Learning Objectives
By the end of this section, you will be able to:
 Simplify expressions with exponents
 Simplify expressions using the Product Property of Exponents
 Simplify expressions using the Power Property of Exponents
 Simplify expressions using the Product to a Power Property
 Simplify expressions by applying several properties
 Multiply monomials
Be Prepared 10.4
Before you get started, take this readiness quiz.
Simplify: $\frac{3}{4}\xb7\frac{3}{4}.$
If you missed the problem, review Example 4.25.
Be Prepared 10.5
Simplify: $(\mathrm{2})(\mathrm{2})(\mathrm{2}).$
If you missed the problem, review Example 3.52.
Simplify Expressions with Exponents
Remember that an exponent indicates repeated multiplication of the same quantity. For example, ${2}^{4}$ means to multiply four factors of $2,$ so ${2}^{4}$ means $2\xb72\xb72\xb72.$ This format is known as exponential notation.
Exponential Notation
This is read $a$ to the ${m}^{\mathrm{th}}$ power.
In the expression ${a}^{m},$ the exponent tells us how many times we use the base $a$ as a factor.
Before we begin working with variable expressions containing exponents, let’s simplify a few expressions involving only numbers.
Example 10.11
Simplify:
 ⓐ$\phantom{\rule{0.2em}{0ex}}{5}^{3}$
 ⓑ$\phantom{\rule{0.2em}{0ex}}{9}^{1}$
Solution
ⓐ  
${5}^{3}$  
Multiply 3 factors of 5.  $5\xb75\xb75$ 
Simplify.  $125$ 
ⓑ  
${9}^{1}$  
Multiply 1 factor of 9.  $9$ 
Try It 10.21
Simplify:
 ⓐ$\phantom{\rule{0.2em}{0ex}}{4}^{3}$
 ⓑ$\phantom{\rule{0.2em}{0ex}}{11}^{1}$
Try It 10.22
Simplify:
 ⓐ$\phantom{\rule{0.2em}{0ex}}{3}^{4}$
 ⓑ$\phantom{\rule{0.2em}{0ex}}{21}^{1}$
Example 10.12
Simplify:
 ⓐ$\phantom{\rule{0.2em}{0ex}}{\left(\frac{7}{8}\right)}^{2}$
 ⓑ$\phantom{\rule{0.2em}{0ex}}{(0.74)}^{2}$
Solution
ⓐ  
${\left(\frac{7}{8}\right)}^{2}$  
Multiply two factors.  $\left(\frac{7}{8}\right)\left(\frac{7}{8}\right)$ 
Simplify.  $\frac{49}{64}$ 
ⓑ  
${(0.74)}^{2}$  
Multiply two factors.  $(0.74)(0.74)$ 
Simplify.  $0.5476$ 
Try It 10.23
Simplify:
 ⓐ$\phantom{\rule{0.2em}{0ex}}{\left(\frac{5}{8}\right)}^{2}$
 ⓑ$\phantom{\rule{0.2em}{0ex}}{(0.67)}^{2}$
Try It 10.24
Simplify:
 ⓐ$\phantom{\rule{0.2em}{0ex}}{\left(\frac{2}{5}\right)}^{3}$
 ⓑ$\phantom{\rule{0.2em}{0ex}}{(0.127)}^{2}$
Example 10.13
Simplify:
 ⓐ$\phantom{\rule{0.2em}{0ex}}{(\mathrm{3})}^{4}$
 ⓑ$\phantom{\rule{0.2em}{0ex}}{\mathrm{3}}^{4}$
Solution
ⓐ  
${\left(\mathrm{3}\right)}^{4}$  
Multiply four factors of −3.  $\left(\mathrm{3}\right)\left(\mathrm{3}\right)\left(\mathrm{3}\right)\left(\mathrm{3}\right)$ 
Simplify.  $81$ 
ⓑ  
${\mathrm{3}}^{4}$  
Multiply two factors.  $(3\xb73\xb73\xb73)$ 
Simplify.  $\mathrm{81}$ 
Notice the similarities and differences in parts ⓐ and ⓑ. Why are the answers different? In part ⓐ the parentheses tell us to raise the (−3) to the 4^{th} power. In part ⓑ we raise only the 3 to the 4^{th} power and then find the opposite.
