Algebra 1 G9

11. Airports  A traveler is walking on a moving walkway in an airport. The See Problem 3.
traveler must walk back on the walkway to get a bag he forgot. The traveler’s
groundspeed is 2 ft > s against the walkway and 6 ft > s with the walkway. What is
the  traveler’s speed off the walkway? What is the speed of the moving walkway?

12. Kayaking  A kayaker paddles upstream from camp to photograph a waterfall and
returns. The kayaker’s speed while traveling upstream and downstream is shown
below. What is the kayaker’s speed in still water? What is the speed of the current?

4 mi/h 7 mi/h

B Apply 13. Money  You have a jar of pennies and quarters. You want to choose 15 coins that

are worth exactly $4.35.

a. Write and solve a system of equations that models the situation.

b. Is your solution reasonable in terms of the original problem? Explain.

Solve each system. Explain why you chose the method you used.

14. 4x + 5y = 3 15. 2x + 7y = -20 16. 5x + 2y = 17
3x - 2y = 8 y = 3x + 7 x - 2y = 8

17. Reasoning  Find A and B so that the system below has the solution (2, 3).

Ax - 2By = 6
3Ax - By = -12

18. Think About a Plan  A tugboat can pull a boat 24 mi downstream in 2 h. Going
upstream, the tugboat can pull the same boat 16 mi in 2 h. What is the speed of the
tugboat in still water? What is the speed of the current?

• How can you use the formula d = rt to help you solve the problem?
• How are the tugboat’s speeds when traveling upstream and downstream related

to its speed in still water and the speed of the current?

Open-Ended  Without solving, decide which method you would use to solve
each system: graphing, substitution, or elimination. Explain.

19. y = 3x - 1 20. 3m - 4n = 1 21. 4s - 3t = 8
y = 4x 3m - 2n = -1 t = -2s - 1

22. Business  A perfume maker has stocks of two perfumes on hand. Perfume A sells
for $15 per ounce. Perfume B sells for $35 per ounce. How much of each should be
combined to make a 3-oz bottle of perfume that can be sold for $63?

STEM 23. Chemistry  In a chemistry lab, you have two vinegars. One is 5% acetic acid, and
one is 6.5% acetic acid. You want to make 200 mL of a vinegar with 6% acetic acid.
How many milliliters of each vinegar do you need to mix together?

Lesson 6-4  Applications of Linear Systems 391

24. Boating  A boat is traveling in a river with a current that has a speed of 1.5 km > h. In
one hour, the boat can travel twice the distance downstream that it can travel
upstream. What is the boat’s speed in still water?

25. Reasoning  A student claims that the best way to solve the system at the right is by y – 3x = 4
substitution. Do you agree? Explain. y – 6x = 12

26. Entertainment  A contestant on a quiz show gets 150 points for every correct
answer and loses 250 points for each incorrect answer. After answering 20 questions,
the contestant has 200 points. How many questions has the contestant answered
correctly? Incorrectly?

C Challenge 27. Number Theory  You can represent the value of any two-digit number with the
expression 10a + b, where a is the tens’ place digit and b is the ones’ place digit. For
example, if a is 5 and b is 7, then the value of the number is 10(5) + 7, or 57. What
two-digit number is described below?

• The ones’ place digit is one more than twice the tens’ place digit.

• The value of the number is two more than five times the ones’ place digit.

28. Mixed Nuts  You want to sell 1-lb jars of mixed peanuts and cashews for $5. You
pay $3 per pound for peanuts and $6 per pound for cashews. You plan to combine
4 parts peanuts and 1 part cashews to make your mix. You have spent $70 on
materials to get started. How many jars must you sell to break even?

Standardized Test Prep

SAT/ACT 29. Last year, one fourth of the students in your class played an instrument. This year,
6 students joined the class. Four of the new students play an instrument. Now, one
third of the students play an instrument. How many students are in your class now?

18 24 30 48
Short
30. Which answer choice shows 2x - y = z correctly solved for y?
Response
y = 2x + z y = 2x - z y = -2x + z y = -2x - z

31. What is an equation of a line passing through the points (3, 1) and (4, 3) written in
slope-intercept form?

Mixed Review

Solve each system using elimination. See Lesson 6-3.
34. 5x + 8y = 40
32. x + 3y = 11 33. 2x + 4y = -12
2x + 3y = 4 -6x + 5y = 2 3x - 10y = -13

Get Ready!  To prepare for Lesson 6-5, do Exercises 35–37.

Solve each inequality. Check your solution. See Lesson 3-4.

35. 3a + 5 7 20 36. 2d - 3 Ú 4d + 2 37. 3(q + 4) … -2q - 8

392 Chapter 6  Systems of Equations and Inequalities

6 Mid-Chapter Quiz MathX

OLMathXL® for School
R SCHO Go to PowerAlgebra.com
L®

FO

Do you know HOW? Write and solve a system of equations to solve each
Solve each system by graphing. Tell whether the problem. Explain why you chose the method you used.
system has one solution, infinitely many solutions,
or no solution. 13. Geometry  The length of a rectangle is 3 times
the width. The perimeter is 44 cm. What are the
1. y = x - 1 dimensions of the rectangle?
y = -3x - 5
2. y3y=-34x4x-=2 -6 14. Farming  A farmer grows only pumpkins and corn
3. y = 3x - 4 on her 420-acre farm. This year she wants to plant
y - 3x = 1 250 more acres of corn than pumpkins. How many
4. y = 3x - 14 acres of each crop should the farmer plant?
y - x = 10
15. Coins  You have a total of 21 coins, all nickels and
Solve each system using substitution. dimes. The total value is $1.70. How many nickels
and how many dimes do you have?
5. y = 2x + 5
y = 6x + 1 16. Business  Suppose you start an ice cream business.
6. x = y + 7 You buy a freezer for $200. It costs you $.45 to make
y - 8 = 2x each single-scoop ice cream cone. You sell each cone
7. 4x + y = 2 for $1.25. How many cones do you need to sell to
3y + 2x = -1 break even?
8. 4x + 9y = 24
y = - 31x + 2 Do you UNDERSTAND?
Reasoning  Without solving, tell which method
Solve each system using elimination. you would choose to solve each system: graphing,
substitution, or elimination. Explain your answer.
9. 2x + 5y = 2
3x - 5y = 53 17. y = 2x - 5
10. 4x + 2y = 34 4y + 8x = 15
10x - 4y = -5
11. 11x - 13y = 89 18. 2y + 7x = 3
-11x + 13y = 107 y - 7x = 9
12. 3x + 6y = 42
-7x + 8y = -109 19. Reasoning  If a system of linear equations has
infinitely many solutions, what do you know
about the slopes and y-intercepts of the graphs
of the equations?

20. Open-Ended  Write a system of equations that you
would solve using substitution.

21. Reasoning  Suppose you write a system of equations
to find a break-even point for a business. You solve
the system and find that it has no solution. What
would that mean in terms of the business?

Chapter 6  Mid-Chapter Quiz 393

6-5 Linear Inequalities MCoamthmemonatCicosreFloStraidtea SSttaannddaarrddss

AM-ARFESI..D91.122.A  G-RraEpIh.4t.h1e2 soGlruatpiohntshteosaoluinteioanrsintoeqaulainliteyarin
itnweoquvarlitaybilnestwasoavahraialfb-plelasnaes .a. h. aAllfs-polaAn-eC.E.D. .AAl.s3o
MMAPF1S,.M91P2.2A, -MCEPD3.,1M.3P 4, MP 6
MP 1, MP 2, MP 3, MP 4, MP 6

Objectives To graph linear inequalities in two variables
To use linear inequalities when modeling real-world situations

You are buying paperback and hardcover Paperback Hardcover
books at a book sale. You can spend $2.50 $4.50
One of these and at most $20. What are the possible
one of those . . . combinations of paperback and hardcover
no, wait. Three of books that you can buy? Explain.
these . . .

MATHEMATICAL

PRACTICES

Lesson A linear inequality in two variables, such as y 7 x - 3, can be formed by replacing the
equal sign in a linear equation with an inequality symbol. A solution of an inequality
Vocabulary in two variables is an ordered pair that makes the inequality true.
• linear inequality
• solution of an Essential Understanding  A linear inequality in two variables has an infinite
number of solutions. These solutions can be represented in the coordinate plane as the
inequality set of all points on one side of a boundary line.

Problem 1 Identifying Solutions of a Linear Inequality

Have you tested Is the ordered pair a solution of y 7 x − 3?
sY oesl.uYtoiou nhsavbeetfeosrteed?
whether ordered A (1, 2) B ( − 3, − 7)
pairs are solutions of y 7 x - 3
equations. Now you will 2 1 - 3 d Write the inequality. S y7x-3
test ordered pairs to see 2 7 -2 ✔
win heqetuhaelirtyt.hey satisfy an d Substitute. S -7 -3 - 3

d Simplify. S -7 7 -6  ✘

(1, 2) is a solution. ( -3, -7) is not a solution.

Got It? 1. a. Is (3, 6) a solution of y … 2 x + 4?
3
b. Reasoning  Suppose an ordered pair is not a solution of

y 7 x + 10. Must it be a solution of y 6 x + 10? Explain.

394 Chapter 6  Systems of Equations and Inequalities

The graph of a linear inequality in two variables consists of all points in the coordinate

plane that represent solutions. The graph is a region called a half-plane that is bounded

by a line. All points on one side of the boundary line are solutions, while all points on

the other side are not solutions.

2 y y ؊x 23x ؉1
y ؊1 y
Ͼ 3 x x Յ
2 2
Each point on a dashed Each point on a solid
Ϫ2 O line is not a solution. A Ϫ2 O 2 line is a solution. A
dashed line is used for Ϫ2 solid line is used for
inequalities with Ͼ or Ͻ.
inequalities with Ն or Յ.

Problem 2 Graphing an Inequality in Two Variables

Why does y = x − 2 What is the graph of y 7 x − 2? y
represent the 2
boundary line? First, graph the boundary line y = x - 2. Since the inequality symbol
For any value of x, the is 7, the points on the boundary line are not solutions. Use a dashed x
corresponding value of y line to indicate that the points are not included in the solution.
is the boundary between Ϫ2 O 2
values of y that are To determine which side of the boundary line to shade, test a point that Ϫ2
greater than x - 2 and is not on the line. For example, test the point (0, 0).
values of y that are less
than x - 2. y7x-2
0 0 - 2 Substitute (0, 0) for (x, y).
0 7 - 2 ✔ (0, 0) is a solution.

Because the point (0, 0) is a solution of the inequality, so are all the points on the same
side of the boundary line as (0, 0). Shade the area above the boundary line.

Got It? 2. What is the graph of y … 12x + 1?

An inequality in one variable can be graphed on a number line or in the coordinate
plane. The boundary line will be a horizontal or vertical line.

Problem 3 Graphing a Linear Inequality in One Variable

Have you graphed What is the graph of each inequality in the coordinate plane?
i nequalities like
these before? A x 7 −1 B y Ú 2
Yes. In Lesson 3-1, you
graphed inequalities in Graph x = -1 using a dashed line. Graph y = 2 using a solid line.
one variable on a number Use (0, 0) as a test point. Use (0, 0) as a test point.
line. Here you graph
them in the coordinate x 7 -1 2y yÚ2 y
plane. Ϫ2 O 2
0 7 -1✔ x 0 Ú 2  ✘ 1 x
Ϫ2 O 2
Shade on the side Shade on the side
of the line that of the line that does
contains (0, 0). not contain (0, 0).

Lesson 6-5  Linear Inequalities 395

Got It? 3. What is the graph of each inequality?
a. x 6 -5 b. y … 2

When a linear inequality is solved for y, the direction of the inequality symbol
determines which side of the boundary line to shade. If the symbol is 6 or …, shade
below the boundary line. If the symbol is 7 or Ú, shade above it.
Sometimes you must first solve an inequality for y before using the method described
above to determine where to shade.

Problem 4 Rewriting to Graph an Inequality
Interior Design  An interior decorator is going to remodel a kitchen. The wall above
the stove and the counter is going to be redone as shown. The owners can spend $420
or less. Write a linear inequality and graph the solutions. What are three possible
prices for the wallpaper and tiles?

Tiled Area Papered Area
3 ft • 4 ft = 12 ft2 3 ft • 8 ft = 24 ft2

Let x = the cost per square foot of the paper. Paper and
Let y = the cost per square foot of the tiles. Tile Costs

Which inequality Write an inequality and solve it for y.
sym bol should
yYoouu m uusset?read the 24x + 12y … 420 Total cost is $420 or less. Tile Cost, y ($) 35
pro blem statement 12y … -24x + 420 Subtract 24x from each side. 30
carefully. Here “$420 y … -2x + 35 Divide each side by 12. 25
or less” means that the 20
solution includes, but Graph y … -2x + 35. The inequality symbol is …, so the boundary line 15 15 25
cannot exceed, $420, is solid and you shade below it. The graph only makes sense in the first 10
so use … . 5
quadrant. Three possible prices per square foot for wallpaper and tile are 0

$5 and $25, $5 and $15, and $10 and $10. 05

Paper Cost, x ($)

Got It? 4. For a party, you can spend no more than $12 on nuts. Peanuts
cost $2>lb. Cashews cost $4>lb. What are three possible combinations of
peanuts and cashews you can buy?

