CChapter 5 hapter 5 RResource Mastersesource Masters · • Contains multiple-choice questions • 2A-2B On-level students • Contains both multiple-choice and free-response questions

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Chapter 5 Chapter 5 Resource MastersResource Masters

ContentsChapter Resources• Family Letter • Are You Ready Worksheets• Diagnostic Test • Pretest

Language Arts Resources• Student Glossary

Practice and Reinforcement• Facts Practice

Leveled Lesson Resources• Explore• Reteach• Skills Practice• Homework Practice• Problem-Solving Practice• Enrich

Technology Resources• Graphing Calculator Activity• Scientific Calculator Activity• Spreadsheet Activity

Assessment Resources• Reflecting on the Chapter• Chapter Quizzes• Vocabulary Test• Chapter Tests• Standardized Test Practice• Extended-Response Test• Student Recording Sheet• Chapter Project Rubric

Answer Pages

ChapterResource Masters

are provided forevery chapter in both

print and digitalformats.

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Copyright © by the McGraw-Hill Companies, Inc. All rights reserved. Permission is granted to reproduce the material contained herein on the condition that such materials be reproduced only for classroom use; be provided to students, teachers, and families without charge; and be used solely in conjunction with Math Connects, Course 3. Any other reproduction, for use or sale, is expressly prohibited without prior written permission of the publisher.

Send all inquiries to:Glencoe/McGraw-Hill8787 Orion PlaceColumbus, OH 43240

ISBN: 978-0-07-892304-3MHID: 0-07-892304-2 Math Connects, Course 3

Printed in the United States of America.

2 3 4 5 6 7 8 9 10 032 18 17 16 15 14 13 12 11 10 09

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CONTENTSCONTENTSTeacher’s Guide . . . . . . . . . . . . . . . . . . . . . . . . . . . v

Chapter 5 ResourcesFamily Letter . . . . . . . . . . . . . . . . . . . . . . 1Are You Ready?

Practice Worksheet . . . . . . . . . . . . . . . . . . . . . 5

AL Review Worksheet . . . . . . . . . . . . . . . . . . 6

BL Apply Worksheet . . . . . . . . . . . . . . . . . . . 7

Diagnostic Test . . . . . . . . . . . . . . . . . . . . . . 8

Pretest . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

Language Arts ResourcesStudent Glossary . . . . . . . . . . . . . . . . . . . . . . 10

Practice and Reinforcement Facts Practice . . . . . . . . . . . . . . . . . . . . . . . . . 11

Lesson Resources

A Powers and ExponentsAL Reteach . . . . . . . . . . . . . . . . . . . . . . . 12

Skills Practice . . . . . . . . . . . . . . . . . . . . . . 13Homework Practice . . . . . . . . . . . . . . . . . . 14Problem-Solving Practice . . . . . . . . . . . . . 15BL Enrich. . . . . . . . . . . . . . . . . . . . . . . . . 16

Scientific Calculator Activity . . . . . . . . . . 17

B Multiply and Divide MonomialsAL Reteach . . . . . . . . . . . . . . . . . . . . . . . 18


C Powers of MonomialsAL Reteach . . . . . . . . . . . . . . . . . . . . . . . 23

Skills Practice . . . . . . . . . . . . . . . . . . . . . . 24

Homework Practice . . . . . . . . . . . . . . . . . . 25Problem-Solving Practice . . . . . . . . . . . . . 26BL Enrich. . . . . . . . . . . . . . . . . . . . . . . . . 27

D PSI: Act It OutAL Reteach . . . . . . . . . . . . . . . . . . . . . . . 28

Skills Practice . . . . . . . . . . . . . . . . . . . . . . 29Homework Practice . . . . . . . . . . . . . . . . . . 30Problem-Solving Practice . . . . . . . . . . . . . 31

Lesson Resources

A Negative ExponentsAL Reteach . . . . . . . . . . . . . . . . . . . . . . . 32


B Scientific NotationAL Reteach . . . . . . . . . . . . . . . . . . . . . . . 37


C Compute with Scientific NotationAL Reteach . . . . . . . . . . . . . . . . . . . . . . . 42


TI-84 Plus Activity . . . . . . . . . . . . . . . . . . 47

Lesson Resources

A Square RootsAL Reteach . . . . . . . . . . . . . . . . . . . . . . . 48

Skills Practice . . . . . . . . . . . . . . . . . . . . . . 49

AL = Approaching Level BL = Beyond Level

Lesson

5-1

Lesson

5-2

Lesson

5-3

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Homework Practice . . . . . . . . . . . . . . . . . . 50Problem-Solving Practice . . . . . . . . . . . . . 51BL Enrich. . . . . . . . . . . . . . . . . . . . . . . . . 52

B : Roots of Non-Perfect SquaresExplore . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

C Estimate Square RootsAL Reteach . . . . . . . . . . . . . . . . . . . . . . . 54


TI-73 Activity . . . . . . . . . . . . . . . . . . . . . . 59

D Compare Real NumbersAL Reteach . . . . . . . . . . . . . . . . . . . . . . . 60


Assessment ResourcesReflecting on Chapter 5 . . . . . . . . . . . . . . . . 65

Chapter Quizzes . . . . . . . . . . . . . . . . . . . . . . 66

Vocabulary Test . . . . . . . . . . . . . . . . . . . . . . . 68

Chapter Tests

AL 1A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

AL 1B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

2A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

2B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

BL 3A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

BL 3B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

Standardized Test Practice . . . . . . . . . . . . . . 81

Extended-Response Test . . . . . . . . . . . . . . . . 83

Extended-Response Rubric . . . . . . . . . . . . . . 84

Student Recording Sheet . . . . . . . . . . . . . . . 85

Chapter Project Rubric . . . . . . . . . . . . . . . . . 86

Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . .A1

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Teacher’s Guide to Using theChapter 5 Resource Masters

The Chapter 5 Resource Masters includes the core materials needed forChapter 5. These materials include information for families, student worksheets, extensions, and assessment options. The answers for these pages appear at the back of this booklet.

All of the materials found in this booklet are included for viewing and printing from the online Teacher Edition.

Family ResourcesFamily Introduction to Course 3 (Available in Chapter 0)

• Talks about the focus of the grade level • Gives Web site information

Family Letter • English and Spanish • Overview of the chapter • Key vocabulary • Provides at home activities

Chapter ResourcesAre You Ready Worksheets • Use after the Are You Ready section in the Student Edition • AL Review: Approaching-level students • Practice: On-level students • BL Apply: Beyond-level students

Chapter Diagnostic Test • Use to test skills needed for success in the upcoming chapter • Retest approaching-level students after the Are You Ready worksheets.

Chapter Pretest • Quick check of the upcoming chapter’s concepts to determine pacing • Use before the chapter to gauge students’ skill level • Use to determine class grouping

NAME ________________________________________ DATE _____________ PERIOD _____

Chapter 5 1

Course 3

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Family Letter5Chapter

Key Vocabulary

Dear Parent or Guardian:

Today we began Chapter 5: Operations on Real Numbers. In this chapter, your student will find powers of numbers and work with scientific notation. We will also be finding square roots and comparing real numbers. Included in this letter are key vocabulary words and activities you can do with your student. You may also wish to log on to glencoe.com for other study help. If you have any questions or comments, feel free to contact me at school.

Sincerely,

_____________________

Operations on Real Numbersbase The common factor of a power.

exponent Tells how many times the base is used as a factor in a power.

irrational number A number that cannot be expressed as the quotient of two integers.

power The product of repeated factors expressed using an exponent and a base.

radical sign The symbol ! used to indicate a positive square root.

real number The sets of rational Real Numbers

WholeNumbers

Rational Numbers

--

Integers

IrrationalNumbers

-

!

numbers and irrational numbers together.

scientific notation A compact way of writing numbers with absolute values that are very large or very small.

square root One of two equal factors of a number.

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Language Arts ResourcesStudent Glossary • Includes key vocabulary terms from the chapter • Students record definitions and/or examples for each term. • Students can use the page as a bookmark as they study the chapter.

Practice and ReinforcementFacts Practice • Quick recall of concepts needed in the upcoming chapter • Use as a timed test to gauge student mastery of prior concepts

Lesson Resources

Explore • Provides additional practice for the activities and exercises

found in the Student Edition • Use as homework for same-day teaching

Reteach • Provides vocabulary, key concepts, additional worked-out

examples, and exercises • Use for students who have been absent

Skills Practice • Focuses on the computational nature of the lesson • Use as an additional practice • Use as homework for second-day teaching

Homework Practice • Mimics the types of problems found in the Practice

and Problem Solving of the Student Edition • Use as an additional practice • Use as homework for second-day teaching

Problem-Solving Practice • Includes word problems that apply the concepts of the lesson • Use as an additional practice • Use as homework for second-day teaching

NAME ________________________________________ DATE _____________ PERIOD _____

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Chapter 5 14

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5-1A

Homework PracticePowers and Exponents

Write each expression using exponents. 1. 3 ! 3 ! m

2. ( 1 " 4 ) ( 1 " 4 ) ( 1 " 4 )

3. 2 ! d ! 5 ! d ! d ! 5 4. p ! (-9) ! p ! (-9) ! p ! q ! q

5. g ! (-7) ! (-7) ! g ! h ! (-7) ! h 6. x ! 1 " 8 ! x ! x ! y ! 1 " 8 ! y ! x

Evaluate each expression.