Try It 10.25
Simplify:
 ⓐ$\phantom{\rule{0.2em}{0ex}}{(\mathrm{2})}^{4}$
 ⓑ$\phantom{\rule{0.2em}{0ex}}{\mathrm{2}}^{4}$
Try It 10.26
Simplify:
 ⓐ$\phantom{\rule{0.2em}{0ex}}{(\mathrm{8})}^{2}$
 ⓑ$\phantom{\rule{0.2em}{0ex}}{\mathrm{8}}^{2}$
Simplify Expressions Using the Product Property of Exponents
You have seen that when you combine like terms by adding and subtracting, you need to have the same base with the same exponent. But when you multiply and divide, the exponents may be different, and sometimes the bases may be different, too. We’ll derive the properties of exponents by looking for patterns in several examples. All the exponent properties hold true for any real numbers, but right now we will only use whole number exponents.
First, we will look at an example that leads to the Product Property.
What does this mean?
How many factors altogether? 

So, we have  
Notice that 5 is the sum of the exponents, 2 and 3.  
We write:  ${x}^{2}\cdot {x}^{3}$ ${x}^{2+3}$ ${x}^{5}$ 
The base stayed the same and we added the exponents. This leads to the Product Property for Exponents.
Product Property of Exponents
If $a$ is a real number and $m,n$ are counting numbers, then
To multiply with like bases, add the exponents.
An example with numbers helps to verify this property.
Example 10.14
Simplify: ${x}^{5}\xb7{x}^{7}.$
Solution
${x}^{5}\xb7{x}^{7}$  
Use the product property, ${a}^{m}\xb7{a}^{n}={a}^{m+n}.$  
Simplify.  ${x}^{12}$ 
Try It 10.27
Simplify: ${x}^{7}\xb7{x}^{8}.$
Try It 10.28
Simplify: ${x}^{5}\xb7{x}^{11}.$
Example 10.15
Simplify: ${b}^{4}\xb7b.$
Solution
${b}^{4}\xb7b$  
Rewrite, $b={b}^{1}.$  ${b}^{4}\xb7{b}^{1}$ 
Use the product property, ${a}^{m}\xb7{a}^{n}={a}^{m+n}.$  
Simplify.  ${b}^{5}$ 
Try It 10.29
Simplify: ${p}^{9}\xb7p.$
Try It 10.30
Simplify: $m\xb7{m}^{7}.$
Example 10.16
Simplify: ${2}^{7}\xb7{2}^{9}.$
Solution
${2}^{7}\xb7{2}^{9}$  
Use the product property, ${a}^{m}\xb7{a}^{n}={a}^{m+n}.$  
Simplify.  ${2}^{16}$ 
Try It 10.31
Simplify: $6\xb7{6}^{9}.$
Try It 10.32
Simplify: ${9}^{6}\xb7{9}^{9}.$
Example 10.17
Simplify: ${y}^{17}\xb7{y}^{23}.$
Solution
${y}^{17}\xb7{y}^{23}$  
Notice, the bases are the same, so add the exponents.  
Simplify.  ${y}^{40}$ 
Try It 10.33
Simplify: ${y}^{24}\xb7{y}^{19}.$
Try It 10.34
Simplify: ${z}^{15}\xb7{z}^{24}.$
We can extend the Product Property of Exponents to more than two factors.
Example 10.18
Simplify: ${x}^{3}\xb7{x}^{4}\xb7{x}^{2}.$^{}
Solution
${x}^{3}\xb7{x}^{4}\xb7{x}^{2}$  
Add the exponents, since the bases are the same.  
Simplify.  ${x}^{9}$ 
Try It 10.35
Simplify: ${x}^{7}\xb7{x}^{5}\xb7{x}^{9}.$
Try It 10.36
Simplify: ${y}^{3}\xb7{y}^{8}\xb7{y}^{4}.$^{}
Simplify Expressions Using the Power Property of Exponents
Now let’s look at an exponential expression that contains a power raised to a power. See if you can discover a general property.
What does this mean?
How many factors altogether? 

So, we have  
Notice that 6 is the product of the exponents, 2 and 3.  
We write:  ${({x}^{2})}^{3}$ ${x}^{2\cdot 3}$ ${x}^{6}$ 
We multiplied the exponents. This leads to the Power Property for Exponents.
Power Property of Exponents
If $a$ is a real number and $m,n$ are whole numbers, then
To raise a power to a power, multiply the exponents.
An example with numbers helps to verify this property.