396 Chapter 6  Systems of Equations and Inequalities

Problem 5 Writing an Inequality From a Graph

cC haonicyeosu? eliminate Multiple Choice  Which inequality represents the graph at the right? y
Y es. The boundary line
is solid and the region y … 2x + 1 y Ú 2x + 1 2
below it is shaded, so y … x + 1 y 6 2x + 1 x
you know the inequality
symbol must be …. You The slope of the line is 2 and the y-intercept is 1, so the equation of the Ϫ2 O 2
can eliminate choices boundary line is y = 2x + 1. The boundary line is solid, so the inequality Ϫ2
C and D. symbol is either … or Ú. The symbol must be … , because the region
below the boundary line is shaded. The inequality is y … 2x + 1.

The correct answer is A.

Got It? 5. You are writing an inequality from a graph. The boundary line is dashed
1
and has slope 3 and y-intercept - 2. The area above the line is shaded. What

inequality should you write?

Lesson Check Do you UNDERSTAND? MATHEMATICAL

Do you know HOW? PRACTICES
1. Is ( -1, 4) a solution of the inequality y 6 2x + 5?
5. Vocabulary  How is a linear inequality in two

variables like a linear equation in two variables?

Graph each linear inequality. How are they different?

2. y … -2x + 3 3. x 6 -1 6. Writing  To graph the inequality y 6 3 x + 3, do you
2
shade above or below the boundary line? Explain.
2y
4. What is an inequality that 7. Reasoning  Write an inequality that describes the
represents the graph at Ϫ2 O 2x
the right? Ϫ2 region of the coordinate plane not included in the
graph of y 6 5x + 1.

Practice and Problem-Solving Exercises MATHEMATICAL

PRACTICES

A Practice Determine whether the ordered pair is a solution of the linear inequality. See Problem 1.

8. y … -2x + 1; (2, 2) 9. x 6 2; ( -1, 0) 10. y Ú 3x - 2; (0, 0)
11. y 7 x - 1; (0, 1) 12. y Ú -25x + 4; (0, 0) 13. 3y 7 5x - 12; ( -6, 1)

Graph each linear inequality. See Problem 2.

14. y … x - 1 15. y Ú 3x - 2 16. y 6 -4x - 1 17. y 7 2x - 6
18. y 6 5x - 5 19. y … 21x - 3 20. y 7 -3x 21. y Ú -x

Lesson 6-5  Linear Inequalities 397

Graph each inequality in the coordinate plane. 24. x 7 -2 See Problems 3 and 4.
22. x … 4 23. y Ú -1 28. 4x - y 7 2 25. y 6 -4
26. -2x + y Ú 3 27. x + 3y 6 15 29. -x + 0.25y … -1.75

30. Carpentry  You budget $200 for wooden planks for outdoor furniture. Cedar costs
$2.50 per foot and pine costs $1.75 per foot. Let x = the number of feet of cedar and
let y = the number of feet of pine. What is an inequality that shows how much of
each type of wood can be bought? Graph the inequality. What are three possible
amounts of each type of wood that can be bought within your budget?

31. Business  A fish market charges $9 per pound for cod and $12 per pound for
flounder. Let x = the number of pounds of cod. Let y = the number of pounds of
flounder. What is an inequality that shows how much of each type of fish the store
must sell today to reach a daily quota of at least $120? Graph the inequality. What
are three possible amounts of each fish that would satisfy the quota?

Write a linear inequality that represents each graph. See Problem 5.

32. y 33. y x 34. 2 y
3 x
2 O
x Ϫ2 O2
Ϫ2
O2

B Apply 35. Think About a Plan  A truck that can carry no more than 6400 lb is being used to
transport refrigerators and upright pianos. Each refrigerator weighs 250 lb and each
piano weighs 475 lb. Write and graph an inequality to show how many refrigerators
and how many pianos the truck could carry. Will 12 refrigerators and 8 pianos
overload the truck? Explain.
• What inequality symbol should you use?
• Which side of the boundary line should you shade?

36. Employment  A student with two summer jobs earns $10 per hour at a cafe and
$8 per hour at a market. The student would like to earn at least $800 per month.

a. Write and graph an inequality to represent the situation.
b. The student works at the market for 60 h per month and can work at most

90 h per month. Can the student earn at least $800 each month? Explain
how you can you use your graph to determine this.

37. Error Analysis  A student graphed y Ú 2x + 3 as shown at the right.
Describe and correct the student’s error.

38. Writing  When graphing an inequality, can you always use (0, 0) as a test ؊3 4 2
point to determine where to shade? If not, how would you choose a test point? 0

C Challenge 39. Music Store  A music store sells used CDs for $5 each and buys used CDs
for $1.50 each. You go to the store with $20 and some CDs to sell. You
want to have at least $10 left when you leave the store. Write and graph an
inequality to show how many CDs you could buy and sell.

398 Chapter 6  Systems of Equations and Inequalities

40. Groceries  At your grocery store, milk normally costs $3.60 per gallon. Ground beef
costs $3 per pound. Today there are specials: Milk is discounted $.50 per gallon,
and ground beef is 20% off. You want to spend no more than $20. Write and graph
a linear inequality to show how many gallons of milk and how many pounds of
ground beef you can buy today.

41. Reasoning  You are graphing a linear inequality of the form y 7 mx + b. The
point (1, 2) is not a solution, but (3, 2) is. Is the slope of the boundary line positive,
negative, zero, or undefined? Explain.

Standardized Test Prep

SAT/ACT 42. What is the equation of the graph shown? y
4
y + x Ú -3 x - y 7 -3
y - x Ú 3 y 7 -x + 3 x
2
43. You secure pictures to your scrapbook using 3 stickers. You started with Ϫ2 O

24 stickers. There are now 2 pictures in your scrapbook. You write the equation
3(x + 2) = 24 to find the number x of additional pictures you can put in your
scrapbook. How many more pictures can you add?

4 8

6 12

Short 44. At Market A, 1-lb packages of rice are sold for the price shown. At BU L K
Response Market B, rice is sold in bulk for the price shown. For each market, write RI CE
a function describing the cost of buying rice in terms of the weight. How
are the domains of the two functions different?

$2.00 $2.00/lb

Mixed Review See Lesson 6-4.
See Lesson 3-6.
45. Small Business  An electrician spends $12,000 on initial costs to start a new See Lesson 6-1.
business. He estimates his expenses at $25 per day. He expects to earn $150 per
day. If his estimates are correct, after how many working days will he break even?

46. What compound inequality represents the phrase “all real numbers that are
greater than 2 and less than or equal to 7”? Graph the solutions.

Get Ready!  To prepare for Lesson 6-6, do Exercises 47–49.

Solve each system by graphing. Tell whether the system has one solution,

infinitely many solutions, or no solution.

47. y = 23x 48. 3x + y = 6 49. x + y = 11
-2x + y = 3 2x - y = 4 x + y = 16

Lesson 6-5  Linear Inequalities 399

6-6 Systems of MCoamthmemonatCicosreFloStraidtea SSttaannddaarrddss
Linear Inequalities
MA-ARFESI..D91.122.A  G-RraEpIh.4..1. 2. t hGersaoplhut.io. n. tsheetstoluatisoynstseemt toof
lainseyasrteimneqouf alilniteieasr inetqwuoalvitaierisabinletswaos vtahreiainbtlesrsaescttihoen
ionftethrseecotirorensopfotnhdeincgorhraeslfp-polnadniensg. half-planes.

MP 1, MP 2, MP 3, MP 4, MP 7

Objectives To solve systems of linear inequalities by graphing
To model real-world situations using systems of linear inequalities

See how many You want to buy at least 6 new ring RING TONES
combinations tones from a Web site, but you cannot
you can find spend more than $15. How many PREMIUM $1.50 BUY NOW
that satisfy this premium ring tones and how many top-10
situation. ring tones can you buy? Explain how you TOP 10 $3.00 BUY NOW
found your answer.

MATHEMATICAL A system of linear inequalities is made up of two or more linear inequalities. A
solution of a system of linear inequalities is an ordered pair that makes all the
PRACTICES

Lesson inequalities in the system true. The graph of a system of linear inequalities is the set of
points that represent all of the solutions of the system.
Vocabulary
• system of linear Essential Understanding  You can graph the solutions of a system of linear
inequalities in the coordinate plane. The graph of the system is the region where the
inequalities graphs of the individual inequalities overlap.
• solution of a

system of linear
inequalities

Problem 1 Graphing a System of Inequalities

Have you seen a What is the graph of the system? y * 2x − 3
problem like this 2x + y + 2
before?
Yes. The solution of a Graph y 6 2x - 3 and 2x + y 7 2.
system of equations is
shown by the intersection The blue region y x The green region
of two lines. The represents solutions of 2 4 represents solutions of
solutions of a system of 2x + y > 2. both inequalities.
inequalities are shown by O2
the intersection of two The yellow region
shaded areas. represents solutions of
y < 2x − 3.

The system’s solutions lie in the green region where the graphs overlap.

400 Chapter 6  Systems of Equations and Inequalities

Problem 2

Check (3, 0) is in the green region. See if (3, 0) satisfies both inequalities.

y 2x - 3 d Write both inequalities. S 2x + y 2

0 2(3) - 3 d Substitute (3, 0) for (x, y). S 2(3) + 0 2

0 6 3 ✔ d Simplify. The solution checks. S 672 ✔

Got It? 1. What is the graph of the system? y Ú -x + 5
-3x + y … -4


You can combine your knowledge of linear equations with your knowledge of
inequalities to describe a graph using a system of inequalities.

Have you seen a Problem 2 Writing a System of Inequalities From a Graph
problem like this
one before? What system of inequalities is represented by the graph below?
Yes. You wrote an y
inequality from a graph
in Lesson 6-5. Now you’ll 2 x
write two inequalities. Ϫ2 O 246

To write a system that is represented by the graph, write an inequality that represents
the yellow region and an inequality that represents the blue region.

y

The red boundary line The blue boundary line
disoyesϭnϪot12inxcϩlud5e.
The region 2 is y ϭ x Ϫ 1. The region
the line,
includes the boundary
only points below. The
inequality is y ϽϪ12x ϩ 5. x line and points above.
The inequality is y Ն x Ϫ 1.
Ϫ 246

The graph shows the intersection of the system y 6 -12x + 5 and y Ú x - 1. y

Got It? 2. a. What system of inequalities is represented by the graph? 2 x
b. Reasoning  In part (a), is the point where the boundary Ϫ2 O 2

lines intersect a solution of the system? Explain.

You can model many real-world situations by writing and graphing systems of linear 401
inequalities. Some real-world situations involve three or more restrictions, so you must
write a system of at least three inequalities.

Lesson 6-6  Systems of Linear Inequalities

Problem 3 Using a System of Inequalities

Time Management  You are planning what to do after school. You can spend at most

6 h daily playing basketball and doing homework. You want to spend less than 2 h
1
playing basketball. You must spend at least 1 2 h on homework. What is a graph

showing how you can spend your time?

• At most 6 h playing basketball To find different ways Write and graph an
you can spend your inequality for each
and doing homework time restriction. Find the region
where all three restrictions
•• ALetslseathsat n1212 h playing basketball are met.
h doing homework

Let x = the number of hours playing basketball. Hours of Homework, y After-School Activities
Let y = the number of hours doing homework. 6 xϽ2
4 xϩyՅ6
Write a system of inequalities. 2 y Ն 112

x + y … 6 At most 6 h of basketball and homework 0
02 4 6 8
x 6 2 Less than 2 h of basketball Hours of Basketball, x
y Ú 1 21
At least 1 1 h of homework
2

Graph the system. Because time cannot be negative,
the graph makes sense only in the first quadrant. The
solutions of the system are all of the points in the shaded
region, including the points on the solid boundary lines.

Got It? 3. You want to build a fence for a rectangular dog run. You want the run to
be at least 10 ft wide. The run can be at most 50 ft long. You have 126 ft of

fencing. What is a graph showing the possible dimensions of the dog run?