7. (-8)4 8. ( 1 " 5 ) 3 9. (- 3 " 5 )

5

10. (-2)3 + 52 11. 34 - 52 12. (-2)5 - (-2)4

13. 43 ÷ 23 14. 53 ! 23 15. 17 + (-3)4

ALGEBRA Evaluate each expression.

16. r3 - s, if r = 5 and s = 4 17. m2 - n3, if m = 6 and n = 2 18. f - g4, if f = 3 and g = -5 19. (x5 - y2)2 + x3, if x = 2 and y = 8 20. Replace with <, >, or = to make a true statement: 24 42. 21. ISLANDS Florida has about 22 ! 32 ! 53 islands (over 10 acres). About how many islands is this?

22. SPACE The radius of Jupiter is about 7.15 ! 104 kilometers. Write this distance is standard form.

Get ConnectedGet Connected For more examples, go to glencoe.com.


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Chapter 5 53

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5-3B

Roots of Non-Perfect SquaresEstimate the side length of each square. 1. Square area = 72 sq units

6 9 12 1530

a. On dot paper, copy and cut out a square like the one shown above. Place one edge of your square on the number line shown above. Between what two consecutive whole numbers is the # $$ 72 , the side length of the square, located? b. Between what two perfect squares is 72 located? c. Estimate the length of a side of the square. Verify your estimate by using a calculator to compute the value of # $$ 72 . 2.

Square area = 128 sq units

a. On dot paper, copy and cut out a square like the one shown above. Place one edge of your square on the number line shown above. Between what two consecutive whole numbers is the # $$ 128 , the side length of the square, located? b. Between what two perfect squares is 128 located? c. Estimate the length of a side of the square. Verify your estimate by using a calculator to compute the value of # $$ 128 .


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Enrich • Provides an extension of the concepts, offers a historical or

multicultural look at the concepts, or widens students’ perspectives on the mathematics

• For use with all levels of students

Technology Activities • Presents ways in which technology can be used with the

concepts in some of the lessons • Use as an alternative approach to teaching the concept • Use as part of the lesson presentation

Assessment Resources

Reflecting on Chapter 5 • Three open-ended questions • Allows students to write about mathematics

Chapter Quizzes • Free-response questions • One quiz for each multi-part lesson

Vocabulary Test • Includes a list of vocabulary words and questions to assess students’ knowledge of

those words • Use in conjunction with one of the Chapter Tests

Chapter Tests • AL 1A-1B Approaching-level students • Contains multiple-choice questions • 2A-2B On-level students • Contains both multiple-choice and free-response questions. • BL 3A-3B Beyond-level students • Contains free-response questions • Tests A and B are the same format with different numbers. • Use when students are absent or for different rows

Standardized Test Practice • Test is cumulative. • Includes multiple-choice and short-response questions

NAME ________________________________________ DATE _____________ PERIOD _____

SCORE _____

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5Chapter

Chapter 5 82

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5. Which number is not a real number? 1 ! 2 , " # –9 , 0.01, " ## 14

F. " # –9 G. 0.01 H. 1 ! 2 I. " ## 14 6. Write a • a • a • b • c • c using exponents. A. 3abc2 B. a3bc2 C. 2a3bc D. 6abc 7. RUNNING The table shows the average number of miles Milo runs each day. Write an equation to find the total miles in any number of days.

F. m = d + 3 G. m = 3d H. d = 3m I. d = m + 3 8. Solve by substitution. y = 2x – 4

y = 6x A. –1, 6 B. 1, –6 C. –1, –6 D. –6, –1 9. GEOMETRY What is the total number of triangles, of any size in the figure below?

10. MOVIES Norton is taking himself and three friends to the movies. The cost of each ticket is $7. He would also like to buy everyone something to eat and/or drink, but he can spend at most $40. Write and solve an inequality to find out how much he can spend per person on snacks.

5. F G H I

6. A B C D

7. F G H I

8. A B C D

9.

10.

Days (d) Total Miles (mi)1 32 63 94 12

Standardized Test Practice (continued)


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1. Give an example that supports the statement that any nonzero number raised to the zero power is 1.

2. Give a real world example where scientific notation is used. Why do you think it is used in this situation?

3. Write Math Briefly explain why the square root of a real number less than zero is not real.

-

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Chapter 5 65

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Chapter 5 16

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5-1A

Enrich

A-Mazing ExponentsSolve the following puzzle by finding the correct path through the boxes. The solution is a famous quote from United States history.

Starting with Box 1, draw an arrow to the box next or diagonal to Box 1 with the expression of the least value. The arrow cannot go to a box that has already been used. The first arrow has been drawn to get you started.When you have finished drawing your path through the boxes, write the box numbers on the lines below. Put the numbers in the order in which they are connected. Then use the chart at the right to convert each box number to a letter.

1

53

6

63

11

44 + 162

16

36 - 63

21

29 + 92

2

132

7

27

12

35

17

83 - 18

22

232

3

43 + 34

8

24 ! 32

13

44

18

162 + 63

23

36 - 35

4

172

9

25 ! 32

14

73

19

192

24

212

5

45 - 93

10

182

15

53 + 35

20

28 + 112

25

23 ! 72

Box Number 1 7Letter G I

Box NumberLetter

H

1 G2 M3 E4 E5 R6 E7 I8 V9 B

10 T11 D12 L13 I14 Y15 R16 E17 E18 E19 O20 G21 T22 A23 M24 V25 I


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Extended-Response Test • Contains performance-assessment tasks • Sample answers are included.

Extended-Response Rubric • The scoring rubric for the Extended-Response Test

Student Recording Sheet • Corresponds with the Test Practice at the end of the

Student Edition chapter

Chapter Project Rubric • The scoring rubric for the Chapter Project found in the

Teacher Edition

Answers

Chapter and Lesson Resources • Chapter Resources, Facts Practice, and Lesson Resources are provided as reduced

pages with answers appearing in black.

Assessments • Full-size answer keys are provided for the assessment masters.

NAME ________________________________________ DATE _____________ PERIOD _____

SCORE _____

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Use this recording sheet with pages 334-335 of the Student Edition.Fill in the correct answer. For gridded-response questions, write your answers in the boxes on the answer grid and fill in the bubbles to match your answers.

Extended ResponseRecord your answers for Exercise 15 on the back of this paper.

1. A B C D

2.

3. F G H I

4. A B C D

5.

6. F G H I

7.

8. A B C D

9. F G H I

10. A B C D

11. F G H I

12.

13. A B C D

14. F G H I

Student Recording Sheet


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NAME ________________________________________ DATE _____________ PERIOD _____

PDF Crx

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.Family Letter5

Chapter

Dear Parent or Guardian:

Today we began Chapter 5: Operations on Real Numbers. In this

chapter, your student will find powers of numbers and work with scientific

notation. We will also be finding square roots and comparing real numbers.

Included in this letter are key vocabulary words and activities you can do

with your student. You may also wish to log on to glencoe.com for other

study help. If you have any questions or comments, feel free to contact me

at school.

Sincerely,

_____________________

base The common factor of a power.

exponent Tells how many times the base is used as a factor in a power.

irrational number A number that cannot be expressed as the quotient of two integers.

power A product of repeated factors using an exponent and a base.

radical sign The symbol ! " used to indicate a positive square root.

real number The sets of rational numbers and irrational

numbers together.

scientific notation A compact way of writing numbers

with absolute values that are very

large or very small.

square root One of two equal factors of a number.

Key Vocabulary

Real Numbers

WholeNumbers

Rational Numbers

--

Integers

IrrationalNumbers

-

!

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Chapter

• Find items in your home that give a quantity in square units on the package. for example, a bag of grass seed lists the square footage the product covers.

• Make a table like the one shown listing each item.

• Estimate the largest square area each item could cover.

• Check your answer by finding the square of the number. How close was your estimate?

• Search online for examples of scientific notation. For example, the distances between stars and planets.

• Make a table listing the examples you found.

• Write a short paragraph describing why you think it is important to use scientific notation to express the distances or sizes you found in the examples.

Online Activity

Hands-On Activity

At-Home Activities

Item Square Units

Square Root

20-lb bag of grass seed

5,000 square feet

?


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.Carta a la familia

Vocabulario clave

5Capítulo

Capítulo 5 3 Course 3

Estimado padre o apoderado:

Hoy comenzamos el Capítulo 5: Operaciones con números reales. En este

capítulo, su estudiante calculará potencias de números y trabajará con

notación científica. Calcularemos raíces además, cuadradas y compararemos

números reales. En esta carta se incluyen palabras del vocabulario clave y

actividades que pueden realizar con su estudiante. Si desean obtener más

ayuda para el estudio, visiten glencoe.com. Si tienen alguna pregunta o desean

hacer algún comentario, pueden contactarme en la escuela.

Sinceramente,

_____________________

base Factor común de una potencia.

exponente Indica cuántas veces se usa la base como factor en una potencia.

número irracional Número que no se puede expresar como el cociente de dos enteros.

potencia Producto de factores repetidos con un exponente y una base.

signo radical El símbolo ! " que se emplea para indicar una raíz cuadrada positiva.

número real Conjunto de los números racionales

e irracionales.

notación científica Manera resumida de escribir números

cuyos valores absolutos son muy

grandes o muy pequeños.

raíz cuadrada Uno de los dos factores iguales de un número.