Example 10.19
Simplify:
 ⓐ$\phantom{\rule{0.2em}{0ex}}{({x}^{5})}^{7}$
 ⓐ$\phantom{\rule{0.2em}{0ex}}{({3}^{6})}^{8}$
Solution
ⓐ  
${\left({x}^{5}\right)}^{7}$  
Use the Power Property, ${\left({a}^{m}\right)}^{n}={a}^{m\xb7n}.$  
Simplify.  ${x}^{35}$ 
ⓑ  
${\left({3}^{6}\right)}^{8}$  
Use the Power Property, ${\left({a}^{m}\right)}^{n}={a}^{m\xb7n}.$  
Simplify.  ${3}^{48}$ 
Try It 10.37
Simplify:
 ⓐ$\phantom{\rule{0.2em}{0ex}}{({x}^{7})}^{4}$
 ⓑ$\phantom{\rule{0.2em}{0ex}}{({7}^{4})}^{8}$
Try It 10.38
Simplify:
 ⓐ$\phantom{\rule{0.2em}{0ex}}{({x}^{6})}^{9}$
 ⓑ$\phantom{\rule{0.2em}{0ex}}{({8}^{6})}^{7}$
Simplify Expressions Using the Product to a Power Property
We will now look at an expression containing a product that is raised to a power. Look for a pattern.
${\left(2x\right)}^{3}$  
What does this mean?  $2x\xb72x\xb72x$ 
We group the like factors together.  $2\xb72\xb72\xb7x\xb7x\xb7x$ 
How many factors of 2 and of $x?$  ${2}^{3}\xb7{x}^{3}$ 
Notice that each factor was raised to the power.  ${\left(2x\right)}^{3}\phantom{\rule{0.2em}{0ex}}\text{is}\phantom{\rule{0.2em}{0ex}}{2}^{3}\xb7{x}^{3}$ 
We write:  ${\left(2x\right)}^{3}$
${2}^{3}\xb7{x}^{3}$ 
The exponent applies to each of the factors. This leads to the Product to a Power Property for Exponents.
Product to a Power Property of Exponents
If $a$ and $b$ are real numbers and $m$ is a whole number, then
To raise a product to a power, raise each factor to that power.
An example with numbers helps to verify this property:
Example 10.20
Simplify: ${(\mathrm{11}x)}^{2}.$
Solution
${\left(\mathrm{11}x\right)}^{2}$  
Use the Power of a Product Property, ${\left(ab\right)}^{m}={a}^{m}{b}^{m}.$  
Simplify.  $121{x}^{2}$ 
Try It 10.39
Simplify: ${(\mathrm{14}x)}^{2}.$
Try It 10.40
Simplify: ${(\mathrm{12}a)}^{2}.$
Example 10.21
Simplify: ${(3xy)}^{3}.$
Solution
${\left(3xy\right)}^{3}$  
Raise each factor to the third power.  
Simplify.  $27{x}^{3}{y}^{3}$ 
Try It 10.41
Simplify: ${(\mathrm{4}xy)}^{4}.$
Try It 10.42
Simplify: ${(6xy)}^{3}.$
Simplify Expressions by Applying Several Properties
We now have three properties for multiplying expressions with exponents. Let’s summarize them and then we’ll do some examples that use more than one of the properties.
Properties of Exponents
If $a,b$ are real numbers and $m,n$ are whole numbers, then
Example 10.22
Simplify: ${({x}^{2})}^{6}{({x}^{5})}^{4}.$
Solution
${({x}^{2})}^{6}{({x}^{5})}^{4}$  
Use the Power Property.  ${x}^{12}\xb7{x}^{20}$ 
Add the exponents.  ${x}^{32}$ 
Try It 10.43
Simplify: ${({x}^{4})}^{3}{({x}^{7})}^{4}.$
Try It 10.44
Simplify: ${({y}^{9})}^{2}{({y}^{8})}^{3}.$
Example 10.23
Simplify: ${(7{x}^{3}{y}^{4})}^{2}.$
Solution
${(7{x}^{3}{y}^{4})}^{2}$  
Take each factor to the second power.  ${(\mathrm{7})}^{2}{({x}^{3})}^{2}{({y}^{4})}^{2}$ 
Use the Power Property.  $49{x}^{6}{y}^{8}$ 
Try It 10.45
Simplify: ${(8{x}^{4}{y}^{7})}^{3}.$
Try It 10.46
Simplify: ${(3{a}^{5}{b}^{6})}^{4}.$
Example 10.24
Simplify: ${(6n)}^{2}(4{n}^{3}).$
Solution
${(6n)}^{2}(4{n}^{3})$  
Raise $6n$ to the second power.  ${6}^{2}{n}^{2}\xb74{n}^{3}$ 
Simplify.  $36{n}^{2}\xb74{n}^{3}$ 
Use the Commutative Property.  $36\xb74\xb7{n}^{2}\xb7{n}^{3}$ 
Multiply the constants and add the exponents.  $144{n}^{5}$ 
Notice that in the first monomial, the exponent was outside the parentheses and it applied to both factors inside. In the second monomial, the exponent was inside the parentheses and so it only applied to the n.