Lesson Check

Do you know HOW? Do you UNDERSTAND? MATHEMATICAL

PRACTICES

1. What is the graph of the system? y 7 3x - 2 4. Vocabulary  How can you determine whether
2y - x … 6
an ordered pair is a solution of a system of
y
2. What system of inequalities linear inequalities?
is represented by the graph at Ϫ2 O
the right? Ϫ2 x 5. Reasoning  Suppose you are graphing a system of
2 two linear inequalities, and the boundary lines for
3. Cherries cost +4>lb. Grapes cost the inequalities are parallel. Does that mean that the
+2.50>lb. You can spend no more system has no solution? Explain.
than $15 on fruit, and you need
at least 4 lb in all. What is a graph 6. Writing  How is finding the solution of a system of
showing the amount of each fruit inequalities different from finding the solution of a
you can buy? system of equations? How is it the same? Explain.

402 Chapter 6  Systems of Equations and Inequalities

Practice and Problem-Solving Exercises MATHEMATICAL

PRACTICES

A Practice Determine whether the ordered pair is a solution of the given system. See Problem 1.

7. (2, 12); 8. (8, 2); 9. ( -3, 17);
y 7 -5x + 2
y 7 2x + 4 3x - 2y … 17 y Ú -3x + 7
y 6 3x + 7 0.3x + 4y 7 9

Solve each system of inequalities by graphing.

10. y 6 2x + 4 11. y 6 2x + 4 12. y 7 2x + 4
-3x - 2y Ú 6 2x - y … 4 2x - y … 4

13. y 7 14x 14. y 6 2x - 3 15. y … -31x + 7
y … -x + 4 y 7 5 y Ú -x + 1

16. x + 2y … 10 17. y Ú -x + 5 18. y … 0.75x - 2
x + 2y Ú 9 y … 3x - 4 y 7 0.75x - 3

19. 8x + 4y Ú 10 20. 2x - 41y 6 1 21. 6x - 5y 6 15
3x - 6y 7 12 4x + 8y 7 4 x + 2y Ú 7

Write a system of inequalities for each graph. See Problem 2.

22. y 23. y

2 x x
2
Ϫ2 O Ϫ2 O
Ϫ2 Ϫ2

24. y 25. y

2 x Ϫ2 O x
Ϫ2 O 2 Ϫ2 2

26. Earnings  Suppose you have a job mowing lawns that pays $12 per hour. You See Problem 3.
also have a job at a clothing store that pays $10 per hour. You need to earn at least
$350 per week, but you can work no more than 35 h per week. You must work a
minimum of 10 h per week at the clothing store. What is a graph showing how
many hours per week you can work at each job?

27. Driving  Two friends agree to split the driving on a road trip from Philadelphia,
Pennsylvania, to Denver, Colorado. One friend drives at an average speed of
60 mi>h. The other friend drives at an average speed of 55 mi>h. They want to drive
at least 500 mi per day. They plan to spend no more than 10 h driving each day. The
friend who drives slower wants to drive fewer hours. What is a graph showing how
they can split the driving each day?

Lesson 6-6  Systems of Linear Inequalities 403

B Apply 28. Think About a Plan  You are fencing in a rectangular area for a garden. You have
only 150 ft of fence. You want the length of the garden to be at least 40 ft. You want
the width of the garden to be at least 5 ft. What is a graph showing the possible
dimensions your garden could have?
• What variables will you use? What will they represent?
• How many inequalities do you need to write?

29. a. Graph the system y 7 3x + 3 and y … 3x - 5.
b. Writing  Will the boundary lines y = 3x + 3 and y = 3x - 5 ever intersect? How

do you know?

c. Do the shaded regions in the graph from part (a) overlap?

d. Does the system of inequalities have any solutions? Explain.

30. Error Analysis  A student graphs the system as shown below. Describe and correct

the student’s error.

y
3

x ≥ ؊2 x

y>2 O2
؊2
1 x ؉ y ≥ 0
2

31. Gift Certificates  You received a $100 gift certificate to a clothing store. The
store sells T-shirts for $15 and dress shirts for $22. You want to spend no more
than the amount of the gift certificate. You want to leave at most $10 of the gift
certificate unspent. You need at least one dress shirt. What are all of the possible
combinations of T-shirts and dress shirts you could buy?

32. a. Geometry  Graph the system of linear inequalities. xÚ2 y I
b. Describe the shape of the solution region. y Ú -3 II 2
c. Find the vertices of the solution region. x+y…4 x
d. Find the area of the solution region. Ϫ2 O 1
y … - 32x - 2 III IV
33. Which region represents the solution of the system? 3y - 9x Ú 6
I III
II IV

Open-Ended  Write a system of linear inequalities with the given characteristic.

34. All solutions are in Quadrant III. 35. There are no solutions.

C Challenge 36. Business  A jeweler plans to produce a ring made of silver and gold. The price of
gold is about $25 per gram. The price of silver is approximately $.40 per gram. She

considers the following in deciding how much gold and silver to use in the ring.
• The total mass must be more than 10 g but less than 20 g.
• The ring must contain at least 2 g of gold.
• The total cost of the gold and silver must be less than $90.
a. Write and graph the inequalities that describe this situation.
b. For one solution, find the mass of the ring and the cost of the gold and silver.

404 Chapter 6  Systems of Equations and Inequalities

37. Solve 0 y 0 Ú x. (Hint: Write two inequalities and then graph them.)

38. Student Art  A teacher wants to post a row of student artwork on a wall that is 20 ft
long. Some pieces are 8.5 in. wide. Other pieces are 11 in. wide. She is going to leave
3 in. of space to the left of each art piece. She wants to post at least 16 pieces of art.
Write and graph a system of inequalities that describes how many pieces of each
size she can post.

PERFORMANCE TASK

Apply What You’ve Learned MATHEMATICAL

PRACTICES

MP 2

Look back at the information on page 363 about Miguel’s exercise program at

the gym. Choose from the following inequalities to complete the sentences

below. In each inequality, x represents the number of minutes Miguel spends

on the stair machine and y represents the number of minutes Miguel spends on

the rowing machine.

x + y … 60 2y … x x - y Ú 30 x + y Ú 30

2y Ú x y Ú 2x x + 2y Ú 30 x + y Ú 60

a. An inequality representing the first condition given by Miguel’s trainer is ? .

b. An inequality representing the second condition given by Miguel’s trainer is ? .

c. An inequality representing the third condition given by Miguel’s trainer is ? .

Lesson 6-6  Systems of Linear Inequalities 405

Concept Byte Graphing Linear CMoamthmemonatCicosreFloStraidtea SSttaannddaarrddss
Inequalities
Use With Lesson 6-6 AM-ARFESI..D91.122.A  G-RraEpIh.4t.h1e2 soGlruatpiohntshteosaoluinteioanrsintoeqaulainliteyarin
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TECHNOLOGY stheet tsoolautsiyosntesmetotof lainseyasrteinmeqoufalliinteieasr.in. e. qualities . . .

MP 5

A graphing calculator can show the solutions of an inequality or a system of X = Plot1 Plot2 Plot3
inequalities. To enter an inequality, press apps and scroll down to select INEQUAL. \Y1 = 3X – 7 Ն
Move the cursor over the = symbol for one of the equations. Notice the inequality \Y2 =
symbols at the bottom of the screen, above the keys labeled F2–F5. Change  \Y3 =
the = symbol to an inequality symbol by pressing alpha followed by one of F2–F5. \Y4 =
\Y5 =
1 \Y6 =
ϭϽ Յ Ͼ
Graph the inequality y * 3x − 7.
1. Move the cursor over the = symbol for Y1. Press alpha and F2 to select

the 6 symbol.
2. Enter the given inequality as Y1.
3. Press graph to graph the inequality.

2

Graph the system. y * −2x − 3
y#x+4

4. Move the cursor over the = symbol for Y1. Press alpha and F2 to select
the 6 symbol. Enter the first inequality as Y1.

5. Then move the cursor over the = symbol for Y2, and press alpha and F5 to select
the Ú symbol. Enter the second inequality as Y2.

6. Press graph to graph the system of inequalities.

Exercises

Use a graphing calculator to graph each inequality. Sketch your graph.

7. y … x 8. y 7 5x - 9 9. y Ú -1 10. y 6 -x + 8

Use a graphing calculator to graph each system of inequalities. Sketch 14. y Ú 2x - 2
your graph. y … 2x - 4

11. y Ú -x + 3 12. y 7 x 13. y Ú -1
y … x + 2 y Ú -2x + 5 y 6 0.5x - 2

406 Concept Byte  Graphing Linear Inequalities

6 Pull It All Together

RMANCPERFOE TASKCompleting the Performance Task

To solve these Look back at your results from the Apply What You’ve Learned section in Lessons 6-2 and
problems you 6-6. Use the work you did to complete the following.
will pull together 1. Solve the problem in the Task Description on page 363 by finding the number of
many concepts
and skills that minutes Ashley should use each exercise machine, and the maximum number of
you have studied minutes Miguel can use the rowing machine. Show all your work and explain each step
about systems of your solution.
of equations and
inequalities. 2. Reflect  Choose one of the Mathematical Practices below and explain how you applied
it in your work on the Performance Task.

MP 1: Make sense of problems and persevere in solving them.

MP 2: Reason abstractly and quantitatively.

MP 4: Model with mathematics.

On Your Own

Brittany uses the ski machine and the treadmill at the gym for an exercise program. Her
trainer wants her on an exercise program that meets these two conditions:
• Brittany will exercise for at least 45 minutes and at most 1 hour and 15 minutes, dividing

her time between the ski machine and the treadmill.

• Brittany will spend at least three times as much time on the treadmill as on the ski
machine.

a. Write a system of inequalities that models the relationships between the amounts of
time Brittany spends on the two machines.

b. Find the minimum number of minutes Brittany can spend on the treadmill.

Chapter 6  Pull It All Together 407

6 Chapter Review

Connecting and Answering the Essential Questions

1 Solving Equations Solving Systems of Equations
and Inequalities (Lessons 6-1, 6-2, and 6-3)
There are several ways
to solve systems of 2 y y=x
equations and inequalities, x y = -3x - 4
including graphing and
using equivalent forms of Ϫ2 2 The solution
equations and inequalities (؊1, ؊1)
within the system. The
number of solutions depends is (-1, -1).
on the type of system.
Linear Inequalities (Lessons 6-5 and 6-6)
y
x

2 Modeling Applying Linear Systems x
You can represent many (Lesson 6-4)
real-world mathematical
problems algebraically. y income
When you need to find two
unknowns, you may be able expenses
to write and solve a system
of equations. break-even point

Chapter Vocabulary • linear inequality (p. 394) • substitution method (p. 372)
• solution of an inequality (p. 394) • system of linear equations (p. 364)
• consistent (p. 365) • solution of a system of linear • system of linear inequalities (p. 400)
• dependent (p. 365)
• elimination method (p. 378) equations (p. 364)
• inconsistent (p. 365) • solution of a system of linear
• independent (p. 365)
inequalities (p. 400)

Choose the correct term to complete each sentence.
1. A system of equations that has no solution is said to be ? .

2. You can solve a system of equations by adding or subtracting the equations in such
a way that one variable drops out. This is called the ? method.

3. Two or more linear equations together form a(n) ? .

408 Chapter 6  Chapter Review

6-1  Solving Systems by Graphing

Quick Review Exercises

One way to solve a system of linear equations is by graphing Solve each system by graphing. Check your answer.
each equation and finding the intersection point of the
graph, if one exists. 4. y = 3x + 13 5. y = -x + 4
y = x - 3 y = 3x + 12

Example y = −2x + 2 6. y = 2x + 3 7. y = 1.5x + 2
y = 0.5x − 3 y = 31x - 2 4.5x - 3y = -9
What is the solution of the system? 8. y = -2x - 21
y = x - 7 9. y = x + 1
2x - 2y = -2
y = - 2x + 2 Slope is - 2; y-intercept is 2.
y = 0.5x - 3 Slope is 0.5; y-intercept is - 3. 10. Songwriting  Jay has written 24 songs to date. He
writes an average of 6 songs per year. Jenna started
The lines appear to intersect at y writing songs this year and expects to write about
(2, -2). Check if (2, -2) makes 2 12 songs per year. How many years from now will
both equations true. Jenna have written as many songs as Jay? Write and
x graph a system of equations to find your answer.
-2 = -2(2) + 2 ✔
Ϫ2 O 2 11. Reasoning  Describe the graph of a system of
-2 = 0.5(2) - 3 ✔ Ϫ2 equations that has no solution.

So, the solution is (2, -2).

6-2  Solving Systems Using Substitution

Quick Review Exercises

You can solve a system of equations by solving one Solve each system using substitution. Tell whether the
equation for one variable and then substituting the system has one solution, infinitely many solutions, or
expression for that variable into the other equation. no solution.