Números reales

Númerosenteros

Números racionales

--

Enteros

Númerosirracionales

-

!

NOMBRE ______________________________________ FECHA ____________ PERÍODO ____


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Capítulo

Capítulo 5 4 Course 3

Actividad manual

Actividad en línea

• Encuentren artículos del hogar cuyos empaques muestren canti-dades en unidades cuadradas. Por ejemplo, una bolsa de semillas de césped indica los pies cuadrados que cubre el producto.

• Hagan una tabla como la que se muestra mostrando cada artículo.

• Estimen la mayor área cuadrada que puede cubrir cada artículo.

• Comprueben sus respuestas cal-culando el cuadrado del número. ¿Qué tan acertadas estuvieron sus aproximaciones?

• Investiguen en línea ejemplos de notación científica. Por ejemplo, las distancias entre los planetas.

• Hagan una tabla enumerando los ejemplos que hallaron.

• Escriban un párrafo corto describiendo por qué creen que es importante usar la notación científica para expresar las distancias o los tamaños que hallaron en los ejemplos.

Actividades para el hogar

ArtículoUnidades cuadradas

Raíz cuadrada

Bolsa de 20 libras de semilla de césped

5,000 pies cuadrados

?

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Are You ReadyAre You Readyfor Chapter 5?for Chapter 5?

Practice


Find each product.

1. 7 ! 7 ! 7

2. 8 ! 8 ! 8 ! 8 ! 8

3. ("6)("5)("5)("6)("5)

4. 3 ! 3 ! 4 ! 4 ! 4

5. 2 ! 9 ! 2 ! 9 ! 9

6. CALLS A call center received 9 ! 3 ! 3 ! 9 ! 3 calls during a campaign season. How many calls did the call center receive?

7. RECREATION The new town recreation center is about 7 ! 5 ! 7 ! 5 ! 5 square feet. About how many square feet are in the recreation center?

Find the prime factorization of each number.

8. "96 9. 42

10. 144 11. 17

12. 54 13. "110

14. 16 15. 156

16. PUMPKINS The table shows the weight of the winning pumpkins in a contest at the state fair. Find the prime factorization of each weight.

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

13.

14.

15.

16.

Pumpkin Weight (lb)Mr. Smith’s 112Ms. Gonzalez’s 98Mrs. Johnson’s 83Mr. Cyzdin’s 72


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ReviewExample 1

Find 6 ! 3 ! 6 ! 3 ! 6.6 ! 3 ! 6 ! 3 ! 6 = 3 ! 3 ! 6 ! 6 ! 6 Commutative Property

= (3 ! 3) ! (6 ! 6 ! 6) Associative Property

= 9 ! 216 Multiply. = 1,944 Simplify.

Example 2

Find (!4)(!4)(!4)(!4).("4)( "4)( "4)( "4) = 256 Multiply.

Exercises

Find each product.

1. 9 ! 9 ! 9

2. ("6)("2)("2)("6)

3. 7 ! 4 ! 7 ! 4 ! 4

4. 8 ! 3 ! 8 ! 8 ! 3

5. 5 ! 5 ! 5

6. ("3)(9)("3)(9)("3)("3)

7. 2 ! 5 ! 2 ! 5 ! 2

8. ("4)(6)("4)(6)

9. DOGHOUSE The doghouse at the local kennel

is about 2 # 4 # 3 # 3 # 4 # 2 square feet. About

how many square feet is the doghouse?

10. SALES A furniture store sold 9 ! 5 ! 9 ! 5 ! 5 dollars

worth of furniture during a one-day sale. How many

dollars was the furniture?

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

001_011_FLCRMC3C05_892304.indd 6001_011_FLCRMC3C05_892304.indd 6 12/15/08 11:53:33 PM12/15/08 11:53:33 PM

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Apply 1. DISTANCE The distance from Eli’s house

to his grandparents’ house is 4 ! 3 ! 4 ! 3 ! 3 miles. How many miles away is the grandparents’ house?

2. TEMPERATURE The table shows the length and width of Florida at its most distant points. Find the prime factorization of each number.

Measurement Distance (mi) Length 500Width 160

3. DOGS The table shows the weights of the dogs in a local dog show. Find the prime factorization of each number.

Dog Weights (lb)Irish Setter 68Jack Russell Terrier 15Great Dane 114Beagle 28

4. HAY Mr. Day feeds the cows on his farm 2 ! 7 ! 5 ! 5 ! 5 pounds of hay per week. How many pounds of hay do they eat per week?

5. NURSERY A nursery owner wants to build a new greenhouse that will have 5 ! 2 ! 5 ! 5 ! 5 square feet. How many square feet is the greenhouse?

6. FUNDRAISER The table shows the amount of money each student raised for the school fundraiser. Find the prime factorization of each number.

Student Money RaisedDorsey $125Danica $88Jazzra $96Alen $150


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Chapter

Diagnostic Test

Find each product.

1. 3 ! 3 ! 3

2. 6 ! 6 ! 6 ! 6 ! 6

3. (9)("5)("5)(9)("5)

4. 7 ! 7 ! 2 ! 2 ! 2

5. 5 ! 3 ! 5 ! 3 ! 3

6. BIRDS A bird call manufacturer sold 8 ! 4 ! 4 ! 8 ! 4 calls in two months. How many calls did the manufacturer sell?

7. RIVER The St. John’s River in Florida is 2 ! 5 ! 31 miles long. About how many miles long is the river?

Find the prime factorization of each number.

8. "82 9. 38

10. 208 11. 11

12. 60 13. "78

14. 20 15. 124

16. HEIGHT The table shows the heights of four sixth grade students. Find the prime factorization of each height.

Student Height (in.)Oksana 59Silvia 62Joseph 64Diego 66

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

13.

14.

15.

16.


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.5

Chapter

Pretest

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

Write each expression using exponents.

1. 8 ! 8 ! 8

2. 4 ! 4 ! 4 ! 4


3. ("5)3

4. 36

Simplify. Express using exponents.

5. 82 ! 84

6. (73)2

Write each expression using a positive exponent.

7. 9"4

8. 6"3

Write each number in standard form.

9. 3.68 # 104

10. 7.924 # 10"5

Find each square root.

11. - $ %% 36

12. $ %% 16 & 25


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Chapter

Student Glossary

This is an alphabetical list of new vocabulary terms you will learn in Chapter 5. Fold the page vertically and use it as a bookmark. As you study the chapter, write each term’s definition or description in as few words as possible.

Vocabulary Word Definition/Description/Example

base

exponent

irrational numbers

monomial

perfect square

power

radical sign

real number

scientific notation

square root

Fold over


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.Facts Practice

Multiply using mental math.

1. 34.6 ! 0.001 2. 5.75 ! 10 3. 0.876 ! 1,000 4. 0.003 ! 100

5. 92.4 ! 0.01 6. 0.05 ! 0.1 7. 6.53 ! 10 8. 118 ! 0.001

9. 9 ! 0.01 10. 3.7 ! 1,000 11. 7.6 ! 100 12. 5.4 ! 0.01

13. 4.82 ! 0.1 14. 12 ! 0.001 15. 15.3 ! 10 16. 88 ! 0.01

17. 1.6 ! 100 18. 35.4 ! 10 19. 26 ! 0.1 20. 2.15 ! 1,000

5Chapter


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5-1A


a. 7 ! 7 ! 7 ! 7

7 ! 7 ! 7 ! 7 = 74 The number 7 is a factor 4 times. So, 7 is the base and 4 is the exponent.

b. y ! y ! x ! y ! x

y ! y ! x ! y ! x = y ! y ! y ! x ! x Commutative Property

= (y ! y ! y) · (x · x) Associative Property

= y3 ! x2 Definition of exponents

To evaluate a power, perform the repeated multiplication to find the product.

Evaluate (-6)4.

(-6)4 = (-6) ! (-6) ! (-6) ! (-6) Write the power as a product.

= 1,296 Multiply.

The order of operations states that exponents are evaluated before multiplication, division, addition, and subtraction.

Evaluate m2 + (n - m)3 if m = -3 and n = 2.

m2 + (n - m)3 = (-3)2 + (2 - (-3))3 Replace m with -3 and n with 2.

= (-3)2 + (5)3 Perform operations inside parentheses.

= (-3 ! -3) + (5 ! 5 ! 5) Write the powers as products.

= 9 + 125 or 134 Add.

Exercises


1. 8 ! 8 ! 8 ! 8 ! 8 2. a ! a ! a ! a ! a ! a 3. 5 ! 5 ! 9 ! 9 ! 5 ! 9 ! 5 ! 5


4. 24 5. (-3)5 6. ( 3 " 4 )

3

ALGEBRA Evaluate each expression if a = 5 and b = -4.

7. a2 + b2 8. (a + b)2 9. a + b2

The product of repeated factors can be expressed as a power. A power consists of a base and an exponent. The exponent tells how many times the base is used as a factor.