Try It 10.47
Simplify: ${(7n)}^{2}(2{n}^{12}).$
Try It 10.48
Simplify: ${(4m)}^{2}(3{m}^{3}).$
Example 10.25
Simplify: ${(3{p}^{2}q)}^{4}{(2p{q}^{2})}^{3}.$
Solution
${(3{p}^{2}q)}^{4}{(2p{q}^{2})}^{3}$  
Use the Power of a Product Property.  ${3}^{4}{({p}^{2})}^{4}{q}^{4}\xb7{2}^{3}{p}^{3}{({q}^{2})}^{3}$ 
Use the Power Property.  $81{p}^{8}{q}^{4}\xb78{p}^{3}{q}^{6}$ 
Use the Commutative Property.  $81\xb78\xb7{p}^{8}\xb7{p}^{3}\xb7{q}^{4}\xb7{q}^{6}$ 
Multiply the constants and add the exponents for each variable. 
$648{p}^{11}{q}^{10}$ 
Try It 10.49
Simplify: ${({u}^{3}{v}^{2})}^{5}{(4u{v}^{4})}^{3}.$
Try It 10.50
Simplify: ${(5{x}^{2}{y}^{3})}^{2}{(3x{y}^{4})}^{3}.$
Multiply Monomials
Since a monomial is an algebraic expression, we can use the properties for simplifying expressions with exponents to multiply the monomials.
Example 10.26
Multiply: $(4{x}^{2})(5{x}^{3}).$
Solution
$(4{x}^{2})(5{x}^{3})$  
Use the Commutative Property to rearrange the factors.  $4\xb7(\mathrm{5})\xb7{x}^{2}\xb7{x}^{3}$ 
Multiply.  $20{x}^{5}$ 
Try It 10.51
Multiply: $(7{x}^{7})(8{x}^{4}).$
Try It 10.52
Multiply: $(9{y}^{4})(6{y}^{5}).$
Example 10.27
Multiply: $\left(\frac{3}{4}\phantom{\rule{0.1em}{0ex}}{c}^{3}d\right)(12c{d}^{2}).$
Solution
$\left(\frac{3}{4}\phantom{\rule{0.1em}{0ex}}{c}^{3}d\right)(12c{d}^{2})$  
Use the Commutative Property to rearrange the factors. 
$\frac{3}{4}\xb712\xb7{c}^{3}\xb7c\xb7d\xb7{d}^{2}$ 
Multiply.  $9{c}^{4}{d}^{3}$ 
Try It 10.53
Multiply: $\left(\frac{4}{5}\phantom{\rule{0.1em}{0ex}}{m}^{4}{n}^{3}\right)(15m{n}^{3}).$
Try It 10.54
Multiply: $\left(\frac{2}{3}\phantom{\rule{0.1em}{0ex}}{p}^{5}q\right)(18{p}^{6}{q}^{7}).$
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Section 10.2 Exercises
Practice Makes Perfect
Simplify Expressions with Exponents
In the following exercises, simplify each expression with exponents.
${10}^{3}$
${\left(\frac{3}{5}\right)}^{2}$
${(0.4)}^{3}$
${(\mathrm{3})}^{5}$
${\mathrm{3}}^{5}$
${\mathrm{2}}^{6}$
${\left(\frac{1}{4}\right)}^{4}$
${0.1}^{4}$
Simplify Expressions Using the Product Property of Exponents
In the following exercises, simplify each expression using the Product Property of Exponents.
${m}^{4}\xb7{m}^{2}$
${y}^{12}\xb7y$
${5}^{10}\xb7{5}^{6}$
$a\xb7{a}^{3}\xb7{a}^{5}$
${y}^{p}\xb7{y}^{3}$
${x}^{p}\xb7{x}^{q}$
Simplify Expressions Using the Power Property of Exponents
In the following exercises, simplify each expression using the Power Property of Exponents.