Example 12. y = 2x - 1 13. -x + y = -13
2x + 2y = 22
What is the solution of the system? y = − 1 x 3x - y = 19
3 14. 2x + y = -12
-4x - 2y = 30 15. 1 y = 7 x + 5
3x + 3y = −18 3 3 3
x - 3y = 5
3x + 3y = -18
Write the second equation. 16. y = x - 7 17. 3x + y = -13
3x - 3y = 21 -2x + 5y = -54
3x + 3( - 1 x) = - 18 Substitute - 1 x for y.
3 3

2x = - 18 Simplify. 18. Business  The owner of a hair salon charges $20
more per haircut than the assistant. Yesterday the
x = -9 Solve for x. assistant gave 12 haircuts. The owner gave 6 haircuts.
The total earnings from haircuts were $750. How
y = - 31( - 9) Sfiursbts etiqtuutaetio-n9. for x in the much does the owner charge for a haircut? Solve by
writing and solving a system of equations.
y=3

The solution is ( - 9, 3).

Chapter 6  Chapter Review 409

6-3 and 6-4  Solving Systems Using Elimination; Applications of Systems

Quick Review Exercises

You can add or subtract equations in a system to eliminate Solve each system using elimination. Tell whether the
a variable. Before you add or subtract, you may have to system has one solution, infinitely many solutions, or
multiply one or both equations by a constant to make no solution.
eliminating a variable possible.
19. x + 2y = 23 20. 7x + y = 6

Example 5x + 10y = 55 5x + 3y = 34

What is the solution of the system? 3x + 2y = 41 21. 5x + 4y = -83 22. 9x + 12 y = 51
5x − 3y = 24 3x - 3y = -12 7x + 13 y = 39

3x + 2y = 41 Multiply by 3. 9x + 6y = 123 23. 4x + y = 21 24. y = 3x - 27
5x - 3y = 24 Multiply by 2. 10x - 6y = 48 -2x + 6y = 9 x - 31 y = 9

19x + 0 = 171 25. Flower Arranging  It takes a florist 3 h 15 min to
x=9 make 3 small centerpieces and 3 large centerpieces.
It takes 6 h 20 min to make 4 small centerpieces
and 7 large centerpieces. How long does it take to
3x + 2y = 41 Write the first equation. make each small centerpiece and each large
3(9) + 2y = 41 Substitute 9 for x. centerpiece? Write and solve a system of equations
y = 7 Solve for y. to find your answer.

The solution is (9, 7).

6-5 and 6-6  Linear Inequalities and Systems of Inequalities

Quick Review Exercises

A linear inequality describes a region of the coordinate Solve each system of inequalities by graphing.
plane with a boundary line. Two or more inequalities form
a system of inequalities. The system’s solutions lie where 26. y Ú x + 4 27. 4y 6 -3x
the graphs of the inequalities overlap. y 6 2x - 1 3
y 6 - 4 x
28. 2x - y 7 0
3x + 2y … -14 29. x + 0.5y Ú 5.5

Example 0.5x + y 6 6.5

What is the graph of the system? y + 2x − 4 30. y 6 10x 31. 4x + 4 7 2y
y " −x + 2 y 7 x - 5 3x - 4y Ú 1

Graph the boundary lines y 32. Downloads  You have 60 megabytes (MB) of space
y = 2x - 4 and y = -x + 2. left on your portable media player. You can choose
For y 7 2x - 4, use a dashed Ϫ2 O x to download song files that use 3.5 MB or video files
boundary line and shade above Ϫ2 2 that use 8 MB. You want to download at least 12 files.
it. For y … -x + 2, use a solid What is a graph showing the numbers of song and
boundary line and shade below. The video files you can download?

green region of overlap contains the

system’s solutions.

410 Chapter 6  Chapter Review

6 Chapter Test athXM

MathXL® for SchoolOL
R SCHO Go to PowerAlgebra.com
L®

FO

Do you know HOW? 13. Gardening  A farmer plans to create a rectangular
Solve each system by graphing. Tell whether the garden that he will enclose with chicken wire. The
system has one solution, infinitely many solutions, garden can be no more than 30 ft wide. The farmer
or no solution. would like to use at most 180 ft of chicken wire.

1. y = 3x - 7 a. Write a system of linear inequalities that models
y = -x + 1 this situation.

2. x + 3y = 12 b. Graph the system to show all possible solutions.
x = y - 8
Write a system of equations to model each situation.
3. x + y = 5 Solve by any method.
x + y = -2
14. Education  A writing workshop enrolls novelists and
Solve each system using substitution. poets in a ratio of 5 : 3. There are 24 people at the
workshop. How many novelists are there? How many
4. y = 4x - 7 poets are there?
y = 2x + 9
STEM 15. Chemistry  A chemist has one solution containing
5. 8x + 2y = -2 30% insecticide and another solution containing
y = -5x + 1 50% insecticide. How much of each solution should
the chemist mix to get 200 L of a 42% insecticide?
6. y + 2x = -1
y - 3x = -16 Do you UNDERSTAND?
16. Open-Ended  Write a system of two linear equations
Solve each system using elimination.
that has no solution.
7. 4x + y = 8
-3x - y = 0 17. Error Analysis  A student concluded that ( -2, -1)
is a solution of the inequality y 6 3x + 2, as shown
8. 2x + 5y = 20 below. Describe and correct the student’s error.
3x - 10y = 37
y < 3x + 2
9. 3x + 2y = -10 -2 <? 3(-1) + 2
2x - 5y = 3 -2 < -1 ✔

Solve each system of inequalities by graphing. 18. Reasoning  Consider a system of two linear
equations in two variables. If the graphs of the
10. y 7 4x - 1 equations are not the same line, is it possible for the
y … -x + 4 system to have infinitely many solutions? Explain.

11. x 7 -3 Reasoning  Suppose you add two linear equations that
-3x + y Ú 6 form a system, and you get the result shown below.
How many solutions does the system have?
12. Garage Sale  You go to a garage sale. All the items
cost $1 or $5. You spend less than $45. Write and 19. x = 8 20. 0 = 4 21. 0 = 0
graph a linear inequality that models the situation.

Chapter 6  Chapter Test 411

6 Common Core Cumulative ASSESSMENT
Standards Review

Some questions on tests ask you Melissa keeps a jar for holding change. TIP 2
to solve a problem that involves The jar holds 21 coins. All of the coins are
a system of equations. Read the quarters and nickels. The total amount in Make sure to answer the
sample question at the right. the jar is $3.85. How many quarters are in question asked. Here
Then follow the tips to answer the jar? you only need to find the
the question. number of quarters.
3
TIP 1 Think It Through
7
When writing an Write a system of equations.
equation, try to use 14 q + n = 21
variables that make 0.25q + 0.05n = 3.85
sense for the problem. 21 Solve the first equation for n and
Instead of using x and y, substitute to find q.
use q for quarters and 0.25q + 0.05n = 3.85
n for nickels. 0.25q + 0.05(21 - q) = 3.85
0.2q + 1.05 = 3.85
q = 14
The correct answer is C.

VLVeooscsacoabnubluarlayry Builder Selected Response

As you solve test items, you must understand Read each question. Then write the letter of the correct
the meanings of mathematical terms. Choose the answer on your paper.
correct term to complete each sentence.
1. A group of students are going on a field trip. If the
A . The (substitution, elimination) method is a way to
solve a system of equations in which you replace group takes 3 vans and 1 car, 22 students can be
one variable with an equivalent expression
containing the other variable. transported. If the group takes 2 vans and 4 cars,

B. A linear (equation, inequality) is a mathematical 28 students can be transported. How many students
sentence that describes a region of the coordinate
plane having a boundary line. can fit in each van?

C. A(n) (x-intercept, y-intercept) is the coordinate of a 2 6
point where a graph intersects the y-axis.
4 10
D . The (area, perimeter) of a figure is the distance
around the outside of the figure. 2. Greg’s school paid $1012.50 for 135 homecoming
T-shirts. How much would it cost the school to
E. A (  function rule, relation) is an equation that purchase 235 T-shirts?
can be used to find a unique range value given a
domain value. $750.00

$1762.50

$2025.00

$2775.00

412 Chapter 6  Common Core Cumulative Standards Review

3. What is the solution of 12(x + 1) = 36? 1 0. Martin used 400 ft of fencing to enclose a rectangular

12  2 area in his backyard. Isabella wants to enclose a

8 -2 similar area that is twice as long and twice as wide as

the one in Martin’s backyard. How much fencing does

4. Which equation describes a line with slope 12 and Isabella need?
y-intercept 4?
800 ft 1600 ft

y = 12x + 4 y = 4x + 12 1200 ft 2000 ft

y = 12(x + 4) y = x + 3 11. At the Conic Company, a new employee’s earnings E,

5. What is the solution of the system 2y x in dollars, can be calculated using the function
of equations shown at the right? 2 E = 0.05s + 30,000, where s represents the employee’s
Ϫ2 total sales, in dollars. All of the new employees earned
(1, -1) Ϫ2
( -1, 1) between $50,000 and $60,000 last year. Which value is
( -1, -1)
(1, 1) in the domain of the function?

$34,000 $430,000

$300,000 $3,400,000

6. The width of Ben’s rectangular family room is 3 ft less 1 2. Hilo’s class fund has $65. The class is having a car
wash to raise more money for a trip. The graph below
than the length. The perimeter is 70 ft. Which equation models the amount of money the class will have if it
can be used to find the length of the room? charges $4 for each car washed.

70 = / - 3 70 = 2(/ - 3)

70 = 2/ - 3 70 = 2(2/ - 3) 100

7. Marisa’s Flower Shop charges $3 per rose plus $16 80

for a delivery. Chris wants to have a bouquet of roses Class Fund, y ($)

delivered to his mother. Which value is in the range of 60

the function that gives the bouquet’s cost in terms of

the number of roses? 40

$16 $34 20

$27 $48 0
0 2 4 6 8 10 12
8. Which number is a solution of 8 7 3x - 1? Number of Cars Washed, x

0 4

3 6 How would the graph change if the class charged
$5 per car washed?
9. The formula for the area A of a trapezoid is
A = 21(b1 + b2)h, where b1 and b2 represent the The y-intercept would increase.
lengths of the bases and h represents the height.
The slope would increase.
Which equation can be used to find the height
The y-intercept would decrease.
of a trapezoid?
The slope would decrease.
h = 2A - b1 - b2
2A 13. A system has two linear equations in two variables.
h = b1 + b2 The graphs of the equations have the same slope but
different y-intercepts. How many solutions does the
h = A1b1 + b22 system have?

2

h = A - 2 0 2
b1 + b2
1 infinitely many

Chapter 6  Common Core Cumulative Standards Review 413

Constructed Response 24. The graph below is the solution of a linear system. How
many solutions does the system have?
14. Rhonda has 25 coins in her pocket. All of the coins are
either dimes or nickels. If Rhonda has a total of $2.30, 4y
how many dimes does she have?

15. What is the value of x in the proportion? 2 x
24
25 = x x Ϫ4 Ϫ2 O
21 - Ϫ2

1 6. An artist is adding a frame to a rectangular painting
that is 12 in. wide and 19 in. long. The frame is 3 in. 2 5. Write a system of inequalities for the graph below.
wide on each side. To the nearest square inch, what is
the area of the painting with the frame?

1 7. What is the solution of 4( -3x + 6) - 1 = -13?

1 8. In a regular polygon, all sides have the same length. y4
Suppose a regular hexagon has a perimeter of 25.2 in.
What is the length of each side in inches? 2

19. The sum of four consecutive integers is 250. What is Ϫ4 Ϫ2 O 2 x
the greatest of these integers? Ϫ2 4

2 0. What is the slope of a line that is perpendicular to the Ϫ4
line with equation y = -5x + 8?

2 1. On a map, Julia’s home is 8.5 in. from the library. If the
map scale is 1 in. : 0.25 mi, how many miles from the 26. The volume V of a cube is given by the formula V = s3,
library does Julia live?
where s represents the length of an edge of the cube.
2 2. The graph shows Jillian’s Distance (blocks) 5
distance from her house as 4 Suppose the edge length is 24 in. What is the volume of
she walks home from school. 3
How many blocks per minute 2 the cube in cubic feet?
does Jillian walk? 1
00 1 2 3 4 5 6 2 7. You plan to mail surveys to different households. A box
23. Sam is ordering pizza. Tony’s of 50 envelopes costs $3.50, and a postage stamp costs
Pizza charges $7 for a large Time (min) $.44. How much will it cost you to mail 400 surveys?

cheese pizza plus $.75 for each additional topping. Extended Response

Maria’s Pizza charges $8 for a large cheese pizza plus 2 8. The vertices of quadrilateral ABCD are A(1, 1), B(1, 5),
C(5, 5), and D(7, 1).
$.50 for each additional topping. For what number of
a. A trapezoid is a four-sided figure with exactly one
toppings will the cost of a large pizza be the same at pair of parallel sides. Is ABCD a trapezoid? Explain
your answer.
either restaurant?
b. You want to transform ABCD into a parallelogram
by only moving point B. A parallelogram is a
four-sided figure with both pairs of opposite sides
parallel. What should be the new coordinates of
point B? Explain.