ReteachPowers and Exponents

Example 1

Example 2

Example 3


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5-1A

Skills PracticePowers and Exponents


1. 2 ! 2 ! 2 ! 2 2. 9 ! 9

3. 7 ! 7 ! 5 ! 5 ! 5 ! 5 4. 3 " 8 ! 3 "

8 ! 3 "

8

5. c ! 1 " 4 ! c ! 1 "

4 ! 1 "

4 6. s ! 6 ! s ! s ! 6 ! 6 ! s

7. 8 ! x ! 2 ! 2 ! 2 ! x ! 8 8. a ! (-4) ! b ! a ! b ! (-4) ! (-4)

9. 1 " 3 ! n ! 4 ! n ! 1 "

3 ! n ! 4 ! 4 10. 9 ! 9 ! x ! w ! x ! y ! w ! 9 ! y


11. 43 12. 25 13. (-8)3

14. ( 3 " 5 )

4 15. 28 - 32 16. 23 ! 52

17. 34 - (-4)2 18. 6 + 26 19. (-3)3 ÷ 32

ALGEBRA Evaluate each expression if g = 2 and h = -3.

20. g4 21. (g + h)3 22. h4 - h3

23. g3 + h2 24. (g - h)2 + h2 25. h4 - (h - g)3


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5-1A

Homework PracticePowers and Exponents


1. 3 ! 3 ! m 2. ( 1 " 4 ) ( 1 "

4 ) ( 1 "

4 )

3. 2 ! d ! 5 ! d ! d ! 5 4. p ! (-9) ! p ! (-9) ! p ! q ! q

5. g ! (-7) ! (-7) ! g ! h ! (-7) ! h 6. x ! 1 " 8 ! x ! x ! y ! 1 "

8 ! y ! x


7. (-8)4 8. ( 1 " 5 )

3 9. (- 3 "

5 )

5

10. (-2)3 + 52 11. 34 - 52 12. (-2)5 - (-2)4

13. 43 ÷ 23 14. 53 ! 23 15. 17 + (-3)4

ALGEBRA Evaluate each expression.

16. r3 - s, if r = 5 and s = 4 17. m2 - n3, if m = 6 and n = 2

18. f - g4, if f = 3 and g = -5 19. (x5 - y2)2 + x3, if x = 2 and y = 8

20. Replace with <, >, or = to make a true statement: 24 42.

21. ISLANDS Florida has about 22 ! 32 ! 53 islands (over 10 acres). About how many islands is this?

22. SPACE The radius of Jupiter is about 7.15 # 104 kilometers. Write this distance is standard form.



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5-1A

1. GEOMETRY The volume of a cube can be found by raising the side length to the third power. What is the volume of the cube below?

14 in.

2. SPORTS In the first round of a local tennis tournament, there are 25 matches. Find the number of matches.

3. PALM TREES There are about 23 ! 3 ! 53 species of palm trees in the whole world. About how many species is this?

4. NATURE A forest fire affected about 34 ! 104 acres of land. About how many acres did the fire affect?

5. BIOLOGY A scientist estimates that after a certain amount of time, there would be 25 ! 33 ! 105 bacteria in a Petri dish. About how many bacteria is this?

6. ACTIVISM A total of 54 ! 73 people have signed a petition. How many people signed the petition?

7. MEASUREMENT There are 106 millimeters in one kilometer. The distance from Dana’s house to her uncle’s house is 44 kilometers. What is this distance in millimeters?

8. DOGS Dedra’s dog weighs 5 ! 24 pounds. What is the weight of Dedra’s dog?

Problem-Solving PracticePowers and Exponents


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5-1A

Enrich

A-Mazing ExponentsSolve the following puzzle by finding the correct path through the boxes. The solution is a famous quote from United States history.

Starting with Box 1, draw an arrow to the box next or diagonal to Box 1 with the expression of the least value. The arrow cannot go to a box that has already been used. The first arrow has been drawn to get you started.

When you have finished drawing your path through the boxes, write the box numbers on the lines below. Put the numbers in the order in which they are connected. Then use the chart at the right to convert each box number to a letter.

1

53

6

63

11

44+ 162

16

36- 63

21

29+ 92

2

132

7

27

12

35

17

83- 18

22

232

3

43+ 34

8

24 ! 32

13

44

18

162+ 63

23

36- 35

4

172

9

25 ! 32

14

73

19

192

24

212

5

45- 93

10

182

15

53+ 35

20

28+ 112

25

23 ! 72

Box Number 1 7Letter G I

Box NumberLetter H

1 G2 M3 E4 E5 R6 E7 I8 V9 B10 T11 D12 L13 I14 Y15 R16 E17 E18 E19 O20 G21 T22 A23 M24 V25 I


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5-1A

The power key on many calculators makes it easier to evaluate expressions with exponents. It is usually labeled yx or .

Evaluate 54.

Enter: 5 4 625

So, 54 = 625.

Evaluate 25 · 43.

Enter: 2 5 4 3 2048

So, 25 · 43 = 2,048.

Exercises


1. 38 2. 524

3. 2 · 63 4. 43 · 27

5. 3 · 25 · 45 6. 53 · 42 · 25

7. 54 - 33 8. 2 · 43 + 34

9. 3 · 53 + 4 · 27 10. 5 · 23 - 3 · 23

11. (4 + 5)2 + 63 · 25 12. (35 - 25) · 55

13. CHALLENGE 10 · 73 + 6 · 23 · 34 - 5 · 43

Scientific Calculator ActivityThe Power Key

Example 1

Example 2


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5-1 ReteachMultiply and Divide MonomialsB


a. 23 ! 22

23 ! 22 = 23 + 2 The common base is 2.

= 25 Add the exponents.

b. 2s6(7s7)

2s6(7s7) = (2 ! 7)(s6 ! s7) Commutative and Associative Properties

= 14(s6 + 7) The common base is s.

= 14s13 Add the exponents.

The Quotient of Powers rule states that to divide powers with the same base, subtract their exponents.

Simplify k8 "

k . Express using exponents.

k8 "

k1 = k8 -1 The common base is k.

= k7 Subtract the exponents.

Simplify (-2)10 ! 56 ! 63

" (-2)6 ! 53 ! 62 .

(-2)10 ! 56 ! 63

" (-2)6 ! 53 ! 62 = ( (-2)10

" (-2)6 ) ! ( 5

6 "

53 ) ! ( 63 "

62 ) Group by common base.

= (–2)4 ! 53 ! 61 Subtract the exponents.

= 16 ! 125 ! 6 or 12,000 Simplify.

Exercises


1. 52 ! 55 2. e2 ! e7 3. 2a5 ! 6a 4. 4x2(–5x6)

5. 79 "

73 6. v14 "

v6 7. 15w7 "

5w2 8. 10m8 "

2m

9. 25 ! 37 ! 43 "

21 ! 35 ! 4 10. 415 ! (-5)6

" 412 ! (-5)4

The Product of Powers rule states that to multiply powers with the same base, add their exponents.

Example 1

Example 2

Example 3


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5-1 Skills PracticeMultiply and Divide MonomialsB


1. 59 · 53 2. 38 · 3 3. c · c6 4. m5 · m2

5. 3x · 4x4 6. (2h7)(7h) 7. -5d6(8d6) 8. (6k5)(-k4)

9. (-w)(-10w3) 10. -7z4(-3z8) 11. bc3(b2c) 12. 3a4 · 6a2

13. 3m3n2(8mn3) 14. 7t5(-6t5) 15. (3ab2)(a2c5) 16. (9p4)(-8p2)

17. 29 !

23 18. 38 !

34 19. 59 !

52 20. 87 !

8

21. b12 !

b5 22. 12n5 !

4n2 23. 14m3 !

7m2 24. 9r8 !

3r4

25. 24t9 !

6t3 26. 18y6

! 2y

27. a4c6 !

a2c 28. 5

10 !

52

Simplify.

29. 48 · 53 · 76

! 46 · 52 · 75 .

30. (-2)9 · (-3)7 · 43

! (-2)5 · (-3)5 · 41 .


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5-1B


1. k8 · k 2. t7 · t6 3. 2w2 · 5w2

4. 3e3 · 7e3 5. 4r4(-4r3) 6. (-3l2w3)(2lw4)

7. (-11w4)(-5w3x4) 8. (-4b6)(-b2c3) 9. (10t4v5)(3t2v5)

10. 59 !

53 11. 38 !

3 12. b6

! b4

13. g15

! g7 14. 18v5

! 9v

15. 24a6 !

6a5

16. y6 ÷ y3 17. n19 !

n11 18. 9521 !

9518

19. Simplify 55 · 63 · 810

! 53 · 6 · 89 .

20. BONUSES A company has set aside 107 dollars for annual employee bonuses. If the company has 104 employees and the money is divided equally among them, how much will each employee receive?

21. CAR LOANS After making a down payment, Mr. Valle will make 62 monthly payments of 63 dollars each to pay for his new car. What is the total of the monthly payments?

Homework PracticeMultiply and Divide Monomials



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5-1

1. SOUND Decibels are units to measure sound. Ordinary conversation is rated at about 60 decibels (or a relative loudness of 106). Thunder is rated at about 120 decibels (or a relative loudness of 1012). How many times greater is the relative loudness of thunder than the relative loudness of ordinary conversation?

2. GEOMETRY Express the area of a square with sides of length 5ab as a monomial.

3. COMPUTERS The byte is the fundamental unit of computer processing. The byte is based on powers of 2, as shown in the table. How many times greater is a gigabyte than a megabyte?

Memory Term Number of Bytesbyte 20 or 1kilobyte 210

megabyte 220

gigabyte 230

4. GEOMETRY The area of the rectangle in the figure is 24a2b3 square units. Find the width of the rectangle.

5. BOOKS A publisher sells 106 copies of a new book. Each book has 102 pages. How many pages total are there in all of the books sold? Write the answer using exponents.

