Learning Objectives
By the end of this section, you will be able to:
 Simplify expressions with exponents
 Simplify expressions using the Product Property for Exponents
 Simplify expressions using the Power Property for Exponents
 Simplify expressions using the Product to a Power Property
 Simplify expressions by applying several properties
 Multiply monomials
Be Prepared 6.4
Before you get started, take this readiness quiz.
Simplify: $\frac{3}{4}\xb7\frac{3}{4}.$
If you missed this problem, review Example 1.68.
Be Prepared 6.5
Simplify: $\left(\mathrm{2}\right)\left(\mathrm{2}\right)\left(\mathrm{2}\right).$
If you missed this problem, review Example 1.50.
Simplify Expressions with Exponents
Remember that an exponent indicates repeated multiplication of the same quantity. For example, ${2}^{4}$ means to multiply 2 by itself 4 times, so ${2}^{4}$ means $2\xb72\xb72\xb72$.
Let’s review the vocabulary for expressions with exponents.
Exponential Notation
This is read $a$ to the ${m}^{th}$ power.
In the expression ${a}^{m}$, the exponent $m$ 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 6.16
Simplify: ⓐ ${4}^{3}$ ⓑ ${7}^{1}$ ⓒ ${\left(\frac{5}{6}\right)}^{2}$ ⓓ ${\left(0.63\right)}^{2}.$
Solution
ⓐ  ${4}^{3}$ 
Multiply three factors of 4.  $4\xb74\xb74$ 
Simplify.  $64$ 
ⓑ  ${7}^{1}$ 
Multiply one factor of 7.  $7$ 
ⓒ  ${\left(\frac{5}{6}\right)}^{2}$ 
Multiply two factors.  $\left(\frac{5}{6}\right)\left(\frac{5}{6}\right)$ 
Simplify.  $\frac{25}{36}$ 
ⓓ  ${\left(0.63\right)}^{2}$ 
Multiply two factors.  $\left(0.63\right)\left(0.63\right)$ 
Simplify.  $0.3969$ 
Try It 6.31
Simplify: ⓐ ${6}^{3}$ ⓑ ${15}^{1}$ ⓒ ${\left(\frac{3}{7}\right)}^{2}$ ⓓ ${\left(0.43\right)}^{2}.$
Try It 6.32
Simplify: ⓐ ${2}^{5}$ ⓑ ${21}^{1}$ ⓒ ${\left(\frac{2}{5}\right)}^{3}$ ⓓ ${\left(0.218\right)}^{2}.$
Example 6.17
Simplify: ⓐ ${\left(\mathrm{5}\right)}^{4}$ ⓑ $\text{\u2212}{5}^{4}.$
Solution
ⓐ  ${\left(\mathrm{5}\right)}^{4}$ 
Multiply four factors of $\mathrm{5}$.  $\left(\mathrm{5}\right)\left(\mathrm{5}\right)\left(\mathrm{5}\right)\left(\mathrm{5}\right)$ 
Simplify.  $625$ 
ⓑ  $\text{\u2212}{5}^{4}$ 
Multiply four factors of 5.  $\text{\u2212}\left(5\xb75\xb75\xb75\right)$ 
Simplify.  $\mathrm{625}$ 
Try It 6.33
Simplify: ⓐ ${\left(\mathrm{3}\right)}^{4}$ ⓑ $\text{\u2212}{3}^{4}.$
Try It 6.34
Simplify: ⓐ ${\left(\mathrm{13}\right)}^{2}$ ⓑ $\text{\u2212}{13}^{2}.$
Notice the similarities and differences in Example 6.17ⓐ and Example 6.17ⓑ! Why are the answers different? As we follow the order of operations in part ⓐ the parentheses tell us to raise the $(\mathrm{5})$ to the 4^{th} power. In part ⓑ we raise just the 5 to the 4^{th} power and then take the opposite.
Simplify Expressions Using the Product Property for 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.
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:
The base stayed the same and we added the exponents. This leads to the Product Property for Exponents.
Product Property for Exponents
If $a$ is a real number, and $m\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}n$ are counting numbers, then
To multiply with like bases, add the exponents.
An example with numbers helps to verify this property.
Example 6.18
Simplify: ${y}^{5}\xb7{y}^{6}.$
Solution
Use the product property, a^{m} · a^{n} = a^{m+n}.  
Simplify. 