414 Chapter 6  Common Core Cumulative Standards Review

Get Ready! CHAPTER

Skills Converting Fractions to Decimals 7

Handbook, Write as a decimal.
page 802

1. 170 2. 652 3. 10800 4. 27 5. 3
11

Lesson 1-2 Using Order of Operations

Simplify each expression.

6. (9 , 3 + 4)2 7. 5 + (0.3)2 8. 3 - (1.5)2 9. 64 , 24
10. 4 , (0.5)2 11. (0.25)42 12. 2(3 + 7)3 13. - 3(4 + 6 , 2)2

Lesson 1-2 Evaluating Expressions

Evaluate each expression for a = −2 and b = 5.

14. (ab)2 15. (a - b)2 16. a3 + b3 17. b - (3a)2

Lesson 2-10 Finding Percent Change

Tell whether each percent change is an increase or decrease. Then find the
percent change. Round to the nearest percent.

18. $15 to $20 19. $20 to $15

20. $600 to $500 21. $2000 to $2100

Lesson 4-6 Understanding Domain and Range

Find the range of each function with domain { −2, 0, 3.5}.

22. f (x) = -2x 2 23. g(x) = 10 - x 3 24. y = 5x - 1

Looking Ahead Vocabulary

25. If you say that a plant has new growth, has the size of the plant changed? What do
you think the growth factor of the plant describes?

26. In a mathematical expression, an exponent indicates repeated multiplication by
the same number. How would you expect a quantity to change when it experiences
exponential growth?

27. Tooth decay occurs when tooth enamel wears away over time. If exponential decay
models the change in the number of dentists in the United States over time, do you
think the number of dentists in the United States is increasing or decreasing?

Chapter 7  Exponents and Exponential Functions 415

CHAPTER Exponents

7 and Exponential
Functions

Download videos VIDEO Chapter Preview 1 Equivalence
connecting math Essential Question:  How can you
to your world.. 7-1 Zero and Negative Exponents represent numbers less than 1 using
7-2 Multiplying Powers With the Same Base exponents?
Interactive! ICYNAM 7-3 More Multiplication Properties of
Vary numbers, ACT I V I TI 2 Properties
graphs, and figures D Exponents Essential Question:  How can you
to explore math ES 7-4 Division Properties of Exponents simplify expressions involving exponents?
concepts.. 7-5 Rational Exponents and Radicals
7-6 Exponential Functions 3 Function
7-7 Exponential Growth and Decay Essential Question:  What are the
7-8 Geometric Sequences characteristics of exponential functions?

The online
Solve It will get
you in gear for
each lesson.

Math definitions VOC ABUL ARY Vocabulary DOMAINS
in English and • The Real Number System
Spanish English/Spanish Vocabulary Audio Online: • Seeing Structure in Expressions
• Linear, Quadratic, and Exponential Models
English Spanish

compound interest, p. 461 interés compuesto

Online access decay factor, p. 462 factor de decremento
to stepped-out
problems aligned exponential decay, p. 462 decremento exponencial
to Common Core
Get and view exponential function, p. 453 función exponencial
your assignments
online. NLINE exponential growth, p. 460 incremento exponencial
ME WO
O geometric sequence, p. 467 progression geométrica
RK
HO growth factor, p. 460 factor incremental

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PERFORMANCE TASK

Common Core Performance Task

Investing Prize Money

Emilio wants to invest $15,820 that he has just received as a prize from a contest
he entered. He researches certificates of deposit (CDs) at two different banks. The
details for his two best options are in the table below. Both CDs earn compound
interest, which you will work with in Lesson 7-7. When interest is compounded, it is
added to the account’s principal to become part of the money that earns interest.

Bank CD Length Annual Frequency of
Bank West 6 years Interest Rate Compounding
First Bank 5 years
3.8% Quarterly
4.3% Monthly

Task Description

Determine which investment will earn Emilio more interest. How much more
interest will he earn with that investment?

Connecting the Task to the Math Practices MATHEMATICAL

As you complete the task, you’ll apply several Standards for Mathematical PRACTICES
Practice.

• You’ll find the value of a compound interest account after one year. (MP 1)

• You’ll write algebraic expressions that will help you understand a formula for
the value of a compound interest account. (MP 7)

• You’ll use a formula to model Emilio’s investment choices. (MP 4)

Chapter 7  Exponents and Exponential Functions 417

7-1 Zero and Negative MCoamthmemonatCicosreFloStraidtea SSttaannddaarrddss
Exponents
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tvoaltuheoss.e.v.aAlulseos .p.r.eAplasorepsrfeopraNre-Rs Nfo.Ar.2
MMAPF1S,.M91P2.2N, -MRNP.31,.2MP 4, MP 7
MP 1, MP 2, MP 3, MP 4, MP 7

Objective To simplify expressions involving zero and negative exponents

Look for a pattern Copy and complete the table. Make a 2x 10x
in the values in the conjecture about how the value of an
table. exponential expression (an expression 24 ϭ ■ 104 ϭ ■
containing an exponent) changes when 23 ϭ ■ 103 ϭ ■
you decrease the exponent by 1. 22 ϭ ■ 102 ϭ ■
What do you think the value of 5−2 is? 21 ϭ ■ 101 ϭ ■
Explain your reasoning. 20ϭ■ 100 ϭ ■
2Ϫ1 ϭ ■ 10Ϫ1 ϭ ■
MATHEMATICAL 2Ϫ2 ϭ ■ 10Ϫ2 ϭ ■

PRACTICES

The patterns you found in the Solve It illustrate the definitions of zero and negative
exponents.

Essential Understanding  You can extend the idea of exponents to include zero
and negative exponents.
Consider 33, 32, and 31. Decreasing the exponents by 1 is the same as dividing by 3. If
you continue the pattern, 30 equals 1 and 3-1 equals 13.

Properties  Zero and Negative Exponents

Zero as an Exponent  For every nonzero number a, a0 = 1.

Examples 40 = 1 ( - 3)0 = 1 (5.14)0 = 1

Negative Exponent  For every nonzero number a and integer n, a-n = a1n.
1 1
Examples 7-3 = 73 (- 5)-2 = ( - 5)2

418 Chapter 7  Exponents and Exponential Functions

Why can’t you use 0 as a base with zero exponents? The first property on the previous
page implies the following pattern.

30 = 1 20 = 1 10 = 1 00 = 1

However, consider the following pattern.

03 = 0 02 = 0 01 = 0 00 = 0

It is not possible for 00 to equal both 1 and 0. Therefore 00 is undefined.

Why can’t you use 0 as a base with a negative exponent? Using 0 as a base with a
negative exponent will result in division by zero, which is undefined.

Problem 1 Simplifying Powers

Can you use the What is the simplified form of each expression?
definition of zero as
an exponent when A 9∙2
the base is a negative
number? 9∙2 = 1 Use the definition of negative exponent.
92
Yes, the definition of zero = 811 Simplify.

as an exponent is true for B ( - 3.6)0 = 1 Use the definition of zero as an exponent.

all nonzero bases.

Got It? 1. What is the simplified form of each expression? d. 6-1
a. 4-3 b. ( - 5)0 c. 3-2 e. ( - 4)-2

An algebraic expression is in simplest form when powers with a variable base are
written with only positive exponents.

Problem 2 Simplifying Exponential Expressions

Which part of the What is the simplified form of each expression?
expression do you
need to rewrite? A 5a3b∙2
( ) 5a3b∙2 = 5a3
The base b has a negative 1 Use the definition of negative exponent.
b2
exponent, so you need to 5a3
rewrite it with a positive = b2 Simplify.

exponent. B 1
x∙5
1
x-5 = 1 , x-5 Rewrite using a division symbol.

# = 1 , x15 Use the definition of negative exponent.

= 1 x 5 Multiply by the reciprocal of x15, which is x5.
= x 5 Identity Property of Multiplication

Got It? 2. What is the simplified form of each expression? 2 n-5
a. x-9 b. n1-3 c. 4c-3b a-3 m2
d. e.

Lesson 7-1  Zero and Negative Exponents 419

When you evaluate an exponential expression, you can simplify the expression before
substituting values for the variables.

How do you simplify Problem 3 Evaluating an Exponential Expression
the expression? What is the value of 3s 3t∙2 for s ∙ 2 and t ∙ ∙ 3?
Use the definition of
negative exponent to Method 1 Simplify first. Method 2  Substitute first.
rewrite the expression
with only positive 3s 3t -2 = 3(s)3 3s 3t-2 = 3(2)3( - 3)-2
exponents. t2

= 3(2)3 = 3(2)3
( - 3)2 ( - 3)2

= 24 = 232 = 24 = 2 2
9 9 3

Got It? 3. What is the value of each expression in parts (a)–(d) for n = -2 and w = 5?
a. n-4w0 b. nw-21 c.
n0 d. 1 1
w6 nw -
e. Reasoning  Is it easier to evaluate n0w0 for n = -2 and w = 3 by

simplifying first or by substituting first? Explain.

Problem 4 Using an Exponential Expression STEM

Population Growth  A population of marine bacteria doubles every hour under

#controlled laboratory conditions. The number of bacteria is modeled by the

expression 1000 2h, where h is the number of hours after a scientist measures the
population size. Evaluate the expression for h ∙ 0 and h ∙ ∙ 3. What does each
value of the expression represent in the situation?

#1000 2h models the Values of the expression Substitute each value of h into
for h = 0 and h = -3 the expression and simplify.
population.

# #1000 2h = 1000 20 Substitute 0 for h.
# = 1000 1 = 1000 Simplify.

The value of the expression for h = 0 is 1000. There were 1000 bacteria at the time the

scientist measured the population.
# #1000 2h = 1000 2-3 Substitute -3 for h.
#
= 1000 1 = 125 Simplify.
8

The value of the expression for h = -3 is 125. There were 125 bacteria 3 h before the
scientist measured the population.

420 Chapter 7  Exponents and Exponential Functions

Got It? 4. A population of insects triples every week. The number of insects is

#modeled by the expression 5400 3w, where w is the number of weeks after

the population was measured. Evaluate the expression for w = -2, w = 0,
and w = 1. What does each value of the expression represent in the situation?

Lesson Check Do you UNDERSTAND? MATHEMATICAL

Do you know HOW? PRACTICES
Simplify each expression.
1. 2-5 7. Vocabulary  A positive exponent shows repeated
2. m0
3. 5s 2t-1 multiplication. What repeated operation does a
4. x4-3
negative exponent show?
Evaluate each expression for a ∙ 2 and b ∙ ∙ 4.
5. a3b-1 8. Error Analysis  A student incorrectly simplified xn
6. 2a-4b0 as shown below. Find and correct the student’s
ear-rnobr0.

xn = anxn
a-nb0 b0
anxn
= 0 undefined

Practice and Problem-Solving Exercises MATHEMATICAL

PRACTICES

A Practice Simplify each expression. See Problem 1.

9. 3-2 10. ( - 4.25)0 11. ( - 5)-2
13. ( - 4)-2
12. - 5-2 16. - 12-1 14. 2-6
19. 1.5-2
15. - 30 17. 1
20
18. 58-1 20. ( - 5)-3

Simplify each expression. See Problem 2.

21. 4ab0 22. 1 23. 5x -4 24. c1-1
25. 3n-2 x-7 27. 3xy-2 28. 7a3bw-2
31. 2s8-3
26. k-4j 0 35. 120t 7u-11 32. 57t -s3
36. 72s-01tm-25
29. c-5d-7 30. c-5d7

33. 6a-d10c -3 34. 2-3x 2z-7

Lesson 7-1  Zero and Negative Exponents 421

Evaluate each expression for r ∙ ∙ 3 and s ∙ 5. 39. 3r See Problem 3.
37. r-3 38. s-3 s-2 40. rs-02
41. 4s-1 42. r 0s-2 43. r-4s 2 44. 2-4r 3s-2

45. Internet Traffic  The number of visitors to a certain Web site triples every month. See Problem 4.

#The number of visitors is modeled by the expression 8100 3m, where m is the

number of months after the number of visitors was measured.
Evaluate the expression m = -4. What does the value of the expression
represent in the situation?

STEM 46. Population Growth  A Galápagos cactus finch population increases

#by half every decade. The number of finches is modeled by the

expression 45 1.5d, where d is the number of decades after the
population was measured. Evaluate the expression for d = -2, d = 0,
and d = 1. What does each value of the expression represent in the
situation?

B Apply Mental Math  Is the value of each expression positive or negative?

47. - 22 48. ( - 2)2 49. ( - 2)3 50. ( - 2)-3

Write each number as a power of 10 using negative exponents.

51. 110 52. 1010 53. 10100 54. 1 Galápagos cactus finch
10,000

55. a. Patterns  Complete the pattern using powers of 5.
1 = ■    511 = ■    510 = ■    51-1 = ■    51-2
52 =■

b. Write 51-4 using a positive exponent.
c. Rewrite a1-n as a power of a.

Rewrite each fraction with all the variables in the numerator.