6. RABBITS Randall has 23 pairs of rabbits on his farm. Each pair of rabbits can be expected to produce 25 baby rabbits in a year. How many baby rabbits will there be on Randall’s farm each year? Write the answer using exponents.

Problem-Solving PracticeMultiply and Divide MonomialsB

6ab


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5-1 EnrichB

Dividing Powers with Different BasesSome powers with different bases can be divided. First, you must be able to write both as powers of the same base. An example is shown below.

39 !

812 = 39 !

(34)2 To find the power of a power, multiply the exponents.

= 39 !

38

= 31 or 3

This method could not have been used to divide 39 !

802 , since 80 cannot be written as a power of 3 using integers.

Simplify each fraction using the method shown above. Express the solution without exponents.

1. 27 !

82 2. 643 !

85 3. 1252 !

253

4. 324 !

164 5. 3433 !

75 6. 814 !

34

7. 1011 !

1,0003 8. 66 !

2162 9. 275 !

94

10. 82 !

22 11. 93 !

33 12. 164 !

83


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5-1 ReteachPowers of MonomialsC

Simplify (53)6.

(53)6 = 53 · 6 Power of a power

= 518 Simplify.

Simplify (-3m2n4)3.

(-3m2n4)3 = (-3)3 · m2 · 3 · n4 · 3 Power of a product

= -27m6n12 Simplify.

Exercises

Simplify.

1. (43)5 2. (42)7 3. (92)4

4. (k4)2 5. [(63)2]2 6. [(32)2]3

7. (5q4r2)5 8. (3y2z2)6 9. (7a4b3c7)2

10. (-4d3e5)2 11. (-5g4h9)7 12. (0.2k8)2

Power of a Power: To find the power of a power, multiply the exponents.

Power of a Product: To find the power of a product, find the power of each factor and multiply.

Example 1

Example 2


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5-1 Skills PracticePowers of MonomialsC

Simplify.

1. (72)3 2. (32)6 3. (83)2 4. (94)2

5. (d7)6 6. (m5)5 7. (h6)3 8. (z7)3

9. [(43)2]2 10. (-5a2b7)7 11. (2m5g11)6 12. [(23)3]2

13. (7a5b6)4 14. (7m3n11)5 15. (-3w3z8)5 16. (-7r4s10)4

GEOMETRY Express the area of each square below as a monomial.

17.

6g 3h 5

18.

13d 5e

GEOMETRY Express the volume of each cube below as a monomial.

19.

7c 5d 2

20.

6r 7s 8


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5-1C

Homework Practice Powers of Monomials

Simplify.

1. (6t5)2 2. (4w9)4 3. (12k6)3 4. (15m8)3

5. (4d3e5)7 6. (-4r6s15)4 7. [(72)2]2 8. [(32)2]3

9. ( 3 ! 5 a6b9)2 10. (4x2)3(3x6)4 11. (0.6p5)3 12. ( 1 !

5 w5x3)

2

GEOMETRY Express the area of each square below as a monomial.

13.

9c6d

14.

14g5h9

15. MEASUREMENT In the Metric System, you would need to have (104)2 grams to equal 1 metric ton. Simplify this measurement by multiplying the exponents, then simplify by finding the actual number of grams needed to equal 1 metric ton.

16. GAMING A video-game designer is using the expression 6n3 in a program to determine points earned, where n is the game level. Simplify the expression for the n2 level.



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5-1 Problem-Solving Practice Powers of MonomialsC

1. DEBATE Charmaine and Aaron are having a debate. Charmaine thinks the answer to their math homework is (42)4, but Aaron says the answer is (44)2. Explain how both Charmaine and Aaron can be correct.

2. LAND Kate was given a square plot of land in which to build. If one side of the plot was (3a)3 feet long, express the area of her plot as a monomial.

(3a)3

3. CRAFTS Numa loves beads and wants to know which amount would be more, a thousand beads or (62)3 beads?

4. TEST The teacher marked Silvano’s problem wrong on his test.

(45)4 = 49

Explain what he did wrong and give the correct answer.

5. WOOD Dmitry calculated that he needs 6s2 square inches of wood for each crate he makes. Simplify the expression when s is replaced by t4.

6. VOLUME Express the volume of the following cube as a monomial.

(4d )2


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5-1

Simplify 8a + 3b - 5b + a.

The monomials 8a and a are like terms, and 3b and -5b are like terms.

8a + 3b - 5b + a Write the polynomial.

= 8a + 3b + (- 5b) + a Definition of subtraction

= (8a + a) + [3b + (-5b)] Group like terms.

= 9a + (-2b) or 9a - 2b Simplify by combining like terms.

Simplify x2 - 2x + 5 + x2 - 1.

x2 + (-2x) + 5 + x2 + (-1) Write the polynomial.

= (x2 + x2) + (-2x) + [5 + (-1)] Group like terms.

= 2x2 + (-2x) + 4 Simplify by combining like terms.

= 2x2 - 2x + 4

Exercises

Simplify each polynomial. If the polynomial cannot be simplified, write in simplest form.

1. 2s + 3d + 3s + 2d 2. 3ƒ - 5g - ƒ - 2g 3. -3h + 6k + 2 - 2k

4. 2e2 - 3e + 6e 5. 2u2 + 5u + 9 - 8u 6. 3r + 2r2 - 4r2

7. 7a2 + 5a + 1 + 3a2 - 8a 8. 2s2 - 7s - 6 - 2s + 4

9. -3d2 + 5d - 8 + 2d2 - d + 5 10. 3x + 5 + 2x + 4y

A monomial is a number, a variable, or a product of numbers and/or variables. An algebraic expression that is the sum or difference of one or more monomials is called a polynomial.

You can simplify polynomials by combining like terms. Like terms must have the same variable and the same power. Thus, 4x2 and -x2 are like terms, while 2x2 and 6x are not.

Enrich

Example 1

Example 2

C

Simplifying Polynomials


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5-1 ReteachProblem-Solving Investigation: Act It OutD

Inola’s dad gave her three quarters on Monday. Each day after that, he gave her twice as many quarters as he gave her on the previous day. By the end of day Thursday, how many quarters had Inola received?

Understand Inola starts with three quarters and receives twice that amount the next day, and then twice that amount the following day, and so on. Use counters, coins, or play money to represent the quarters.

Plan Place the counters neatly; you can use different colors to show the difference from one day to the next.

Solve Find the sum of the counters once they are all placed.

Monday

Tuesday

Wednesday

Thursday

3 + 6 + 12 + 24 = 45

Inola received 45 quarters by the end of day Thursday.

Check The number of quarters can be written as the product of 3 and a multiple of 2. 3 + 3(21) + 3(22) + 3(23) = 45.

Exercises

For Exercises 1 –3, solve each problem using the act it out strategy.

1. TILES Alistair has a red square tile, a blue square tile, a green square tile, and a yellow square tile. How many different ways can he arrange the tiles so that they form a larger square?

2. MONEY Ari wants to buy a comic book that costs $0.65. If he uses exact change, how many different combinations of nickels, dimes, and quarters can he use?

3. NUMBER LINE In a math class game, players are using a number line on the floor. Grace starts at zero and moves forward 7 numbers on her first turn and moves backward 4 numbers on her second turn. If this pattern continues, how many turns will it take for her to move forward to 16?

Example 1


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5-1 Skills PracticeProblem-Solving Investigation: Act It OutD

For Exercises 1–7, use the act it out strategy to solve.

1. A piece on a game board moves forward 8 spaces on its first turn and moves backward 3 spaces on its second turn. If the pattern continues, how many turns will it take for the piece to move at least 30 spaces?

2. How many ways can you arrange 3 paintings in a row on a wall?

3. Livia cut a paper into fourths, then cut each of those pieces into fourths, and then cut each of those pieces into fourths. How many pieces did she then have?

4. A piece on a game board moves forward 6 spaces on its first turn and moves backward 5 spaces on its second turn. If the pattern continues, how many turns will it take for the piece to move at least 10 spaces?

5. Basi is taller than Brody, who is taller than Hektor, who is taller than Kavi. How many different ways can they stand in line so that the tallest person is always last?

6. How many different combinations of quarters, nickels, dimes, and pennies can be used to make $0.25?

7. Roll a number cube 10 times and record the results. Repeat 3 times. Using your results, is there any way to predict which number the number cube will land?

Roll 1 Roll 2 Roll 3 Roll 4 Roll 5 Roll 6 Roll 7 Roll 8 Roll 9 Roll 10

Set 1

Set 2

Set 3


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5-1

Mixed Problem Solving

For Exercises 1 and 2, use the act it out strategy.

1. BILLS Joaquin bought a DVD for $21. He gave the cashier two $20 bills. How many different combinations of $1, $5, and $10 bills can the cashier give him for change?

2. TENNIS Felix, Lolita, Tetsuo, Kaveri, and Maxine are on the school tennis team. When ranked from first to fifth, how many ways can they be ranked if Maxine is always first and Felix is always ranked above Tetsuo?

Use any strategy to solve Exercises 3–6. Some strategies are shown below.

PROBLEM-SOLVING STRATEGIES

• Act it out.

• Work backward.

• Look for a pattern.

• Choose an operation.