Try It 6.35
Simplify: ${b}^{9}\xb7{b}^{8}.$
Try It 6.36
Simplify: ${x}^{12}\xb7{x}^{4}.$
Example 6.19
Simplify: ⓐ ${2}^{5}\xb7{2}^{9}$ ⓑ $3\xb7{3}^{4}.$
Solution
 ⓐ
Use the product property, a^{m} · a^{n} = a^{m+n}. Simplify.  ⓑ
Use the product property, a^{m} · a^{n} = a^{m+n}. Simplify.
Try It 6.37
Simplify: ⓐ $5\xb7{5}^{5}$ ⓑ ${4}^{9}\xb7{4}^{9}.$
Try It 6.38
Simplify: ⓐ ${7}^{6}\xb7{7}^{8}$ ⓑ $10\xb7{10}^{10}.$
Example 6.20
Simplify: ⓐ ${a}^{7}\xb7a$ ⓑ ${x}^{27}\xb7{x}^{13}.$
Solution
 ⓐ
Rewrite, a = a^{1}. Use the product property, a^{m} · a^{n} = a^{m+n}. Simplify.  ⓑ
Notice, the bases are the same, so add the exponents. Simplify.
Try It 6.39
Simplify: ⓐ ${p}^{5}\xb7p$ ⓑ ${y}^{14}\xb7{y}^{29}.$
Try It 6.40
Simplify: ⓐ $z\xb7{z}^{7}$ ⓑ ${b}^{15}\xb7{b}^{34}.$
We can extend the Product Property for Exponents to more than two factors.
Example 6.21
Simplify: ${d}^{4}\xb7{d}^{5}\xb7{d}^{2}.$
Solution
Add the exponents, since bases are the same.  
Simplify. 
Try It 6.41
Simplify: ${x}^{6}\xb7{x}^{4}\xb7{x}^{8}.$
Try It 6.42
Simplify: ${b}^{5}\xb7{b}^{9}\xb7{b}^{5}.$
Simplify Expressions Using the Power Property for 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:
We multiplied the exponents. This leads to the Power Property for Exponents.
Power Property for Exponents
If $a$ is a real number, and $m\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}n$ are whole numbers, then
To raise a power to a power, multiply the exponents.
An example with numbers helps to verify this property.
Example 6.22
Simplify: ⓐ ${({y}^{5})}^{9}$ ⓑ ${({4}^{4})}^{7}.$
Solution
ⓐ
Use the power property, (a^{m})^{n} = a^{m·n}.  
Simplify. 
ⓑ
Use the power property.  
Simplify. 
Try It 6.43
Simplify: ⓐ ${({b}^{7})}^{5}$ ⓑ ${({5}^{4})}^{3}.$
Try It 6.44
Simplify: ⓐ ${({z}^{6})}^{9}$ ⓑ ${({3}^{7})}^{7}.$
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. Can you find this 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 and ${\left(2x\right)}^{3}$ is ${2}^{3}\xb7{x}^{3}$.
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 for 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 6.23
Simplify: ⓐ ${(\mathrm{9}d)}^{2}$ ⓑ ${(3mn)}^{3}.$
Solution
 ⓐ
Use Power of a Product Property, (ab)^{m} = a^{m}b^{m}. Simplify.  ⓑ
Use Power of a Product Property, (ab)^{m} = a^{m}b^{m}. Simplify.
Try It 6.45
Simplify: ⓐ ${(\mathrm{12}y)}^{2}$ ⓑ ${(2wx)}^{5}.$
Try It 6.46
Simplify: ⓐ ${(5wx)}^{3}$ ⓑ ${(\mathrm{3}y)}^{3}.$
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 $a\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}b$ are real numbers, and $m\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}n$ are whole numbers, then
Product Property  $\begin{array}{ccc}\hfill {a}^{m}\xb7{a}^{n}& =\hfill & {a}^{m+n}\hfill \end{array}$ 
Power Property  $\begin{array}{ccc}\hfill {({a}^{m})}^{n}& =\hfill & {a}^{m\xb7n}\hfill \end{array}$ 
Product to a Power  $\begin{array}{ccc}\hfill {(ab)}^{m}& =\hfill & {a}^{m}{b}^{m}\hfill \end{array}$ 
All exponent properties hold true for any real numbers $m\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}n$. Right now, we only use whole number exponents.