56. ba-2 57. 4g 58. 5m6 59. 11d8c45e-2
h3 3n

60. Think About a Plan  Suppose your drama club’s budget doubles every year. This

year the budget is $500. How much was the club’s budget 2 yr ago?
• What expression models what the budget of the club will be in 1 yr?
In 2 yr? In y years?
• What value of y can you substitute into your expression to find the budget of the

club 2 yr ago?

# 61. Copy and complete the table at the right. n 3 ■■ 5 ■
8
62. a. Simplify an a-n.
b. Reasoning  What is the mathematical n؊1 ■ 6 1 ■ 0.5
7
relationship between an and a-n? Explain.

422 Chapter 7  Exponents and Exponential Functions

63. Open-Ended  Choose a fraction to use as a value for the variable a. Find the values
of a-1, a2, and a-2.

STEM 64. Manufacturing  A company is making metal rods with a target diameter of 1.5 mm.
A rod is acceptable when its diameter is within 10-3 mm of the target diameter.
Write an inequality for the acceptable range of diameters.

65. Reasoning  Are 3x-2 and 3x 2 reciprocals? Explain.

C Challenge Simplify each expression. 67. ( - 5)2 - (0.5)-2 6 + 5m-2
m2 3-3
( ) 66. rt--74bw-18 0 2x-5y 3 r 2y 5 ( )68.
70. n2 , 2n 71. 2-1 - 1 + 5 1
69. 23150 - 6m22 3-2 22

72. For what value or values of n is n-3 = 1n1 25?

Standardized Test Prep

SAT/ACT 73. What is the simplified form of -6( -6)-1? 4y D
2C
74. Segment CD represents the flight of a bird that passes through the points O2 x
(1, 2) and (5, 4). What is the slope of a line that represents the flight of a 4
second bird that flew perpendicular to the first bird?

75. What is the solution of the equation 1.5(x - 2.5) = 3?

76. What is the simplified form of 0 3.5 - 4.7 0 + 5.6?

77. What is the y-intercept of the graph of 3x - 2y = -8?

Mixed Review

Solve each system by graphing. See Lesson 6-6.
78. y 7 3x + 4
y … -3x + 1 79. y … -2x + 1 80. y Ú 0.5x
y 6 2x - 1 y…x+2

Write an equation in slope-intercept form for the line with the given slope m See Lesson 5-3.

and y-intercept b.

81. m = -1, b = 4 82. m = 5, b = -2 83. m = 52, b = - 3

84. m = - 131, b = - 17 85. m = 95, b = 1 86. m = 1.25, b = -3.79
3

Get Ready!  To prepare for Lesson 7-2, do Exercises 87–91.

Simplify each expression. #89. 8.2 105 #90. 3 10-3 #See Lesson 7-1.

# # 87. 6 104 88. 7 10-2 91. 3.4 105

Lesson 7-1  Zero and Negative Exponents 423

Concept Byte Multiplying MCoamthmemonatCicosreFloStraidtea SSttaannddaarrddss
Powers
Use With Lesson 7-2 Prepares for NM-ARFNS..A9.112  .ENx-pRlaNin.1h.1o wEtxhpeladinefhinoiwtiotnhe
doef ftihneitimoneaonfinthgeomf reaatnioinngaloefxrpaotinoennatlsefxoplloonwesntfsrofmollows
ACTIVITY ferxotmenedxintegntdhiengprtohpeeprtrioepseortfieinsteogf einrteexgpeorneexnptosnteonthsotsoe
tvhaolusesv.a.lu. es . . .

MP 7

You can use patterns to find a shortcut for multiplying powers.

1

Copy and complete each statement in Exercises 1–8.

# # # # 1. 22 22 = 2 2 2 2 = 24 = 22+■
# # # # 2. 32 32 = 3 3 3 3 = 34 = 32+■
# # # # # 3. 33 32 = 3 3 3 3 3 = 3■ = 3■+■
# # # # # 4. 43 42 = 4 4 4 4 4 = 4■ = 4■+■
# # # 5. 51 52 = 5 5 5 = 5■ = 5■+■
# 6. 63 63 = ■ = 6■ = 6■+■
# 7. 72 76 = ■ = 7■ = 7■+■
# 8. 103 107 = ■ = 10■ = 10■+■

9. a. Look for a Pattern  What pattern do you see in your answers to Exercises 1–8?

# b. Predict  Use your pattern to predict the solution to 75 76 = 7■.
# c. Generalize  Use your pattern to predict the value of xn xm.

You can find a similar pattern when multiplying powers with negative exponents.

2

Copy and complete each statement in Exercises 10–15.
# # # 10. 22
2-1 = 2 2 1 = 21 = 22 + ■
# # # # # # 11. 24 2

2-2 = 2 2 2 2 1 1 = 2■ = 24 + ■
# # # # # 12. 33 2 2

3-2 = 3 3 3 1 1 = 3■ = 33 + ■
# # # # # # 13. 4-3 3 3

43 = 1 1 1 4 4 4 = 4■ = 4■+■
# 14. 8-4 86 = ■ = 8■ = 8■+■444

# 15. 12-3 127 = ■ = 12■ = 12■+■

16. a. Look for a Pattern  What pattern do you see in your answers to Exercises 10–15?

# b. Predict  Use your pattern to predict the solution to 95 9-7 = 9■.
# c. Generalize  Use your pattern to predict the value of xn x-m.

424 Concept Byte  Multiplying Powers

7-2 Multiplying Powers MCoamthmemonatCicosreFloStraidtea SSttaannddaarrddss
With the Same Base
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tohfeinptreogpeerretixepsoonfeintsegtoertheoxpseonveanlutsesto. .th. ose values . . .

MP 1, MP 2, MP 3, MP 4, MP 7

Objective To multiply powers with the same base

Notice the length Scientists estimate that there are about 1020 stars in the universe.
of the beach is in A cubic meter of beach sand contains about 109 grains of sand.
kilometers. How
will you find the Suppose all of the sand from the world’s beaches is combined into one
number of cubic
meters of beach large beach, as shown below. Are there more stars in the universe or
sand?
grains of sand on the world’s beaches? Explain your reasoning.
MATHEMATICAL
1m Not to scale
PRACTICES 100 m
100,000 km

All of the numbers in the Solve It are powers of 10. In this lesson, you will learn a
method for multiplying powers that have the same base.

Essential Understanding  You can use a property of exponents to multiply

#powers with the same base.

You can write a product of powers with the same base, such as 34 32, using
one exponent.

# # # # # #34 32 = (3 3 3 3) (3 3) = 36
#Notice that the sum of the exponents in the expression 34 32 equals the exponent of 36.
#In general, an equation such as 34 32 = 36 can be written using variables:
#am an = am+n.

Here’s Why It Works  You can use repeated multiplication to rewrite a product

# # # # # # # # # # #of powers.
am an = ((a++a)+c+* a) ((a++a)+c+*a ) = a (a++c)+a+=*am+n


m factors of a n factors of a m + n factors of a

Lesson 7-2  Multiplying Powers With the Same Base 425

When can you use Problem 1 Multiplying Powers
the property for
multiplying powers? #What is each expression written using each base only once?
You can use the property
for multiplying powers A 124 123 = 124+3 Add the exponents of the powers with the same base.
when the bases of the = 127 Simplify the exponent.
powers are the same.
B ( - 5)-2( - 5)7 = ( - 5)-2+7 Add the exponents of the powers with the same base.

= ( - 5)5 Simplify the exponent.

Got It? 1. What is each expression written using each base only once?
# # # a. 83 86 b. (0.5)-3(0.5)-8
c. 9-3 92 96

When variable factors have more than one base, be careful to combine only those
powers with the same base.

Problem 2 Multiplying Powers in Algebraic Expressions

Which parts of the # # #What is the simplified form of each expression?
expression can you
combine? A 4z5 9z-12 = (4 9)1z5 z-122   Commutative and associative properties of multiplication
You can group the
coefficients and multiply. = 36 1z5 + (-12) 2   Multiply the coefficients. Add the exponents of the powers
You can also write any with the same base.
powers that have the
same base with a single = 36z-7   Simplify the exponent.
exponent.
= 3z67   Rewrite using a positive exponent.

# # # # # B 2a a2 21 b4 2 Commutative and associative properties
9b4 3a2 = (2 9 3)1a of multiplication

# a = a1
= 541a1 a221b42 Multiply the coefficients. Write a as a1.

= 541a1+221b42 Add exponents of powers with the same base.

= 54a3b4 Simplify.

Got It? 2. What is the simplified form of each expression in parts (a)–(c)?
# # # # # # a. 5x 4 x 9 3x b. -4c 3 7d 2 2c-2
c. j 2 k-2 12j
# # d. Reasoning  Explain how to simplify the expression x a x b x c.

426 Chapter 7  Exponents and Exponential Functions

Brian Reardon
hsm11a1se_0703_a01726.ai
10/13/2008

You can use the property for multiplying powers with the same base to multiply two
numbers written in scientific notation.

Recall that you can use powers of 10 to make writing very large and very small numbers
more convenient. In scientific notation, you can write any number as a * 10b, where
1 … |a| 6 10. For example, 256,000 is written in scientific notation as 2.56 * 105.

Problem 3 Multiplying With Scientific Notation STEM

Chemistry  At 20°C, one cubic meter of water has a mass
of about 9.98 × 105 g. Each gram of water contains
about 3.34 × 1022 molecules of water. About how
many molecules of water does the droplet of
water shown below contain?

1 m3

V = 1.13 × 10–7 m3

How do you find grams molecules Use unit analysis.
the number of cubic meters grams
molecules? # #moleculesofwaterϭcubicmetersؒ ؒ

Use unit analysis. Divide = 11.13 * 10-72 19.98 * 1052 13.34 * 10222 Substitute.

out the common units. # # # # Commutative and

= (1.13 9.98 3.34) * 110-7 105 10222 associative properties
of multiplication

≈ 37.7 * 10-7+5+22 Multiply. Add exponents.

= 37.7 * 1020 Simplify.

= 3.77 * 1021 Write in scientific notation.

The droplet contains about 3.77 * 1021 molecules of water.

Got It? 3. About how many molecules of water are in a swimming pool that holds
200 m3 of water? Write your answer in scientific notation.

Exponents can also be expressed as fractions. Fractional exponents are called rational

#exponents.

Recall that 32 means 3 3, which equals 9. You can write the same expression using
rational exponents: 921 . The equation 912 = b indicates that b is the positive number that

#when multiplied by itself, equals 9.
912 = 3 since 3 3 = 9.

In general, am1 = b means that b multiplied as a factor m times equals a.

Lesson 7-2  Multiplying Powers With the Same Base 427

What number Problem 4 Simplifying Expressions With Rational Exponents
multiplied by itself
Simplify the expression 8141 .
## #4 times equals 81?
8141 Find the number that when multiplied by itself four times gives 81.
9 9 = 81 and
(3 3)(3 3) = 81. # # # 8114 = 3 3 3 3 3 = 81

Got It? 4. Simplify each expression. c. 6412
a. 1641 b. 2713

# # ## # #You can also have expressions like 923 , which means 912 921 912. Consider each factor

individually. Because 912 = 3, you know 921 912 912 = 3 3 3 = 27. So, 923 = 27.

Problem 5 Simplifying Expressions With Rational Exponents

How can the Simplify the expression 6432.
fractional exponent
be rewritten? # #6423 = 6412 6421 6412 Rewrite the expression.
Exponents can be # # = 8 8 8
rewritten as multiple Substitute 8 for 6421.
factors if the base of
each exponential factor = 512 Simplify.
is the same.

Got It? 5. Simplify each expression. c. 1634
a. 2523 b. 2723

You can use the properties of multiplying powers with the same base to simplify
expressions with rational exponents.

Property  Multiplying Powers With the Same Base

Words  To multiply powers with the same base, add the exponents.
#Algebra  am an = am+n, where a ≠ 0 and m and n are rational numbers
# #Examples  413413 41 + 1 432 b7 b-4 = b7+(-4) = b3
= 3 3 =

428 Chapter 7  Exponents and Exponential Functions

Problem 6 Simplifying Expressions With Rational Exponents
( # )( # )Simplify the expression 2a32 3b14 a31 5b21 .
Why must like # # ( # )( # )= (2 3 5) a32 a13 b14 b21 Commutative and associative properties of multiplication
variables be grouped ( # )( # )= 30 a23
together? a13 b 1 b12 Simplify.
4
To simplify by adding
( )( )= 30 a33 b34
exponents, the bases

must be the same.

Add exponents that have the same base.

= 30ab34 Simplify.

Got It? 6. Simplify each expression.