${({x}^{2})}^{7}$
${({a}^{3})}^{2}$
${({2}^{8})}^{3}$
${({y}^{12})}^{8}$
${({y}^{3})}^{x}$
${({7}^{a})}^{b}$
Simplify Expressions Using the Product to a Power Property
In the following exercises, simplify each expression using the Product to a Power Property.
${(7x)}^{2}$
${(9n)}^{3}$
${(5ab)}^{3}$
${(5abc)}^{3}$
Simplify Expressions by Applying Several Properties
In the following exercises, simplify each expression.
${({y}^{4})}^{3}\xb7{({y}^{5})}^{2}$
${({b}^{7})}^{5}\xb7{({b}^{2})}^{6}$
${(2y)}^{3}(6y)$
${(4b)}^{2}{(3b)}^{3}$
${(3{y}^{2})}^{4}$
${(2m{n}^{4})}^{5}$
${(\mathrm{10}{u}^{2}{v}^{4})}^{3}$
${\left(\frac{7}{9}\phantom{\rule{0.1em}{0ex}}p{q}^{4}\right)}^{2}$
${(5{r}^{2})}^{3}{(3r)}^{2}$
${(4{x}^{3})}^{3}{(2{x}^{5})}^{4}$
${\left(\frac{1}{2}\phantom{\rule{0.1em}{0ex}}{x}^{2}{y}^{3}\right)}^{4}{(4{x}^{5}{y}^{3})}^{2}$
${\left(\frac{1}{3}\phantom{\rule{0.1em}{0ex}}{m}^{3}{n}^{2}\right)}^{4}{(9{m}^{8}{n}^{3})}^{2}$
${(2p{q}^{4})}^{3}{(5{p}^{6}q)}^{2}$
Multiply Monomials
In the following exercises, multiply the following monomials.
$(\mathrm{10}{y}^{3})(7{y}^{2})$
$(\mathrm{6}{c}^{4})(\mathrm{12}c)$
$\left(\frac{1}{4}\phantom{\rule{0.1em}{0ex}}{a}^{5}\right)(36{a}^{2})$
$(6{m}^{4}{n}^{3})(7m{n}^{5})$
$\left(\frac{5}{8}\phantom{\rule{0.1em}{0ex}}{u}^{3}v\right)(24{u}^{5}v)$
$\left(\frac{2}{3}\phantom{\rule{0.1em}{0ex}}{x}^{2}y\right)\left(\frac{3}{4}\phantom{\rule{0.1em}{0ex}}x{y}^{2}\right)$
$\left(\frac{3}{5}\phantom{\rule{0.1em}{0ex}}{m}^{3}{n}^{2}\right)\left(\frac{5}{9}\phantom{\rule{0.1em}{0ex}}{m}^{2}{n}^{3}\right)$
Everyday Math
Email Janet emails a joke to six of her friends and tells them to forward it to six of their friends, who forward it to six of their friends, and so on. The number of people who receive the email on the second round is ${6}^{2},$ on the third round is ${6}^{3},$ as shown in the table. How many people will receive the email on the eighth round? Simplify the expression to show the number of people who receive the email.
Round  Number of people 

$1$  $6$ 
$2$  ${6}^{2}$ 
$3$  ${6}^{3}$ 
$\dots $  $\dots $ 
$8$  $?$ 
Salary Raul’s boss gives him a $\text{5\%}$ raise every year on his birthday. This means that each year, Raul’s salary is $1.05$ times his last year’s salary. If his original salary was $\text{\$40,000}$, his salary after $1$ year was $\text{\$40,000}(1.05),$ after $2$ years was $\text{\$40,000}{(1.05)}^{2},$ after $3$ years was $\text{\$40,000}{(1.05)}^{3},$ as shown in the table below. What will Raul’s salary be after $10$ years? Simplify the expression, to show Raul’s salary in dollars.
Year  Salary 

$1$  $\text{\$40,000}(1.05)$ 
$2$  $\text{\$40,000}{(1.05)}^{2}$ 
$3$  $\text{\$40,000}{(1.05)}^{3}$ 
$\dots $  $\dots $ 
$10$  $?$ 
Writing Exercises
Explain why ${\mathrm{5}}^{3}={(\mathrm{5})}^{3}$ but ${\mathrm{5}}^{4}\ne {(\mathrm{5})}^{4}.$
Explain why ${x}^{3}\xb7{x}^{5}$ is ${x}^{8},$ and not ${x}^{15}.$
Self Check
ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ After reviewing this checklist, what will you do to become confident for all objectives?