏Šช๏Šฐ๏Šจ๏Šช๏Šฐ๏Šช๏Šฌ๏Šฎ

Substituting this into the original expression gives ๐‘ฅ๐‘ฆร—๐‘ฅ๐‘ฆ๐‘ฅ๐‘ฆ=๐‘ฅ๐‘ฆ๐‘ฅ๐‘ฆ.๏Šช๏Šช๏Šจ๏Šช๏Šช๏Šฉ๏Šฌ๏Šฎ๏Šช๏Šฉ

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: ๐‘ฅ๐‘ฆ๐‘ฅ๐‘ฆ=๏€พ๐‘ฅ๐‘ฅ๏Šร—๏€พ๐‘ฆ๐‘ฆ๏Š.๏Šฌ๏Šฎ๏Šช๏Šฉ๏Šฌ๏Šช๏Šฎ๏Šฉ

We can then simplify the final two quotients by using the quotient rule for exponents, which states that for any integers ๐‘š and ๐‘› and any nonzero value of ๐‘ฅ, we have ๐‘ฅ๐‘ฅ=๐‘ฅ.๏‰๏Š๏‰๏Šฑ๏Š

We assume that ๐‘ฅ and ๐‘ฆ are nonzero and then apply this rule to get ๏€พ๐‘ฅ๐‘ฅ๏Šร—๏€พ๐‘ฆ๐‘ฆ๏Š=๏€น๐‘ฅ๏…ร—๏€น๐‘ฆ๏…=๐‘ฅ๐‘ฆ.๏Šฌ๏Šช๏Šฎ๏Šฉ๏Šฌ๏Šฑ๏Šช๏Šฎ๏Šฑ๏Šซ๏Šจ๏Šซ

Letโ€™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 ๐‘š and ๐‘› such that ๐‘šโ‰ฅ๐‘› and any nonzero value of ๐‘ฅ, we have ๐‘ฅ๐‘ฅ=๐‘ฅ.๏‰๏Š๏‰๏Šฑ๏Š

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