# # a. 2c 53 2c 51 b. n31 n34
( # )( # ) ( # )( # ) c. b23 c52 b49 c190 d. 2 1 1 3
3j 3 7m 4 3j 6 7m 2

Lesson Check #Do you UNDERSTAND? MATHEMATICAL
PRACTICES
Do you know HOW?
5. Writing  Can x 8 y3 be written as a single power?
# 1. What is 84 88 written using each base only once?
# 2. What is the simplified form of 2n23 3n43? Explain your reasoning.

3. What is 13 * 105218 * 1042 written in 6. Reasoning  Suppose a * 10m and b * 10n are two
numbers in scientific notation. Is their product
scientific notation? ab * 10m+n always, sometimes, or never a number in

4. Measurement  The diameter of a penny is about #scientific notation? Justify your answer.
1.9 * 10-5 km. It would take about 2.1 * 109 pennies
placed end to end to circle the equator once. What is 7. Error Analysis  Your friend says 4a21 3a51 = 7a71.
the approximate length of the equator? Explain your friend’s error. What is the correct answer?

Practice and Problem-Solving Exercises MATHEMATICAL

PRACTICES

A Practice Rewrite each expression using each base only once. See Problem 1.
# # # 8. 73 74
9. ( - 6)12 ( - 6)5 ( - 6)2 # #10. 96 9-4 9-2
# # # # 11. 22 27 20 #13. (-8)5 (-8)-5
12. 5-2 5-4 58

Simplify each expression. #15. 5c4 c6 See Problem 2.
14. m3m4
18. 15x5213y6213x22 #16. 4t-5 2t-3
17. 1 x5y2 21 x-6y 2 # # #19. -m2 4r 3 12r-4 5m

Write each answer in scientific notation. See Problem 3.

STEM 20. Biology  A human body contains about 2.7 * 104 microliters (mL) of blood for each
pound of body weight. Each microliter of blood contains about 7 * 104 white blood
cells. About how many white blood cells are in the body of a 140-lb person?

Lesson 7-2  Multiplying Powers With the Same Base 429

STEM 21. Astronomy  The distance light travels in one second (one light-second) is about
1.86 * 105 mi. Saturn is about 475 light-seconds from the sun. About how many
miles from the sun is Saturn?

Simplify each expression. See Problem 4.
22. 8 31
23. 625 1 24. 1000 1
4 3

Simplify each expression. See Problem 5.
25. 16 43
26. 9 5 27. 64 7
2 3

Simplify each expression. ( # )( # )29. ( # )( # )See Problem 6.
( # )( # ) 28. 8b23 9t 15 8b35 9t
3 7d 3 2g 5 2g 3 7d 5 30. 4r 2 5s 2 5s 5 4r 3
5 2 6 2 6 5 7 7 5

B Apply Complete each equation.
# 31. 52 5■ = 511 #32. m■ m-4 = m-9 #33. 2■ 212 = 21
# 34. a■ a4 = 1 #35. #36. x3y■ x■ = y2
a 2 a■ = a65
3

37. Think About a Plan  A liter of water contains about 3.35 * 1025 molecules. The
Mississippi River discharges about 1.7 * 107 L of water every second. About how

many molecules does the Mississippi River discharge every minute? Write your

answer in scientific notation.

• How can you use unit analysis to help you find the answer?

• What properties can you use to make the calculation easier?
#
38. When you simplify an algebraic expression like c 3 c 1 , you know that the bases of
5 2

the expressions must be the same. You also need to rewrite the exponents so that

they have a common denominator.

# a. Explain why you need to find the common denominator to simplify.31
b. Simplify the expression c 5 c 2 .

Simplify each expression. Write each answer in scientific notation.

39. 19 * 107213 * 10-162 40. 10.5 * 10-6210.3 * 10-22 41. 10.2 * 105214 * 10-122

STEM 42. Chemistry  In chemistry, a mole is a unit of measure equal to 6.02 * 1023 atoms of a
substance. The mass of a single neon atom is about 3.35 * 10-23 g. What is the mass
of 2 moles of neon atoms? Write your answer in scientific notation.

Simplify each expression. 44. 8m 1 1 m 1 + 2) 45. -4x313x3 - 10x2
3 3
# 43. a4 1a-3

46. a. Open-Ended  Write y6 as a product of two powers with the same base in four
different ways. Use only positive exponents.

b. Write y6 as a product of two powers with the same base in four different ways,
using negative or zero exponents in each product.

c. Reasoning  How many ways can you write y 6 as the product of two powers?
Explain your reasoning.

430 Chapter 7  Exponents and Exponential Functions

C Challenge Simplify each expression. # #48. 2n 2n+2 2 # # #49. 314 2y 32 2x
#52. 5x+1 51-x
# # 47. 3x 32-x 32 51. (t + 3)54(t + 3)52

50. (a + b)2(a + b)-3

53. Nature  A book shows an enlarged photo of a carpenter bee. A carpenter bee is
about 6 * 10-3 m long. The photo is 13.5 cm long. About how many times as long as
a carpenter bee is the photo?

Standardized Test Prep

SAT/ACT ( )( )54. What is the simplified form of2x1y24x 1 y 5 ?
2 3 4 6

6x 12y 2 6xy 8x 12y 7 8x 34y 3
3 9 2

55. What is the x-intercept of the graph of 5x - 3y = 30?

-10 - 6 6 10

56. At the Athens Olympics, the winning time for the women’s 100-m hurdles was
2.06 * 10-1 min. Which number is another way to express this time in minutes?

0.206 20.6 206 * 101 206 * 10-2

57. What is the solution of 4x - 5 = 2x + 13? 32
3 4 9

Extended 58. Bill’s company packages its circular mirrors in boxes with square bottoms, as r
Response shown at the right. Show your work for each answer.

a. What is an expression for the area of the bottom of the box?

b. If the mirror has a radius of 4 in., what is the area of the bottom of the box?

c. The area of the bottom of a second box is 196 in.2. What is the diameter of the

largest mirror the box can hold?

Mixed Review

Solve each system. See Lesson 6-3.

59. 2x + 3y = 12 60. 2x - y = -3 61. 2x + y = 15
-3x + y = -7 x-y=1 - 12x + y = 5

Find the third, seventh, and tenth terms of the sequence described by each rule. See Lesson 4-7.

62. A(n) = 10 + (n - 1)(4) 63. A(n) = -5 + (n - 1)(2) 64. A(n) = 1.2 + (n - 1)( -4)

Get Ready!  To prepare for Lesson 7-3, do Exercises 65–68.

Simplify each expression. See Lesson 7-1.
65. ( - 2)-4 66. 5xy0
67. 4m-1n2 68. - 3x y1 -21 z 6

3

Lesson 7-2  Multiplying Powers With the Same Base 431

Concept Byte MCoamthmemonatCicosreFloStraidtea SSttaannddaarrddss

Use With Lesson 7-3 Powers of Powers Prepares for NM-ARFNS..A9.112  .ENx-pRlaNin.1h.1o wEtxhpelain
hdeofwintithioendeofinthiteiomneoafntihnegmofearantiinognaolferxaptioonneanl ts
ACTIVITY
and Powers of Products efoxlploownesnftrsomfolleoxwtesnfdroinmg ethxteepnrdoinpgertthieesporof pinetretigeesr
oexf pinotneegnetrsetxoptohnoesnetsvatoluethso.s.e. values . . .
MP 7

You can use patterns to find a shortcut for simplifying a power raised to a power or a
product raised to a power.

1

Copy and complete each statement in Exercises 1–9.
# # 1. (45)2 = 45 45 = 4■+■ = 45 ■ = 4■
# # # 2. (36)3 = 36 36 36 = 3■+■+■ = 36 ■ = 3■
# # # # 3. (58)4 = 58 58 58 58 = 5■+■+■+■ = 58 ■ = 5■
# # # 4. (413)3 = 413 413 413 = 4■+■+■ = 431 ■ = 4■
# # # # 5. (521)4 = 512 521 512 521 = 5■+■+■+■ = 521 ■ = 5■
# # 6. (a4)2 = a4 a4 = a■+■ = a4 ■ = a■
# # # 7. (n2)3 = ■ ■ ■ = n■+■+■ = n2 ■ = n■
# # # # 8. (x 5)4 = ■ ■ ■ ■ = x■+■+■+■ = x5 ■ = x■
# # # # 9. (a41)4 = ■
■ ■ ■ = a■ + ■ + ■ + ■ = a 1 ■ = a■
4

10. a. Look for a Pattern  What pattern do you see in your answers to Exercises 1–9?
b. Predict  Use your pattern to simplify (y11)33.

2

Copy and complete each statement in Exercises 11–15.

# # # 11. (3n)2 = 3n 3n = (3 3)(n n) = 3■n■
# # # # # # 12. (2x)3 = 2x 2x 2x = (2 2 2)(x x x) = 2■x■
# # # 13. (ab)2 = ab ab = (a a)(b b) = a■b■
# # # # # # 14. (xy)3 = xy xy xy = (■ ■ ■)(■ ■ ■) = x■y■
# # # # # # # # # 15. (pq)4 = ■ ■ ■ ■ = (■ ■ ■ ■)(■ ■ ■ ■) = p■q■

16. a. Look for a Pattern  What pattern do you see in your answers to
Exercises 11–15?

b. Predict  Use your pattern to simplify (rs)20.

432 Concept Byte  Powers of Powers and Powers of Products

7-3 More Multiplication MCoamthmemonatCicosreFloStraidtea SSttaannddaarrddss
Properties of Exponents
MN-ARFNS..A9.112  .ENx-pRlaNin.1h.1o wEtxhpeladinefhinoiwtiotnheofdtehfienmitieoannoinfgthoef
Objectives To raise a power to a power mraetiaonnianlgeoxfporanteionntsalfoelxlopwonsefnrotsmfoelxlotewnsdfinrogmtheextpernodpienrgties
To raise a product to a power tohfeinptreogpeerretixepsoonfeintsegtoertheoxpseonveanlutsest,oatllhoowseinvgaflouresa,
anlolotawtionng for raandoictaaltsioinn tfeorrmrasdoifcaralstiionntaelremxspofneranttiso.nal
eMxPpo1n,eMntPs. 2, MP 3, MP 4, MP 7
MP 1, MP 2, MP 3, MP 4, MP 7

The radius of a bubble made by the bubble machine on the right is

2.5 times as large as the radius of a bubble made by the bubble

Make a plan. What machine on the left. What is the volume of a bubble made by the
do you need to 4
know before you machine on the right? Explain your reasoning. (Hint: V ‫؍‬ 3 Pr3 )
can use the volume
formula? radius = x in.

MATHEMATICAL

PRACTICES

In the Solve It, the expression for the volume of the larger bubble involves a product
raised to a power. In this lesson, you will use properties of exponents to simplify similar
expressions.

Essential Understanding  You can use properties of exponents to simplify a
power raised to a power or a product raised to a power.

You can use repeated multiplication to simplify a power raised to a power.

# # (x 5)2 = x 5 x 5 = x 5+5 = x 5 2 = x 10

#Notice that (x 5)2 = x 5 2. Raising a power to a power is the same as raising the base to

the product of the exponents.

Property  Raising a Power to a Power

Words  To raise a power to a power, multiply the exponents.

Algebra  (am)n = amn, where a ≠ 0 and m and n are rational numbers
# #Examples  (54)2 = 54 2 = 58
(m3)5 = m3 5 = m15
# # (a23)3 = a23 3 = a92
(x21)35 = x12 3 = x3
5 10

Lesson 7-3  More Multiplication Properties of Exponents 433

Problem 1 Simplifying a Power Raised to a Power

Should you add #A   What is the simplified form of (n4)7?
or multiply the (n4)7 = n4 7 Multiply exponents when raising a power to a power.
exponents to simplify = n28 Simplify.
the expression?
You multiply the   What is the simplified form of (x )2 1 ?
exponents when raising a #B
power to a power. 32

(x )2 1 = x 2 1 Multiply exponents when raising a rational power to a rational power.
3 2
32

= x 2 = x 1 Simplify.
6 3

Got It? 1. What is the simplified form of each expression in parts (a)–(d)?
a. (p 5)4 b. (p 4)5 c. )1 1 )1 1
(p d. (p
24 42
e. Reasoning Is (am)n = (an)m true for all integers m and n? Explain.

Use the order of operations when you simplify an exponential expression.

Problem 2 Simplifying an Expression With Powers
What is the simplified form of y 3(y 25)−2?

What is the first step You multiply exponents when #y 3(y 25)-2=y3y5(-2)
in simplifying the raising a power to a power. 2
expression?
By the order of You add exponents when = y 3y -120
operations, you simplify multiplying powers with the
powers before you same base. = y3+(-5)
multiply.
Write the expression using
only positive exponents. = y-2

= 1
y2

Got It? 2. What is the simplified form of each expression? c. (s -5)-12(s 32)
a. x 2(x 6)-4 b. w -2(w 53)3

You can use repeated multiplication to simplify an expression like (4m12)3.