Example 6.24
Simplify: ⓐ ${({y}^{3})}^{6}{({y}^{5})}^{4}$ ⓑ ${\left(\mathrm{6}{x}^{4}{y}^{5}\right)}^{2}.$
Solution
ⓐ  ${({y}^{3})}^{6}{({y}^{5})}^{4}$ 
Use the Power Property.  ${y}^{18}\xb7{y}^{20}$ 
Add the exponents.  ${y}^{38}$ 
ⓑ  ${\left(\mathrm{6}{x}^{4}{y}^{5}\right)}^{2}$ 
Use the Product to a Power Property.  ${(\mathrm{6})}^{2}{({x}^{4})}^{2}{({y}^{5})}^{2}$ 
Use the Power Property.  ${(\mathrm{6})}^{2}$ 
Simplify.  $36{x}^{8}{y}^{10}$ 
Try It 6.47
Simplify: ⓐ ${({a}^{4})}^{5}{({a}^{7})}^{4}$ ⓑ ${\left(\mathrm{2}{c}^{4}{d}^{2}\right)}^{3}.$
Try It 6.48
Simplify: ⓐ ${\left(\mathrm{3}{x}^{6}{y}^{7}\right)}^{4}$ ⓑ ${({q}^{4})}^{5}{({q}^{3})}^{3}.$
Example 6.25
Simplify: ⓐ ${(5m)}^{2}(3{m}^{3})$ ⓑ ${(3{x}^{2}y)}^{4}{(2x{y}^{2})}^{3}.$
Solution
ⓐ  ${(5m)}^{2}(3{m}^{3})$ 
Raise $5m$ to the second power.  ${5}^{2}{m}^{2}\xb73{m}^{3}$ 
Simplify.  $25{m}^{2}\xb73{m}^{3}$ 
Use the Commutative Property.  $25\xb73\xb7{m}^{2}\xb7{m}^{3}$ 
Multiply the constants and add the exponents.  $75{m}^{5}$ 
ⓑ  ${(3{x}^{2}y)}^{4}{(2x{y}^{2})}^{3}$ 
Use the Product to a Power Property.  $({3}^{4}{x}^{8}{y}^{4})({2}^{3}{x}^{3}{y}^{6})$ 
Simplify.  $(81{x}^{8}{y}^{4})(8{x}^{3}{y}^{6})$ 
Use the Commutative Property.  $81\xb78\xb7{x}^{8}\xb7{x}^{3}\xb7{y}^{4}\xb7{y}^{6}$ 
Multiply the constants and add the exponents.  $648{x}^{11}{y}^{10}$ 
Try It 6.49
Simplify: ⓐ ${(5n)}^{2}(3{n}^{10})$ ⓑ ${({c}^{4}{d}^{2})}^{5}{(3c{d}^{5})}^{4}.$
Try It 6.50
Simplify: ⓐ ${({a}^{3}{b}^{2})}^{6}{(4a{b}^{3})}^{4}$ ⓑ ${(2x)}^{3}(5{x}^{7}).$
Multiply Monomials
Since a monomial is an algebraic expression, we can use the properties of exponents to multiply monomials.
Example 6.26
Multiply: $\left(3{x}^{2}\right)\left(\mathrm{4}{x}^{3}\right).$
Solution
$\left(3{x}^{2}\right)\left(\mathrm{4}{x}^{3}\right)$  
Use the Commutative Property to rearrange the terms.  $3\xb7\left(\mathrm{4}\right)\xb7{x}^{2}\xb7{x}^{3}$ 
Multiply.  $\mathrm{12}{x}^{5}$ 
Try It 6.51
Multiply: $\left(5{y}^{7}\right)\left(\mathrm{7}{y}^{4}\right).$
Try It 6.52
Multiply: $\left(\mathrm{6}{b}^{4}\right)\left(\mathrm{9}{b}^{5}\right).$
Example 6.27
Multiply: $\left(\frac{5}{6}{x}^{3}y\right)\left(12x{y}^{2}\right).$
Solution
$\left(\frac{5}{6}{x}^{3}y\right)\left(12x{y}^{2}\right)$  
Use the Commutative Property to rearrange the terms.  $\frac{5}{6}\xb712\xb7{x}^{3}\xb7x\xb7y\xb7{y}^{2}$ 
Multiply.  $10{x}^{4}{y}^{3}$ 
Try It 6.53
Multiply: $\left(\frac{2}{5}{a}^{4}{b}^{3}\right)\left(15a{b}^{3}\right).$
Try It 6.54
Multiply: $\left(\frac{2}{3}{r}^{5}s\right)\left(12{r}^{6}{s}^{7}\right).$
Media
Access these online resources for additional instruction and practice with using multiplication properties of exponents:
Section 6.2 Exercises
Practice Makes Perfect
Simplify Expressions with Exponents
In the following exercises, simplify each expression with exponents.
