Lesson Explainer: Simplifying Monomials: Quotient Rule | Nagwa Lesson Explainer: Simplifying Monomials: Quotient Rule | Nagwa

Lesson Explainer: Simplifying Monomials: Quotient Rule Mathematics • First Year of Preparatory School

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 12𝑥 and 4𝑥. We simplify this quotient by recalling that powers are defined by repeated multiplication. So, 𝑥=𝑥×𝑥×𝑥 and 𝑥=𝑥×𝑥. We can use this to rewrite the quotient: 12𝑥4𝑥=12×𝑥×𝑥×𝑥4×𝑥×𝑥.

At this point, it is worth noting that 𝑥 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 𝑥 if 𝑥 is 0. In cases like this, we can say that 𝑥 is nonzero since the quotient 12𝑥4𝑥 would be undefined otherwise.

Thus, when 𝑥0, we can cancel the shared factors of 𝑥. We can use commutativity and associativity of multiplication (noting that division is equivalent to multiplication by the inverse) to rewrite the expression as follows: 12×𝑥×𝑥×𝑥4×𝑥×𝑥=124×𝑥×𝑥×𝑥𝑥×𝑥=3×𝑥×𝑥×𝑥𝑥×𝑥1=3𝑥.

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: 12×𝑥×𝑥×𝑥4×𝑥×𝑥=312×𝑥×𝑥×𝑥14×𝑥×𝑥=3𝑥

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

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

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

Rule: Quotient Rule for Exponents

For any integers 𝑚 and 𝑛 with 𝑚𝑛 and any nonzero value of 𝑥, we have 𝑥𝑥=𝑥.

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 𝑥𝑥.

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 𝑚 and 𝑛 and any nonzero value of 𝑥, we have 𝑥𝑥=𝑥.

In this case, we have that 𝑚=7 and 𝑛=6. We substitute these values into the quotient rule for exponents and evaluate to get 𝑥𝑥=𝑥=𝑥.

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

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 𝑥 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 𝑙, we must have 𝑙×4𝑥=24𝑥.

We note that 𝑥0 since the width and area of the rectangle are nonzero; so, we can divide both sides of the equation by 4𝑥 to get 𝑙×4𝑥4𝑥=24𝑥4𝑥.

On the left-hand side of the equation, we can cancel the shared factors of 4𝑥 to get 𝑙=24𝑥4𝑥.

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 𝑚 and 𝑛 and any nonzero value of 𝑥, we have 𝑥𝑥=𝑥.

Since 𝑥=𝑥, we can apply this rule with 𝑚=2 and 𝑛=1 as follows: 24𝑥4𝑥=244×𝑥𝑥=6𝑥=6𝑥=6𝑥.

Hence, the length of the rectangle is given by 6𝑥.

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 10𝑥𝑦 and 4𝑥𝑦, we can rewrite the quotient as 10𝑥𝑦4𝑥𝑦=10×𝑥×𝑦4×𝑥×𝑦=104×𝑥𝑥×𝑦𝑦.

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 𝑥 and 𝑦 are nonzero. We get 104×𝑥𝑥×𝑦𝑦=52×𝑥×𝑦=52𝑥𝑦.

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 12𝑥𝑦6𝑥𝑦.

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: 12𝑥𝑦6𝑥𝑦=126×𝑥𝑥×𝑦𝑦.

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

We assume that 𝑥 and 𝑦 are nonzero, and we note that 𝑦=𝑦. Thus, 126×𝑥𝑥×𝑦𝑦=(2)×𝑥𝑥×𝑦𝑦=(2)𝑥𝑦=2𝑥𝑦.

Hence, 12𝑥𝑦6𝑥𝑦=2𝑥𝑦.

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 10𝑥𝑦 cm2 and whose width is 5𝑥𝑦 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 𝑙, we must have 𝑙×5𝑥𝑦=10𝑥𝑦.

We can divide both sides of the equation by 5𝑥𝑦 to get 𝑙×5𝑥𝑦5𝑥𝑦=10𝑥𝑦5𝑥𝑦.

Assuming that both 𝑥 and 𝑦 are nonzero, we can cancel the shared factor of 5𝑥𝑦 in the numerator and denominator on the left-hand side of the equation to get 𝑙=10𝑥𝑦5𝑥𝑦.

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: 10𝑥𝑦5𝑥𝑦=105×𝑥𝑥×𝑦𝑦.

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

We assume that 𝑥 and 𝑦 are nonzero, and we note that 𝑥=𝑥 and 𝑦=𝑦. Thus, 105×𝑥𝑥×𝑦𝑦=2×𝑥𝑥×𝑦𝑦=2𝑥𝑦=2𝑥𝑦.

Any nonzero number raised to the power of 0 is 1; thus, 𝑙=2𝑥.

Since the lengths are given in centimetres, we can say that the length of the rectangle is 2𝑥 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 𝑥𝑦×𝑥𝑦𝑥𝑦.

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 𝑥 and nonnegative integers 𝑚 and 𝑛, we have 𝑥×𝑥=𝑥.

This means that we can simplify the numerator of the expression by adding the exponents of the variables. We have 𝑥𝑦×𝑥𝑦=𝑥×𝑥×𝑦×𝑦=𝑥×𝑦=𝑥𝑦.

Substituting this into the original expression gives 𝑥𝑦×𝑥𝑦𝑥𝑦=𝑥𝑦𝑥𝑦.

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: 𝑥𝑦𝑥𝑦=𝑥𝑥×𝑦𝑦.

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

We assume that 𝑥 and 𝑦 are nonzero and then apply this rule to get 𝑥𝑥×𝑦𝑦=𝑥×𝑦=𝑥𝑦.

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 𝑚 and 𝑛 such that 𝑚𝑛 and any nonzero value of 𝑥, we have 𝑥𝑥=𝑥.

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