Polynomials
51 Divide Monomials
Learning Objectives
By the end of this section, you will be able to:
 Simplify expressions using the Quotient Property for Exponents
 Simplify expressions with zero exponents
 Simplify expressions using the quotient to a Power Property
 Simplify expressions by applying several properties
 Divide monomials
Before you get started, take this readiness quiz.
Simplify Expressions Using the Quotient Property for Exponents
Earlier in this chapter, we developed the properties of exponents for multiplication. We summarize these properties below.
If are real numbers, and are whole numbers, then
Product Property  
Power Property  
Product to a Power 
Now we will look at the exponent properties for division. A quick memory refresher may help before we get started. You have learned to simplify fractions by dividing out common factors from the numerator and denominator using the Equivalent Fractions Property. This property will also help you work with algebraic fractions—which are also quotients.
If are whole numbers where ,
As before, we’ll try to discover a property by looking at some examples.
Consider  and  
What do they mean?  
Use the Equivalent Fractions Property.  
Simplify. 
Notice, in each case the bases were the same and we subtracted exponents.
When the larger exponent was in the numerator, we were left with factors in the numerator.
When the larger exponent was in the denominator, we were left with factors in the denominator—notice the numerator of 1.
We write:
This leads to the Quotient Property for Exponents.
If is a real number, , and are whole numbers, then
A couple of examples with numbers may help to verify this property.
Simplify: ⓐ ⓑ
To simplify an expression with a quotient, we need to first compare the exponents in the numerator and denominator.
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Since 9 > 7, there are more factors of x in the numerator. Use the Quotient Property, . Simplify.  ⓑ
Since 10 > 2, there are more factors of x in the numerator. Use the Quotient Property, . Simplify.
Notice that when the larger exponent is in the numerator, we are left with factors in the numerator.
Simplify: ⓐ ⓑ
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Simplify: ⓐ ⓑ
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Simplify: ⓐ ⓑ
To simplify an expression with a quotient, we need to first compare the exponents in the numerator and denominator.
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Since 12 > 8, there are more factors of b in the denominator. Use the Quotient Property, . Simplify.  ⓑ
Since 5 > 3, there are more factors of 3 in the denominator. Use the Quotient Property, . Simplify. Simplify.
Notice that when the larger exponent is in the denominator, we are left with factors in the denominator.
Simplify: ⓐ ⓑ
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Simplify: ⓐ ⓑ
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Notice the difference in the two previous examples:
 If we start with more factors in the numerator, we will end up with factors in the numerator.
 If we start with more factors in the denominator, we will end up with factors in the denominator.
The first step in simplifying an expression using the Quotient Property for Exponents is to determine whether the exponent is larger in the numerator or the denominator.
Simplify: ⓐ ⓑ
 ⓐ Is the exponent of larger in the numerator or denominator? Since 9 > 5, there are more in the denominator and so we will end up with factors in the denominator.
Use the Quotient Property, . Simplify.  ⓑ Notice there are more factors of in the numerator, since 11 > 7. So we will end up with factors in the numerator.
Use the Quotient Property, . Simplify.
Simplify: ⓐ ⓑ
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Simplify: ⓐ ⓑ
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Simplify Expressions with an Exponent of Zero
A special case of the Quotient Property is when the exponents of the numerator and denominator are equal, such as an expression like . From your earlier work with fractions, you know that:
In words, a number divided by itself is 1. So, , for any , since any number divided by itself is 1.
The Quotient Property for Exponents shows us how to simplify when and when by subtracting exponents. What if ?
Consider , which we know is 1.
Write as .  
Subtract exponents.  
Simplify. 
Now we will simplify in two ways to lead us to the definition of the zero exponent. In general, for :
We see simplifies to and to 1. So .
If is a nonzero number, then .
Any nonzero number raised to the zero power is 1.
In this text, we assume any variable that we raise to the zero power is not zero.
Simplify: ⓐ ⓑ
The definition says any nonzero number raised to the zero power is 1.
ⓐ Use the definition of the zero exponent. 

