7.2: Division with Exponents
- Page ID
- 137930
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- Simplify expressions using the Quotient Property of Exponents
- Simplify expressions with zero exponents
- Simplify expressions using the Quotient to a Power Property
- Simplify expressions by applying several properties
- Divide monomials
Be Prepared 10.9
Before you get started, take this readiness quiz.
Simplify:
If you missed the problem, review Example 4.19.
Be Prepared 10.10
Simplify:
If you missed the problem, review Example 10.23.
Be Prepared 10.11
Simplify:
If you missed the problem, review Example 4.23.
Simplify Expressions Using the Quotient Property of Exponents
Earlier in this chapter, we developed the properties of exponents for multiplication. We summarize these properties here.
Summary of Exponent Properties for Multiplication
If are real numbers and are whole numbers, then
Now we will look at the exponent properties for division. A quick memory refresher may help before we get started. In Fractions you learned that fractions may be simplified by dividing out common factors from the numerator and denominator using the Equivalent Fractions Property. This property will also help us work with algebraic fractions—which are also quotients.
Equivalent Fractions Property
If are whole numbers where then
As before, we'll try to discover a property by looking at some examples.
Notice that in each case the bases were the same and we subtracted the exponents.
- When the larger exponent was in the numerator, we were left with factors in the numerator and in the denominator, which we simplified.
- When the larger exponent was in the denominator, we were left with factors in the denominator, and in the numerator, which could not be simplified.
We write:
Quotient Property of Exponents
If is a real number, and are whole numbers, then
A couple of examples with numbers may help to verify this property.
When we work with numbers and the exponent is less than or equal to we will apply the exponent. When the exponent is greater than , we leave the answer in exponential form.
Example 10.45
Simplify:
- ⓐ
- ⓑ
- Answer
To simplify an expression with a quotient, we need to first compare the exponents in the numerator and denominator.
ⓐ Since 10 > 8, there are more factors of in the numerator. Use the quotient property with . Simplify. ⓑ Since 9 > 2, there are more factors of 2 in the numerator. Use the quotient property with Simplify.
Notice that when the larger exponent is in the numerator, we are left with factors in the numerator.
Try It 10.89
Simplify:
- ⓐ
- ⓑ
Try It 10.90
Simplify:
- ⓐ
- ⓑ
Example 10.46
Simplify:
- ⓐ
- ⓑ
- Answer
To simplify an expression with a quotient, we need to first compare the exponents in the numerator and denominator.
ⓐ Since 15 > 10, there are more factors of in the denominator. Use the quotient property with Simplify. ⓑ Since 5 > 3, there are more factors of 3 in the denominator. Use the quotient property with Simplify. Apply the exponent.
Notice that when the larger exponent is in the denominator, we are left with factors in the denominator and in the numerator.
Try It 10.91
Simplify:
- ⓐ
- ⓑ
Try It 10.92
Simplify:
- ⓐ
- ⓑ
Example 10.47
Simplify:
- ⓐ
- ⓑ
- Answer
ⓐ Since 9 > 5, there are more 's in the denominator and so we will end up with factors in the denominator. Use the Quotient Property for 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 for Simplify.
Try It 10.93
Simplify:
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- ⓑ
Try It 10.94
Simplify:
- ⓐ
- ⓑ
Simplify Expressions with Zero Exponents
A special case of the Quotient Property is when the exponents of the numerator and denominator are equal, such as an expression like From earlier work with fractions, we know that
In words, a number divided by itself is So for any (), since any number divided by itself is
The Quotient Property of Exponents shows us how to simplify when and when by subtracting exponents. What if ?
Now we will simplify in two ways to lead us to the definition of the zero exponent.
Consider first which we know is
Write 8 as . | |
Subtract exponents. | |
Simplify. |
We see simplifies to a and to . So .
Zero Exponent
If is a non-zero number, then
Any nonzero number raised to the zero power is
In this text, we assume any variable that we raise to the zero power is not zero.
Example 10.48
Simplify:
- ⓐ
- ⓑ
- Answer
The definition says any non-zero number raised to the zero power is
ⓐ Use the definition of the zero exponent. 1 ⓑ Use the definition of the zero exponent. 1
Try It 10.95
Simplify:
- ⓐ
- ⓑ
Try It 10.96
Simplify:
- ⓐ
- ⓑ
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. | 1 |
This tells us that any non-zero expression raised to the zero power is one.
Example 10.49
Simplify:
- Answer
Use the definition of the zero exponent. 1
Try It 10.97
Simplify:
Try It 10.98
Simplify:
Example 10.50
Simplify:
- ⓐ
- ⓑ
- Answer
ⓐ The product is raised to the zero power. Use the definition of the zero exponent. ⓑ Notice that only the variable is being raised to the zero power. Use the definition of the zero exponent. Simplify.