3. PUMPKINS Mr. Greene harvested pumpkins for selling at four markets. He sold one-fifth of his crop at the first market, 40 at the second, 25% of the remaining at the third, and twice what he sold at the second at the forth market. If Mr. Greene has one pumpkin remaining, how many pumpkins did he sell?

4. CHORES Kimberley has the choice of washing the car, mowing the lawn, or raking leaves on Saturday and baking a cake, washing the dishes, or doing the laundry on Sunday. In how many ways can she choose one chore for each day?

5. FUNDRAISER The drama club is selling 100 T-shirts for $15 each for a fundraiser. The T-shirts cost a total of $623. If they sell all the T-shirts, how much money will be raised for the drama club?

6. NEWS Tuan told good news to two friends. They each told three friends, and each of their friends told three friends. How many people had heard good news at this point?

Homework PracticeProblem-Solving Investigation: Act It OutD



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5-1

For Exercises 1–6, use the act it out strategy to solve.

Problem-Solving PracticeProblem-Solving Investigation: Act It OutD

1. PHOTOGRAPHY Maura has six photos that she has taken framed and hanging in a row on the wall. If she wants to rearrange them so that the middle two photos stay in place, how many different ways can she arrange the photos?

2. TEAMS There are 5 players on a basketball team. If Evan always plays in the point guard position, and Holman always plays in the power forward position, how many different ways can the coach arrange Mohe, Alki, and Shahid in the center, small forward, and off-guard positions?

3. MONEY Elaine wants to buy an apple that costs $0.55. How many different combinations of quarters, nickels, and dimes can be used to make $0.55?

4. AGES Parvin is older than Jan, who is older than Meg, who is older than Laurie, who is older than Vicky, who is older than Leslie. How many different ways can they stand in line so that the youngest person is always first, and the oldest person is always last?

5. E-MAILS Nina received two E-mails on Monday. Every day after that she received one more than twice as many as the day before. How many E-mails did she receive on Thursday?

6. MONEY Brian wants to buy a muffin that costs $0.80. How many different combinations of nickels and dimes can be used to make $0.80?


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A


ReteachNegative Exponents


a. 7!3

7!3 = 1 " 73 Definition of negative exponent

b. a!4

a!4 = 1 " a4 Definition of negative exponent


a. 5!4

5!4 = 1 " 54 Definition of negative exponent

= 1 " 625

54 = 5 # 5 # 5 # 5

b. (!3)!5

(!3)!5 = 1 " (!3)5 Definition of negative exponent

= 1 " !243

(!3)5 = (!3) # (!3) # (!3) # (!3) # (!3)

Example 3

Write 1 " 65 as an expression using a negative exponent.

1 " 65 = 6!5 Definition of negative exponent

Simplify. Express using positive exponents.

a. x!3 " x5

x!3# x5 = x(!3) + 5 Product of Powers

= x2 Add the exponents.

b. w!5 #

w!7

w!5 "

w!7 = w!5 ! (!7) Quotient of Powers

= w2 Subtract the exponents.

Exercises


1. a!8 2. 6!3 3. n!4


4. 7!2 5. 9!3 6. (!2)!5

Write each fraction as an expression using a negative exponent.

7. 1 " 57 8. 1 "

36 9. 1 " x8


10. 4!2 # 4!4 11. r!3 # r5 12. h!2 "

h4

Any nonzero number to the zero power is 1. Any nonzero number to the negative n power is the multiplicative inverse of the number to the nth power.

Example 1

Example 2

Example 4


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Skills PracticeNegative Exponents


1. 4!5 2. 5!7 3. m!9 4. s!6

5. f!3 6. (!2)!6 7. (!4)!3 8. w!12


9. (!5)!5 10. 3!2 11. 8!3 12. (!9)!4


13. 1 " 123 14. 1 "

81 15. 1 "

t6 16. 1 "

88


17. 2!6 # 23 18. s!5 # s7 19. m8 "

m!4 20. 109 "

108

21. y!3 # y3 22. s!5 # s7 23. x6 "

x!3 24. 68 "

68

25. 3!5 "

3!3 26. e!3 "

e!2 27. n!6 "

n4 28. j!2

" j!2


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Homework PracticeNegative Exponents


1. 8!5 2. 3!9 3. z!2 4. p!4


5. (!6)!5 6. 8!4 7. 2!9 8. (!7)!3


9. 1 " 29 10. 1 "

64 11. 1 "

e5 12. 1 " 74


13. 65 "

62 14. n!2 # n!3 15. w3 "

w!1 16. k!4 "

k!6

17. ROADS A state highway that is 44 miles long runs parallel to a smaller country road that is 42 miles long. How many times longer than the country road is the state highway? Write the answer as a number with a positive exponent.

18. FUNDRAISERS The hospital spent 95 dollars on new medical equipment this year. Last year, they spent 97 dollars. How many times more money did they spend last year than this year?

19. MEASUREMENT 1 milligram is equal to 10!3 grams. Write this number using a positive exponent.

20. DISTANCE A long-distance runner runs 25 miles one week and 27 miles the next week. How many times farther did he run in the second week than in the first week?



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A


1. MOTHS A Polyphemus Moth caterpillar weighs about 1 !

642 times less when it first becomes a larva than it does when it is fully grown. Write this number using a negative exponent.

2. WEIGHT The length of one common termite is about 30"2 meters. Write this number using a positive exponent.

3. MONEY The school system spent 38 dollars on fuel for buses and school vehicles per week last year. This year, they spent 310 dollars per week. How many times more did they spend per week this year than last year?

4. MEASUREMENT The table converts the size of each measurement to kilograms. Write each number using a positive exponent.

Amount Amount in Kilograms1 centigram 10-5

1 decigram 10-4

1 dekagram 10-2

5. SCIENCE Electrons are smaller than 10-18 meters. Write this number using a positive exponent.

6. MONEY A bank loans a new business 67 dollars to get started. If the business pays back 65 dollars per year, how many years will it take to pay off the loan? Write your answer using a positive exponent.

Problem-Solving PracticeNegative Exponents


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Solve the problems below to find one group of people who use exponents every day on the job. For each exercise, determine which box matches the value shown. Write the letter of that box in the appropriate blank below.

1 ! 2 " 7 " 7 " 7 ( 2 !

7 )

3 2 ! 7 " 2 !

7 " 2 !

7 0.8 " 0.8 0.82 1 !

8 " 1 !

8

A 1. S C 2. T

3 " 3 " 3 " 3 3#4 1 ! 3 " 1 !

3 " 1 !

3 " 1 !

3 3 " 3 " 3 " 4 ( 3 !

4 )

3 3 !

4 " 3 !

4 " 3 !

4

O 3. I R 4. E

5 " 5 " 5 53 1 ! 5 " 1 !

5 " 1 !

5 2 " 2 " 2 " 2 24 2 !

5 " 2 !

5 " 2 !

5 " 2 !

5

N 5. S T 6. S

5 " 6 " 6 5 ! 62 5 !

6 " 1 !

6 9 " 9 " 9 " 9 9#4 1 !

9 " 1 !

9 " 1 !

9 " 1 !

9

F 7. I E 8. S

3 " 3 " 3 33 1 ! 3 " 1 !

3 " 1 !

3 8 " 8 " 8 " 8 1 !

84 1 ! 8 " 1 !

8 " 1 !

8 " 1 !

8

T 9. N L 10. S

A group of people who use exponents every day are:

1. 2. 3. 4. 5. 6. 7. 8. 9. 10.

Enrich


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ReteachScientific Notation

Write 8.65 ! 107 in standard form.

8.65 ! 107 = 8.65 ! 10,000,000 107 = 10 ! 10 ! 10 ! 10 ! 10 ! 10 ! 10 or 10,000,000

= 86,500,000 The decimal point moves 7 places to the right.

Write 9.2 ! 10–3 in standard form.

9.2 ! 10-3 = 9.2 ! 0.001 The decimal point moves 3 places to the left.

= 0.0092

Write 76,250 in scientific notation.

76,250 = 7.625 ! 10,000 The decimal point moves 4 places.

= 7.625 ! 104 Since 76,250 is >1, the exponent is positive.

Write 0.00157 in scientific notation.

0.00157 = 1.57 ! 0.001 The decimal point moves 3 places.

= 1.57 ! 10–3 Since 0.00157 is <1, the exponent is negative.

Exercises


1. 5.3 ! 101 2. 9.4 ! 103

3. 7.07 ! 105 4. 2.6 ! 10-3

5. 8.651 ! 10-2 6. 6.7 ! 10-6

Write each number in scientific notation.

7. 561 8. 14

9. 56,400,000 10. 0.752

11. 0.0064 12. 0.000581

B

A number in scientific notation is written as the product of a factor that is at least one but less than ten and a power of ten.