ⓑ Use the definition of the zero exponent. 
Simplify: ⓐ ⓑ
ⓐ 1 ⓑ 1
Simplify: ⓐ ⓑ
ⓐ 1 ⓑ 1
Now that we have defined the zero exponent, we can expand all the Properties of Exponents to include whole number exponents.
What about raising an expression to the zero power? Let’s look at . We can use the product to a power rule to rewrite this expression.
Use the product to a power rule.  
Use the zero exponent property.  
Simplify. 
This tells us that any nonzero expression raised to the zero power is one.
Simplify: ⓐ ⓑ
ⓐ  
Use the definition of the zero exponent.  
ⓑ  
Use the definition of the zero exponent. 
Simplify: ⓐ ⓑ
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Simplify: ⓐ ⓑ
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Simplify Expressions Using the Quotient to a Power Property
Now we will look at an example that will lead us to the Quotient to a Power Property.
This means:  
Multiply the fractions.  
Write with exponents. 
Notice that the exponent applies to both the numerator and the denominator.
We write:  
This leads to the Quotient to a Power Property for Exponents.
If and are real numbers, , and is a counting number, then
To raise a fraction to a power, raise the numerator and denominator to that power.
An example with numbers may help you understand this property:
Simplify: ⓐ ⓑ ⓒ
ⓐ
Use the Quotient Property, .  
Simplify. 
ⓑ
Use the Quotient Property, .  
Simplify. 
ⓒ
Raise the numerator and denominator to the third power. 
Simplify: ⓐ ⓑ ⓒ
ⓐⓑⓒ
Simplify: ⓐ ⓑ ⓒ
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Simplify Expressions by Applying Several Properties
We’ll now summarize all the properties of exponents so they are all together to refer to as we simplify expressions using several properties. Notice that they are now defined for whole number exponents.
If are real numbers, and are whole numbers, then
Product Property  
Power Property  
Product to a Power  
Quotient Property  
Zero Exponent Definition  
Quotient to a Power Property 
Simplify:
Multiply the exponents in the numerator.  
Subtract the exponents. 
Simplify:
Simplify:
Simplify:
Multiply the exponents in the numerator.  
Subtract the exponents.  
Simplify. 
Simplify:
1
Simplify:
1
Simplify:
Remember parentheses come before exponents. Notice the bases are the same, so we can simplify inside the parentheses. Subtract the exponents. 

Multiply the exponents. 
Simplify:
Simplify:
Simplify:
Here we cannot simplify inside the parentheses first, since the bases are not the same.
Raise the numberator and denominator to the third power using the Quotient to a Power Property, . 

Use the Powert Property and simplify. 
Simplify:
Simplify:
Simplify:
Raise the numberator and denominator to the fourth power, using the Quotient to a Power Property, . 

Raise each factor to the fourth power.  
Use the Power Property and simplify. 
Simplify:
Simplify:
Simplify:
Use the Power Property, .  
Add the exponents in the numerator.  
Use the Quotient Property, . 
Simplify:
Simplify:
Simplify:
Use the Product to a Power Property, .  
Use the Power Property, .  
Add the exponents in the denominator.  
Use the Quotient Property, .  
Simplify. 
Simplify:
Simplify:
Divide Monomials
You have now been introduced to all the properties of exponents and used them to simplify expressions. Next, you’ll see how to use these properties to divide monomials. Later, you’ll use them to divide polynomials.
Find the quotient:
Rewrite as a fraction.  
Use fraction multiplication.  
Simplify and use the Quotient Property. 
Find the quotient:
Find the quotient:
Find the quotient:
Use fraction multiplication.  
Simplify and use the Quotient Property.  
Multiply. 
Find the quotient:
Find the quotient:
Find the quotient:
Use fraction multiplication.  
Simplify and use the Quotient Property.  
Multiply. 
Find the quotient:
Find the quotient:
Once you become familiar with the process and have practiced it step by step several times, you may be able to simplify a fraction in one step.
Find the quotient:
Be very careful to simplify by dividing out a common factor, and to simplify the variables by subtracting their exponents.
Simplify and use the Quotient Property. 
Find the quotient:
Find the quotient:
In all examples so far, there was no work to do in the numerator or denominator before simplifying the fraction. In the next example, we’ll first find the product of two monomials in the numerator before we simplify the fraction. This follows the order of operations. Remember, a fraction bar is a grouping symbol.
Find the quotient:
Simplify the numerator.  
Simplify. 
Find the quotient:
Find the quotient:
Access these online resources for additional instruction and practice with dividing monomials:
Key Concepts
 Quotient Property for Exponents:
 If is a real number, , and are whole numbers, then:
 If is a real number, , and are whole numbers, then:
 Zero Exponent
 If is a nonzero number, then .
 If is a nonzero number, then .
 Quotient to a Power Property for Exponents:
 If and are real numbers, and is a counting number, then:
 To raise a fraction to a power, raise the numerator and denominator to that power.
 If and are real numbers, and is a counting number, then:
 Summary of Exponent Properties
 If are real numbers and are whole numbers, then
 If are real numbers and are whole numbers, then
Practice Makes Perfect
Simplify Expressions Using the Quotient Property for Exponents
In the following exercises, simplify.
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Simplify Expressions with Zero Exponents
In the following exercises, simplify.
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Simplify Expressions Using the Quotient to a Power Property
In the following exercises, simplify.
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Simplify Expressions by Applying Several Properties
In the following exercises, simplify.
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Divide Monomials
In the following exercises, divide the monomials.
Mixed Practice
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Everyday Math
Memory One megabyte is approximately bytes. One gigabyte is approximately bytes. How many megabytes are in one gigabyte?
Memory One gigabyte is approximately bytes. One terabyte is approximately bytes. How many gigabytes are in one terabyte?
Writing Exercises
Jennifer thinks the quotient simplifies to . What is wrong with her reasoning?
Maurice simplifies the quotient by writing . What is wrong with his reasoning?
Answers will vary.
When Drake simplified and he got the same answer. Explain how using the Order of Operations correctly gives different answers.
Robert thinks simplifies to 0. What would you say to convince Robert he is wrong?
Answers will vary.
Self Check
ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ On a scale of 110, how would you rate your mastery of this section in light of your responses on the checklist? How can you improve this?