Try It 10.99
Simplify:
- ⓐ
- ⓑ
Try It 10.100
Simplify:
- ⓐ
- ⓑ
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 see that is
This leads to the Quotient to a Power Property for Exponents.
Quotient to a Power Property of 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:
Example 10.51
Simplify:
- ⓐ
- ⓑ
- ⓒ
- Answer
ⓐ Use the Quotient to a Power Property, . Simplify. ⓑ Use the Quotient to a Power Property, . Simplify.
ⓒ | |
Raise the numerator and denominator to the third power. |
Try It 10.101
Simplify:
- ⓐ
- ⓑ
- ⓒ
Try It 10.102
Simplify:
- ⓐ
- ⓑ
- ⓒ
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.
Summary of Exponent Properties
If are real numbers and are whole numbers, then
Example 10.52
Simplify:
- Answer
Multiply the exponents in the numerator, using the
Power Property.Subtract the exponents.
Try It 10.103
Simplify:
Try It 10.104
Simplify:
Example 10.53
Simplify:
- Answer
Multiply the exponents in the numerator, using the
Power Property.Subtract the exponents. Zero power property
Try It 10.105
Simplify:
Try It 10.106
Simplify:
Example 10.54
Simplify:
- Answer
Remember parentheses come before exponents, and the
bases are the same so we can simplify inside the
parentheses. Subtract the exponents.Simplify. Multiply the exponents.
Try It 10.107
Simplify:
Try It 10.108
Simplify:
Example 10.55
Simplify:
- Answer
Here we cannot simplify inside the parentheses first, since the bases are not the same.
Raise the numerator and denominator to the third power
using the Quotient to a Power Property,Use the Power Property,
Try It 10.109
Simplify:
Try It 10.110
Simplify:
Example 10.56
Simplify:
- Answer
Raise the numerator and denominator to the fourth
power using the Quotient to a Power Property.Raise each factor to the fourth power, using the Power
to a Power Property.Use the Power Property and simplify.
Try It 10.111
Simplify:
Try It 10.112
Simplify:
Example 10.57
Simplify:
- Answer
Use the Power Property. Add the exponents in the numerator, using the Product Property. Use the Quotient Property.
Try It 10.113
Simplify:
Try It 10.114
Simplify:
Divide Monomials
We have now seen all the properties of exponents. We'll use them to divide monomials. Later, you'll use them to divide polynomials.
Example 10.58
Find the quotient:
- Answer
Rewrite as a fraction. Use fraction multiplication to separate the number
part from the variable part.Use the Quotient Property.
Try It 10.115
Find the quotient:
Try It 10.116
Find the quotient:
When we divide monomials with more than one variable, we write one fraction for each variable.
Example 10.59
Find the quotient:
- Answer
Use fraction multiplication. Simplify and use the Quotient Property. Multiply.
Try It 10.117
Find the quotient:
Try It 10.118
Find the quotient:
Example 10.60
Find the quotient:
- Answer
Use fraction multiplication. Simplify and use the Quotient Property. Multiply.
Try It 10.119
Find the quotient:
Try It 10.120
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.
Example 10.61
Find the quotient:
- Answer
Simplify and use the Quotient Property. Be very careful to simplify by dividing out a common factor, and to simplify the variables by subtracting their exponents.
Try It 10.121
Find the quotient:
Try It 10.122
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.
Example 10.62
Find the quotient:
- Answer
Remember, the fraction bar is a grouping symbol. We will simplify the numerator first.
Simplify the numerator. Simplify, using the Quotient Rule.
Try It 10.123
Find the quotient:
Try It 10.124
Find the quotient:
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Section 10.4 Exercises
Practice Makes Perfect
Simplify Expressions Using the Quotient Property of Exponents
In the following exercises, simplify.
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.
Simplify Expressions by Applying Several Properties
In the following exercises, simplify.
Divide Monomials
In the following exercises, divide the monomials.
Mixed Practice
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- ⓒ
- ⓓ
- ⓐ
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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 megabyte is approximately bytes. One terabyte is approximately bytes. How many megabytes are in one terabyte?
Writing Exercises
Vic thinks the quotient simplifies to What is wrong with his reasoning?
Mai simplifies the quotient by writing What is wrong with her reasoning?
When Dimple simplified and she got the same answer. Explain how using the Order of Operations correctly gives different answers.
Roxie thinks simplifies to What would you say to convince Roxie she is wrong?
Self Check
ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ On a scale of 1–10, how would you rate your mastery of this section in light of your responses on the checklist? How can you improve this?