# #(4m12)3 = 4m21 4m12 4m12
# # # # # = 4 4 4 m21 m12 m12

= 43m 3
2

= 64m 3
2

Notice that (4m21)3 = 43m32. This example illustrates another property of exponents.

434 Chapter 7  Exponents and Exponential Functions

Property  Raising a Product to a Power

Words  To raise a product to a power, raise each factor to the power and multiply.
Algebra  (ab)n = anbn, where a ≠ 0, b ≠ 0, and n is a rational number

Examples  (3x)4 = 34x 4 = 81x 4 (4b)32 = 423b 3 = 8b 3
2 2



Problem 3 Simplifying a Product Raised to a Power

Multiple Choice  Which expression represents the area of the square?

10x 3 25x 5
5x 6 25x 6

How do you find the (5x 3)2 = 52(x 3)2 Raise each factor to the second power. 5x3
area of the square? = 52x 6 Multiply the exponents of a power raised to a power.
The area of a square = 25x 6 Simplify.
with side length s is s 2.
Square the side length The correct answer is D.
of the square to find the
area.

Got It? 3. What is the simplified form of each expression? c. (3g 4)-2
a. (7m9)3 b. (2z)-4

Problem 4 Simplifying an Expression With Products

What is the exponent What is the simplified form of (n12)10(4mn -32)3?
of m?
It has an implied (n21)10(4mn -32)3 = (n21)1043m3(n -32)3 Raise each factor of 4mn- 2 to the third power.
exponent of 1. Similar to 3
coefficients, exponents
of 1 don’t need to be = n543m3n -2 Multiply the exponents of a power raised to a power.
written.
= 43m3n5n -2 Commutative Property of Multiplication

= 43m3n5+(-2) Add the exponents of powers with the same base.

= 64m3n3 Simplify.

Got It? 4. What is the simplified form of each expression? c. (6ab)3(5a -3)2
a. (x -2)2(3xy 5)4 b. (3c 52)4(c 2)3

You can use the property of raising a product to a power to solve problems involving
scientific notation. For example, to simplify the expression (3 * 108)2, you raise both
3 and 108 to the second power. Then multiply the two powers.

Lesson 7-3  More Multiplication Properties of Exponents 435

Problem 5 Raising a Number in Scientific Notation to a Power STEM

Aircraft  The expression 12mv2 gives the kinetic energy, in joules, of an object with a
mass of m kg traveling at a speed of v meters per second. What is the kinetic energy
of an experimental unmanned jet with a mass of 1.3 × 103 kg traveling at a speed of
about 3.1 × 103 m>s?
#12 m v 2
= 1 (1.3 * 103)(3.1 * 103)2 Substitute the values for m
How do you raise a # # # #= 2 and v into the expression.
number in scientific 1.3 103 3.12 (103)2 Raise the two factors to the second power.
notation to a power? 1 1.3 103 3.12 106 Multiply the exponents of a power raised to a power.
A number written in 2 1.3 3.12 103 106 Use the Commutative Property of Multiplication.
scientific notation is a 1.3 3.12 103+6 Add exponents of powers with the same base.
product. Use the property # # # #= 1
for raising a product to # # # #= 2
a power. 1
# # #= 2
1
2

= 6.2465 * 109 Simplify. Write in scientific notation.

The aircraft has a kinetic energy of about 6.2 * 109 joules.

Got It? 5. What is the kinetic energy of an aircraft with a mass of 2.5 * 105 kg traveling
at a speed of 3 * 102 m>s?

Lesson Check Do you UNDERSTAND? MATHEMATICAL

Do you know HOW? PRACTICES
Simplify each expression.
1. (n3)6 7. Vocabulary  Compare and contrast the property
3. (3a21)4
2. (b -7)3 for raising a power to a power and the property for
4. (9x 21)2(x2)5
multiplying powers with the same base.

Simplify each expression. Write each answer in 8. Error Analysis  One student simplified x 5 + x 5 to
scientific notation. x 10. A second student simplified x 5 + x 5 to 2x 5.
Which student is correct? Explain.

5. (4 * 105)2 6. (2 * 10-3)5 9. Open-Ended  Write four different expressions that are
equivalent to (x 23)3.

Practice and Problem-Solving Exercises MATHEMATICAL

PRACTICES

A Practice Simplify each expression. See Problems 1 and 2.

10. (n 8)4 11. (n 4)8 12. (c 2) 1 13. (x 25)10
14. (w 7)-1 4
18. (a32)3c 4
15. (x )3 -21 16. d(d -2)-9 17. (z 8)0z 1
2
5

19. (c 3)91(d 3)0 20. (t 2)-2(t 2)-5 21. (m 3)-1(x )1 1

34

436 Chapter 7  Exponents and Exponential Functions

Simplify each expression. See Problems 3 and 4.
25. (36g 4)-12
22. (3n -6)-4 23. (7a)-2 24. (5y 21)4 29. (y 2z -3)16(y 3)2
26. (2x 61)3x 2 27. (2y 97)-3 28. (r 52s)5 33. (2j 2k 4)-5(k -1j 7)6
30. (3b -2)2(a2b4)3 31. 4j 2k 6(2j 11)3k 5 32. (mg 4)-1(mg 4)

Simplify. Write each answer in scientific notation. 36. (2 * 10-10)3 See Problem 5.
40. (3.5 * 10-4)3 37. (2 * 10-3)3
34. (3 * 105)2 35. (4 * 102)5 41. (2.37 * 108)3
38. (7.4 * 104)2 39. (6.25 * 10-12)-2

42. Geometry  The radius of a cylinder is 7.8 * 10-4 m. The height of the cylinder is
3.4 * 10-2 m. What is the volume of the cylinder? Write your answer in scientific
notation. (Hint: V = pr 2h)

B Apply Complete each equation.

43. (b2)■ = b8 44. (m ■) 1 = m -12 45. (x ■)7 = x 6
3

46. (n9)■ = n 47. (y -4)■ = y 1 48. 7(c 1)■ = 7c 2
49. (5x ■)2 = 25x -4 2 3

50. (3x 3y ■)3 = 27x 9 51. (m 2n 3)■ = 1
m6n9

52. Think About a Plan  How many times the volume of the small cube is 3x
the volume of the large cube?
6x
• What expression can you write for the volume of the small cube? For
the volume of the large cube?

• What property of exponents can you use to simplify the volume
expressions?

Simplify each expression.

53. 32(3x)3 54. (4.1)5(4.1)-5 55. (b 61)3b 1
6

56. ( - 5x)2 + 5x 2 57. ( - 2a23b)3(ab31)3 58. (2x -3)2(0.2x)2

59. 4xy 204( - y)-3 60. (103)4(4.3 * 10-8) 61. (37)2(3-4)3

62. Reasoning Simplify (x 2)3 and x 23. Are the expressions equivalent? Explain.

63. a. E rror Analysis  What mistake did the student make in simplifying the (2 + 3)2 = 22 + 32
expression at the right? =4+9
= 13
b. What is the correct simplified form of the expression?

STEM 64. Wind Energy  The power generated by a wind turbine depends on the wind speed.
The expression 800v 3 gives the power in watts for a certain wind turbine at wind
speed v in meters per second. If the wind speed triples, by what factor does the
power generated by the wind turbine increase?

65. Can you write the expression 49x2y 2z2 using only one exponent? Show how or
explain why not.

Lesson 7-3  More Multiplication Properties of Exponents 437

STEM 66. a. Geography  Earth has a radius of about 6.4 * 106 m. What is the 6.4 ϫ 106 m
approximate surface area of Earth? Use the formula for the surface area
of a sphere, S.A. = 4pr 2. Write your answer in scientific notation.

b. Oceans cover about 70% of the surface of the Earth. About how many

square meters of Earth’s surface are covered by ocean water?
c. The oceans have an average depth of 3790 m. Estimate the volume of

water in Earth’s oceans.

C Challenge Solve each equation. Use the fact that if ax = ay, then x = y.

67. 56 = 25x 68. 3x = 274 69. 8 1 = 2x
3

70. 4x = 212 71. 32x = 94 72. 2x = 1
32

73. Reasoning  How many different ways are there to rewrite the expression 16x4 using
only the property of raising a product to a power? Show the ways.

PERFORMANCE TASK MATHEMATICAL

Apply What You’ve Learned PRACTICES
MP 1
Look back at the information on page 417 about the two CDs that Emilio is
considering for the investment of his prize money.

When interest is compounded quarterly, interest is added to the account’s

principal every three months (one-quarter of a year). The bank calculates the
1
interest using a rate that is 4 of the annual interest rate.

a. How many times per year is the interest from the Bank West investment
compounded? What is the interest rate used to calculate the interest?

b. Suppose Emilio chooses the Bank West CD. Confirm the values in the table below,
and then complete the table. What is the value of the CD after one year?

Bank West CD

Quarter Starting Principal Interest Earned Ending Principal

1 $15,820.00 $150.29 $15,970.29

2 $15,970.29 ■ ■

3■ ■■
4■ ■■

c. If the Bank West CD earned simple interest instead of compound interest, what
would the value of Emilio’s CD be after one year?

438 Chapter 7  Exponents and Exponential Functions

7-4 Division Properties of CMoamthmemonatCicosreFloStraidtea SSttaannddaarrddss
Exponents
NM-ARFNS..A9.112  .ENx-pRlaNin.1h.1o wEtxhpeladinefhinoiwtiotnheofdtehfienmitieoannoinfgthoef
Objectives To divide powers with the same base mraetiaonnianlgeoxfporanteionntsalfoelxlopwonsefnrotsmfoelxlotewnsdfinrogmtheextpernodpienrgties
To raise a quotient to a power tohfeinptreogpeerretixepsoonfeintsegtoertheoxpseonveanlutsest,oatllhoowseinvgaflouresa,
anlolotawtionng for raandoictaaltsioinn tfeorrmrasdoifcaralstiionntaelremxspofneranttiso.nal
eMxPpo1n,eMntPs. 2, MP 3, MP 4, MP 7
MP 1, MP 2, MP 3, MP 4, MP 7

Solve a simpler A machine makes wooden dowels by removing material from a block
problem first. Use of wood as shown in the diagram. What percent of the wood does
a value for x to the machine remove from the original piece of wood to form the
understand all of dowel? Explain how you found your answer. (Hint: What is the
the relationships volume of the dowel?)
in this problem.
ᐉx ᐉ
MATHEMATICAL x x

PRACTICES

In the Solve It, the expression for the volume of the dowel involves a quotient raised to
a power.

Essential Understanding  You can use properties of exponents to divide powers
with the same base.

You can use repeated multiplication to simplify quotients of powers with the same base.
Expand the numerator and the denominator. Then divide out the common factors.

45 ϭ 4ؒ4ؒ4ؒ4ؒ4 ϭ 42
43 4ؒ4ؒ4

This example suggests the following property of exponents.

Property  Dividing Powers With the Same Base

Words To divide powers with the same base, subtract the exponents.
Algebra
am = a m - n, where a ≠ 0 and m and n are rational numbers
an

Examples 26 = 26 - 2 = 24 x4 = x4-7 = x-3 = 1 s43 = s3 - 1 = s3 - 2 = s14
22 x7 x3 s21 4 2 4 4

Lesson 7-4  Division Properties of Exponents 439

Problem 1 Dividing Algebraic Expressions

How are the What is the simplified form of each expression?
properties for
dividing powers and A x52
multiplying powers x2
similar? 5
x2 = x25-2 Subtract exponents when dividing powers with the same base.
For both properties, the
x2 = 1
bases of the powers must
x 2 Simplify.
be the same. Dividing
B mm25nn 4
a power is the same as 3

multiplying by a negative m2n4 = m 2 - 5n 4 - 3 Subtract exponents when dividing powers with the same base.
m5n3 = m -3n 1 Simplify the exponents.
exponent.

= n Rewrite using positive exponents.
m3

Got It? 1. What is the simplified form of each expression? d. aa-53bb27 e. x 4y -1z 8
a. yy 4231 b. dd372 c. kk6jj52 x 4y -5z

You can use the property of dividing powers with the same base to divide numbers in
scientific notation.

Problem 2 Dividing Numbers in Scientific Notation
Demographics  Population density describes the number of people per unit area.
During one year, the population of Angola was 1.21 × 107 people. The area of Angola
is 4.81 × 105 mi2. What was the population density of Angola that year?

• The population The population Write the ratio of population to area.
• The area density

1.21 * 107 = 1.21 * 107-5 Subtract exponents when dividing powers
4.81 * 105 4.81 with the same base.
Simplify the exponent.
= 1.21 * 102 Divide. Round to the nearest thousandth.
4.81
≈ 0.252 * 102

= 25.2 Write in standard notation.

The population density of Angola was about 25.2 people per square mile.

Got It? 2. During one year, Honduras had a population of 7.33 * 106 people. The
area of Honduras is 4.33 * 104 mi2. What was the population density of
Honduras that year?

440 Chapter 7  Exponents and Exponential Functions