ⓐ ${3}^{5}$ ⓑ ${9}^{1}$ ⓒ ${\left(\frac{1}{3}\right)}^{2}$ ⓓ ${\left(0.2\right)}^{4}$
ⓐ ${2}^{6}$ ⓑ ${14}^{1}$ ⓒ ${\left(\frac{2}{5}\right)}^{3}$ ⓓ ${\left(0.7\right)}^{2}$
ⓐ ${\left(\mathrm{6}\right)}^{4}$ ⓑ $\text{\u2212}{6}^{4}$
ⓐ $\text{\u2212}{\left(\frac{1}{4}\right)}^{4}$ ⓑ ${\left(\frac{1}{4}\right)}^{4}$
ⓐ $\text{\u2212}{0.5}^{2}$ ⓑ ${\left(\mathrm{0.5}\right)}^{2}$
Simplify Expressions Using the Product Property for Exponents
In the following exercises, simplify each expression using the Product Property for Exponents.
${d}^{3}\xb7{d}^{6}$
${n}^{19}\xb7{n}^{12}$
ⓐ ${4}^{5}\xb7{4}^{9}$ ⓑ ${8}^{9}\xb78$
ⓐ $y\xb7{y}^{3}$ ⓑ ${z}^{25}\xb7{z}^{8}$
$w\xb7{w}^{2}\xb7{w}^{3}$
${a}^{4}\xb7{a}^{3}\xb7{a}^{9}$
${m}^{x}\xb7{m}^{3}$
${y}^{a}\xb7{y}^{b}$
Simplify Expressions Using the Power Property for Exponents
In the following exercises, simplify each expression using the Power Property for Exponents.
ⓐ ${({m}^{4})}^{2}$ ⓑ ${({10}^{3})}^{6}$
ⓐ ${\left({y}^{3}\right)}^{x}$ ⓑ ${\left({5}^{x}\right)}^{y}$
Simplify Expressions Using the Product to a Power Property
In the following exercises, simplify each expression using the Product to a Power Property.
ⓐ ${(6a)}^{2}$ ⓑ ${(3xy)}^{2}$
ⓐ ${\left(\mathrm{4}m\right)}^{3}$ ⓑ ${(5ab)}^{3}$
Simplify Expressions by Applying Several Properties
In the following exercises, simplify each expression.
ⓐ ${\left({y}^{2}\right)}^{4}\xb7{\left({y}^{3}\right)}^{2}$ ⓑ ${(10{a}^{2}b)}^{3}$
ⓐ ${(\mathrm{2}{r}^{3}{s}^{2})}^{4}$ ⓑ ${\left({m}^{5}\right)}^{3}\xb7{\left({m}^{9}\right)}^{4}$
ⓐ ${(\mathrm{10}{q}^{2}{p}^{4})}^{3}$ ⓑ ${\left({n}^{3}\right)}^{10}\xb7{\left({n}^{5}\right)}^{2}$
ⓐ ${(3x)}^{2}(5x)$ ⓑ ${(5{t}^{2})}^{3}{(3t)}^{2}$
ⓐ ${\left(5a\right)}^{2}{\left(2a\right)}^{3}$ ⓑ ${(\frac{1}{2}{y}^{2})}^{3}{(\frac{2}{3}y)}^{2}$
ⓐ ${\left(4b\right)}^{2}{\left(3b\right)}^{3}$ ⓑ ${(\frac{1}{2}{j}^{2})}^{5}{(\frac{2}{5}{j}^{3})}^{2}$
ⓐ ${\left(\frac{2}{5}{x}^{2}y\right)}^{3}$ ⓑ ${\left(\frac{8}{9}x{y}^{4}\right)}^{2}$
ⓐ ${({m}^{2}n)}^{2}{(2m{n}^{5})}^{4}$ ⓑ ${(3p{q}^{4})}^{2}{(6{p}^{6}q)}^{2}$
Multiply Monomials
In the following exercises, multiply the monomials.