Example 1

Example 2

Example 3

"""#

Example 4

"" #

"" "# """

"# "


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B


1. 6.7 ! 101 2. 6.1 ! 104

3. 1.6 ! 103 4. 3.46 ! 102

5. 2.91 ! 105 6. 8.651 ! 107

7. 3.35 ! 10-1 8. 7.3 ! 10-6

9. 1.49 ! 10-7 10. 4.0027 ! 10-4

11. 5.2277 ! 10-3 12. 8.50284 ! 10-2


13. 34 14. 273

15. 79,700 16. 6,590

17. 4,733,800 18. 2,204,000,000

19. 0.00916 20. 0.29

21. 0.00000571 22. 0.0008331

23. 0.0121 24. 0.00000018

Skills PracticeScientific Notation


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Homework PracticeScientific Notation


1. 9.03 ! 102 2. 7.89 ! 103 3. 4.115 ! 105 4. 3.201 ! 106

5. 5.1 ! 10-2 6. 7.7 ! 10-5 7. 3.85 ! 10-4 8. 1.04 ! 10-3


9. 4,400 10. 75,000 11. 69,900,000 12. 575,000,000

13. 0.084 14. 0.0099 15. 0.000000515 16. 0.0000307

17. Which number is greater: 3.5 ! 104 or 2.1 ! 106?

18. Which number is less: 7.2 ! 107 or 9.9 ! 105?

19. POPULATION The table lists the populations of five countries. List the countries from least to greatest population.

20. SOLAR SYST EM Pluto is 3.67 ! 109 miles from the Sun. Write this number in standard form.

21. MEASUREMENT One centimeter is equal to about 0.0000062 mile. Write this number in scientific notation.

22. DISASTERS In 2005, Hurricane Katrina caused over $125 billion in damage in the southern United States. Write $125 billion in scientific notation.

B

Country PopulationAustralia 2 ! 107

Brazil 1.9 ! 108

Egypt 7.7 ! 107

Luxembourg 4.7 ! 105

Singapore 4.4 ! 106



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Problem-Solving PracticeScientific NotationB

1. MEASUREMENT There are about 25.4 millimeters in one inch. Write this number in scientific notation.

2. POPULATION In the year 2000, the population of Rahway, New Jersey, was 26,500. Write this number in scientific notation.

3. MEASUREMENT There are 5,280 feet in one mile. Write this number in scientific notation.

4. PHYSICS The speed of light is about 1.86 ! 105 miles per second. Write this number in standard notation.

5. COMPUTERS A CD can store about 650,000,000 bytes of data. Write this number in scientific notation.

6. SPACE The diameter of the Sun is about 1.39 ! 109 meters. Write this number in standard notation.

7. ECONOMICS The U.S. Gross Domestic Product in the year 2004 was1.17 ! 1013 dollars. Write this number in standard notation.

8. MASS The mass of planet Earth is about 5.98 ! 1024 kilograms. Write this number in standard notation.


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Enrich

Scientific Notation and SpaceWhat travels faster than jets, spaceships, and sound waves? Light does. The speed of light is about 3 ! 108 meters per second (3 ! 105 kilometers per second). Because distances in space are so large, they are often discussed in terms of light-years, or the distance a photon of light would travel in a year.

1 light-year = speed of light in meters per second ! number of seconds in a year.

There are 365 ! 24 ! 60 ! 60 = 31,536,000 " 3.15 ! 107 seconds in a year.

1 light-year " (3 ! 108) ! (3.15 ! 107) = 9.45 ! 1015 meters = 9.45 ! 1012 kilometers

When performing operations with numbers in scientific notation, it is often helpful to consider the decimal part and the power of ten separately.

(2.3 ! 103) (1.4 ! 102) = (2.3 ! 1.4) ! (103 ! 102) = 3.22 ! (10 ! 10 ! 10) ! (10 ! 10) = 3.22 ! 105

Use the information above and the following tables to answer Exercises 1–6.

1. How long does it take a photon of light to travel from the Sun to Earth?

2. How long does it take a photon of light to travel from the Sun to Pluto?

3. How far is Alpha Centauri from Earth in kilometers?

4. The Pleiades Cluster is about how many times as far from Earth as Alpha Centauri?

5. If you see Sirius in the night sky, how long ago was that light emitted from the star?

6. The diameter of Jupiter is how many times the diameter of Earth?

B

ObjectDistance from Sun

(km)

Diameter (km)

Mercury 5.7 ! 107 5.9 ! 103

Venus 1.07 ! 108 1.2 ! 104

Earth 1.5 ! 108 1.3 ! 104

Mars 2.3 ! 108 6.8 ! 103

Jupiter 7.8 ! 108 1.43 ! 105

Saturn 1.4 ! 109 1.2 ! 105

Uranus 2.9 ! 109 5.1 ! 104

Neptune 4.5 ! 109 5 ! 104

Pluto 5.9 ! 109 2.4 ! 103

ObjectDistance

from Earth (light-years)

Alpha Centauri 4.27Sirius (Dog star) 8.7Arcturus 36Pleiades Cluster 400Betelgeuse 520Deneb 1,600Crab Nebula 4,000Center of Milky Way Galaxy

38,000


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Example 1

Evaluate (3.4 ! 105)(2.3 ! 103). Express the result in scientific notation.

(3.4 ! 105)(2.3 ! 103) = (3.4 ! 2.3)(105 ! 103) Commutative and Associative Properties

= (7.82)(105 ! 103) Multiply 3.4 by 2.3.

= 7.82 ! 105 + 3 Product of Powers

= 7.82 ! 108 Add the exponents.

Example 2

Evaluate 2.325 ! 104 "

3.1 ! 102 . Express the result in scientific notation.

2.325 ! 104 "

3.1 ! 102 = ( 2.325 "

3.1 ) ( 104

" 102 ) Associative Property

= (0.75) ( 104 "

102 ) Divide 2.325 by 3.1.

= 0.75 ! 104 – 2 Quotient of Powers

= 0.75 ! 102 Subtract the exponents.

= 0.75 ! 102 Write 0.75 ! 102 in scientific notation.

= 7.5 ! 10 Since the decimal point moved 1 place to the right, subtract 1 from the exponent.

Example 3

Evaluate (5.24 ! 105) + (8.65 ! 106). Express the result in scientific notation.

(5.24 ! 105) + (8.65 ! 106) = (5.24 ! 105) + (86.5 ! 105) Write 8.65 ! 106 as 86.5 ! 105.

= (5.24 + 86.5) ! 105 Distributive Property

= 91.74 ! 105 Add 5.24 and 86.5.

= 9.174 ! 106 Write 91.74 ! 105 in scientific notation.

Exercises

Evaluate each expression. Express the result in scientific notation.

1. (6.7 ! 104)(2.9 ! 105) 2. (4.3 ! 104) + (5.21 ! 105)

3. 5.46 ! 105 "

8.4 ! 103 4. (9.6 ! 105) – (3.7 ! 103)

CReteachCompute with Scientific Notation

You can use the Product of Powers and Quotient of Powers properties to multiply and divide numbers written in scientific notation.

#


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1. (5.8 ! 105)(6.4 ! 102) 2. (3.92 ! 106)(2.2 ! 104)

3. 2.952 ! 106 "

3.6 ! 103 4. 2.052 ! 107 "

5.4 ! 104

5. (6.9 ! 107) + (2.12 ! 105) 6. (1.78 ! 104) + (5.35 ! 103)

7. (8.4 ! 107) – (6.3 ! 106) 8. (9.62 ! 105) – (2.58 ! 103)

9. 6.256 ! 108 "

6.8 ! 104 10. 2.888 ! 105 "

7.22 ! 102

11. (3.68 ! 103)(2.4 ! 106) 12. (7.2 ! 107)(1.82 ! 102)

13. (6.78 ! 104) – (4.13 ! 102) 14. 3.024 ! 106 "

4.8 ! 102

15. (5.9 ! 108) + (2.6 ! 106) 16. (3.45 ! 107)(1.68 ! 104)

17. (8.33 ! 103) + (4.1 ! 105) 18. (6.82 ! 105) – (3.11 ! 104)

Skills PracticeCompute with Scientific NotationC


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Homework PracticeCompute with Scientific Notation


1. (7.3 ! 108)(2.4 ! 103) 2. 4.62 ! 107 "

1.2 ! 104

3. 8.64 ! 106 "

4.32 ! 103 4. (5.32 ! 108) – (4.6 ! 106)

5. (9.67 ! 106) + (3.45 ! 105) 6. (4.5 ! 103)(1.6 ! 105)

7. (2.82 ! 109) + (6.3 ! 107) 8. (3.64 ! 106) – (2.18 ! 104)

9. 2.144 ! 107 "

3.2 ! 104 10. (7.2 ! 107)(1.82 ! 102)

11. (9.8 ! 105) – (6.7 ! 103) 12. (6.98 ! 105) + (1.65 ! 107)

13. (2.46 ! 107)(1.78 ! 102) 14. 3.936 ! 105 "

2.4 ! 102

15. MARS The diameter of Mars is about 6.8 ! 103 kilometers. The diameter of Earth is about 1.2763 ! 104 kilometers. About how much greater is Earth’s diameter than the diameter of Mars?

16. WAREHOUSE A factory builds a new warehouse that is approximately 1.28 ! 105 square feet. Later, they add on 1.13 ! 103 more square feet for offices. Use scientific notation to write the total size of the new building.

C



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1. OCEAN Humpback whales are known to weigh as much as 8 ! 104 pounds. The tiny krill they eat weigh only 2.1875 ! 10"3 pounds. How many times greater than krill are humpback whales?

2. MEASUREMENT One inch is equal to 1.5782 ! 10"5 miles. One centimeter is equal to 6.2137 ! 10"6 miles. How many miles greater is one inch than one centimeter?

3. MONUMENT The Statue of Liberty is about 1.5108 ! 102 feet tall from the base to the torch. The pedestal is 1.54 ! 102 feet tall. How tall is the Statue of Liberty from the foundation of the pedestal to the top of the torch?