$\left(\mathrm{10}{x}^{5}\right)\left(\mathrm{3}{x}^{3}\right)$
$(\mathrm{6}{c}^{4})(\mathrm{12}c)$
$(\frac{1}{4}{d}^{5})(36{d}^{2})$
$(6{m}^{4}{n}^{3})(7m{n}^{5})$
$\left(\frac{5}{8}{x}^{3}y\right)\left(24{x}^{5}y\right)$
$\left(\frac{3}{5}{m}^{3}{n}^{2}\right)\left(\frac{5}{9}{m}^{2}{n}^{3}\right)$
Mixed Practice
In the following exercises, simplify each expression.
${\left({y}^{4}\right)}^{3}\xb7{\left({y}^{5}\right)}^{2}$
${\left({b}^{7}\right)}^{5}\xb7{\left({b}^{2}\right)}^{6}$
${(3{y}^{2})}^{4}$
${(2m{n}^{4})}^{5}$
${(\mathrm{10}{u}^{2}{v}^{4})}^{3}$
${\left(\frac{7}{9}p{q}^{4}\right)}^{2}$
${(5{r}^{2})}^{3}{(3r)}^{2}$
${(4{x}^{3})}^{3}{(2{x}^{5})}^{4}$
${\left(\frac{1}{3}{m}^{3}{n}^{2}\right)}^{4}{\left(9{m}^{8}{n}^{3}\right)}^{2}$
${(2p{q}^{4})}^{3}{(5{p}^{6}q)}^{2}$
Everyday Math
Email Kate emails a flyer to ten of her friends and tells them to forward it to ten of their friends, who forward it to ten of their friends, and so on. The number of people who receive the email on the second round is ${10}^{2}$, on the third round is ${10}^{3}$, as shown in the table below. How many people will receive the email on the sixth round? Simplify the expression to show the number of people who receive the email.
Round  Number of people 

1  10 
2  ${10}^{2}$ 
3  ${10}^{3}$ 
…  … 
6  ? 
Salary Jamal’s boss gives him a 3% raise every year on his birthday. This means that each year, Jamal’s salary is 1.03 times his last year’s salary. If his original salary was $35,000, his salary after 1 year was $\text{\$}\mathrm{35,000}\left(1.03\right)$, after 2 years was $\text{\$}\mathrm{35,000}{\left(1.03\right)}^{2}$, after 3 years was $\text{\$}\mathrm{35,000}{\left(1.03\right)}^{3}$, as shown in the table below. What will Jamal’s salary be after 10 years? Simplify the expression, to show Jamal’s salary in dollars.
Year  Salary 

1  $\text{\$}\mathrm{35,000}\left(1.03\right)$ 
2  $\text{\$}\mathrm{35,000}{\left(1.03\right)}^{2}$ 
3  $\text{\$}\mathrm{35,000}{\left(1.03\right)}^{3}$ 
…  … 
10  ? 
Clearance A department store is clearing out merchandise in order to make room for new inventory. The plan is to mark down items by 30% each week. This means that each week the cost of an item is 70% of the previous week’s cost. If the original cost of a sofa was $1,000, the cost for the first week would be $\text{\$}\mathrm{1,000}\left(0.70\right)$ and the cost of the item during the second week would be $\text{\$}\mathrm{1,000}{\left(0.70\right)}^{2}$. Complete the table shown below. What will be the cost of the sofa during the fifth week? Simplify the expression, to show the cost in dollars.
Week  Cost 

1  $\text{\$}\mathrm{1,000}\left(0.70\right)$ 
2  $\text{\$}\mathrm{1,000}{\left(0.70\right)}^{2}$ 
3  
4  … 
5  ? 
Depreciation Once a new car is driven away from the dealer, it begins to lose value. Each year, a car loses 10% of its value. This means that each year the value of a car is 90% of the previous year’s value. If a new car was purchased for $20,000, the value at the end of the first year would be $\text{\$}\mathrm{20,000}\left(0.90\right)$ and the value of the car after the end of the second year would be $\text{\$}\mathrm{20,000}{\left(0.90\right)}^{2}$. Complete the table shown below. What will be the value of the car at the end of the eighth year? Simplify the expression, to show the value in dollars.
Year  Cost 

1  $\text{\$}\mathrm{20,000}\left(0.90\right)$ 
2  $\text{\$}\mathrm{20,000}{\left(0.90\right)}^{2}$ 
3  
…  … 
8  ? 
Writing Exercises
Explain why $\text{\u2212}{5}^{3}={(\mathrm{5})}^{3}$ but $\text{\u2212}{5}^{4}\ne {(\mathrm{5})}^{4}$.
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 goals?