4. FUNDRAISER The table shows the amount of money raised by each region for cancer awareness. How much money did the North and South raise together?

5. TURKEYS When the National Wild Turkey Federation was formed in 1973, there were only about 1.3 ! 106 wild turkeys in North America. Now there are over 7 ! 106 wild turkeys in North America. About how many more turkeys are there now than there were in 1973?

6. MONEY A bank starts the day with 2.93 ! 104 dollars in the vault. At the end of the day, the bank has 3.5 ! 105 dollars in the vault. How much more money is in the vault at the end of the day than there was in the morning?

Problem-Solving PracticeCompute with Scientific NotationC

Region Amount Raised ($)East 1.46 ! 104

North 2.38 ! 104

South 6.75 ! 103

West 8.65 ! 103


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Planet PlacementThe diagram below shows the planets of our solar system. Use it to help you answer the exercises below.

Sun Mercury Venus Earth Mars Jupiter Saturn Uranus Neptune

1. Mercury is about 3.5984 ! 107 miles from the Sun. Earth is about 9.2957 ! 107 miles from the Sun. About how far is Mercury from Earth?

2. Saturn is about 8.8819 ! 108 miles from the Sun. How much farther from the Sun is Saturn than Earth?

3. Uranus is about 1.784 ! 109 miles from the Sun. About how far is Uranus from Mercury?

4. Pluto is considered to be “dwarf planet” in our solar system. That is, it is massive enough to be rounded by its own gravity. Pluto is about 2.78631 ! 109 miles farther from the Sun than Saturn. About how far from the Sun is Pluto?

5. About how many times farther from the Sun is Pluto than Mercury?

EnrichC


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A graphing calculator can be used to simplify expressions written in scientific notation. You can choose whether you want the answer given as a decimal or given in scientific notation.

Evaluate the expression (2 ! 102)(1.5 ! 103). Write your answer as a decimal.

Enter: 2 [EE] 2 ! 1.5 [EE] 3 300000

So, (2 ! 102)(1.5 ! 103) = 300,000

Evaluate the expression (4.1 ! 103) + (2.7 ! 103). Write your answer in scientific notation.

Enter: [QUIT]

Enter: 4.1 [EE] 3 2.7 [EE] 3 6.8E3

So, (4.1 ! 103) + (2.7 ! 103) = 6.8 ! 103.

Exercises

Evaluate each expression. Write each answer as a decimal.

1. (3.2 ! 102)(4.3 ! 102) 2. 5.4 ! 105 "

2 ! 103 3. (8.1 ! 103) + (5 ! 102)

Evaluate each expression. Write each answer in scientific notation.

4. (7.6 ! 104) – (2.7 ! 103) 5. (6 ! 102)(5 ! 103) 6. 6.9 ! 104 "

2.3 ! 101

TI-84 Plus ActivityScientific NotationC

Example 1

Example 2


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5-3A

ReteachSquare Roots

A square root of a number is one of its two equal factors. A radical sign, ! " , is used to indicate a positive square root. Every positive number has both a negative and positive square root.


! " 1 Find the positive square root of 1; 12 = 1.

- ! "" 16 Find the negative square root of 16; (-4)2 = 16.

± ! "" 0.25 Find both square roots of 0.25; 0.52 = 0.25.

! "" -49 There is no real square root because no number times itself is equal to -49.

Solve a2 = 4 ! 9 . Check your solution(s).

a2 = 4 # 9 Write the equation.

a = ± ! " 4 # 9 Definition of square root

a = 2 # 3 or - 2 #

3 Check 2 #

3 · 2 #

3 = 4 #

9 and (- 2 #

3 ) (- 2 #

3 ) = 4 #

9 .

The equation has two solutions, 2 # 3 and - 2 #

3 .

ExercisesFind each square root.

1. ! " 4 2. ! " 9

3. - ! "" 49 4. - ! "" 25

5. ± ! "" 0.01 6. - ! "" 0.64

7. ! "" 9 # 16

8. ! "" -1 # 25

ALGEBRA Solve each equation. Check your solution(s).

9. x2 = 121 10. a2 = 3,600

11. p2 = 81 # 100

12. t2 = 121 # 196

Examples

Example 5


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5-3A

Skills PracticeSquare Roots


1. ! "" 16 2. - ! " 9

3. ! "" 36 4. ! "" 196

5. ! "" 121 6. - ! "" 81

7. - ! "" 0.04 8. ! """ -289

9. ± ! "" 0.81 10. - ! "" 400

11. ! "" 16 # 49

12. ! "" 49 # 100


13. s2 = 81 14. t2 = 36

15. x2 = 49 16. 256 = z2

17. 900 = y2 18. 1,024 = h2

19. c2 = 49 # 64

20. a2 = 25 # 121

21. 1 # 100

= d2 22. 144 # 169

= r2

23. b2 = 9 # 441

24. x2 = 121 # 400


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5-3A

Homework PracticeSquare Roots


1. ! "" 36 2. - ! "" 144 3. - ! "" 9 # 16

4. ! "" 1.96

5. ± ! "" 2.25 6. ± ! "" 121 # 289

7. ! "" -81 # 100

8. ± ! """ 0.0025

9. - ! "" 0.49 10. - ! "" 3.24 11. - ! "" 25 # 441

12. ± ! "" 361


13. h2 = 121 14. 324 = a2 15. x2 = 81 # 169

16. 0.0196 = m2 17. ! " y = 6 18. ! " z = 8.4

19. GARDENING Moesha has 196 pepper plants that she wants to plant in square formation. How many pepper plants should she plant in each row?

20. RESTAURANTS A new restaurant has ordered 64 tables for its outdoor patio. If the manager arranges the tables in a square formation, how many will be in each row?

GEOMETRY The formula for the perimeter of a square is P = 4s, where s is the length of a side. Find the perimeter of each square.

21.

Area =144 square

inches

22. Area =

81 square feet

23.

Area =324 square

meters



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5-3A

Problem-Solving PracticeSquare Roots

1. PLANNING Rosy wants a large picture window put in the living room of her new house. The window is to be square with an area of 49 square feet. How long should each side of the window be?

2. GEOMETRY If the area of a square is 81 square meters, how many meters long is each side?

3. ART A miniature portrait of George Washington is square and has an area of 169 square centimeters. How long is each side of the portrait?

4. BAKING Cody is baking a square cake for his friend’s wedding. When served to the guests, the cake will be cut into square pieces 1 inch on a side. The cake should be large enough so that each of the 121 guests gets one piece. How long should he make each side of the cake?

5. ART Cara has 196 marbles that she is using to make a square formation. How many marbles should be in each row?

6. GARDENING Tate is planning to put a square garden with an area of 289 square feet in his back yard. What will be the length of each side of the garden?

7. HOME IMPROVEMENT Basil has 324 square paving stones that he plans to use to construct a square patio. How many paving stones will make up the width of the patio?

8. GEOMETRY If the area of a square is 529 square inches, what is the length of a side of the square?


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5-3A

Enrich

Cube RootsA square root is just one of many kinds of roots. Another kind of root is the cube root. Just as the number 9 is a perfect square because it is a square of a whole number, the number 27 is a perfect cube because it is the cube of a whole number.

Square Root

The square root of 9 is 3 because 3 ! 3 = 9.In symbols, we can write: " # 9 = 3.

Cube Root

A cube root of 27 is 3 because 3 ! 3 ! 3 = 27.In symbols, we can write: 3 " ## 27 = 3.

For Exercises 1 and 2, discuss the problems with a classmate before writing your answer.

1. What does " ## -9 mean? Does this make sense? Explain.

2. What does 3 " ## -27 mean? Does this make sense? Explain.

3. Complete the following table to list some perfect cubes and their cube roots.

Perfect Cube -1 8 64 125 1,000

Cube Root -2 1 6 7 8

Find each cube root.

4. 3 " ## 343 5. 3

" ## 27 $ 64

6. 3 " #### 0.000008

Solve each equation. Check your solution.

7. 3 " # x = -5 8. y3 = 216 9. z3 = -0.512

10. PACKAGING FunTime Woodworkers manufacture letter blocks to be used by young children. Each block is a cube measuring 1 inch on each side. FunTime wants to package the blocks in containers that are perfect cubes, and their marketing research recommends that each package contain at least three full sets of 36 blocks. What is the smallest perfect cube box that will fit all the blocks?


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5-3B Roots of Non-Perfect Squares

Estimate the side length of each square. 1. Square area = 72 sq units

6 9 12 1530

a. On dot paper, copy and cut out a square like the one shown above. Place one edge of your square on the number line shown above. Between what two consecutive whole numbers is the ! "" 72 , the side length of the square, located?

b. Between what two perfect squares is 72 located?

c. Estimate the length of a side of the square. Verify your estimate by using a calculator to compute the value of ! "" 72 .

2. Square area = 128 sq units

a. On dot paper, copy and cut out a square like the one shown above. Place one edge of your square on the number line shown above. Between what two consecutive whole numbers is the ! "" 128 , the side length of the square, located?

b. Between what two perfect squares is 128 located?

c. Estimate the length of a side of the square. Verify your estimate by using a calculator to compute the value of ! "" 128 .


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5-3 ReteachEstimate Square Roots

Most numbers are not perfect squares. You can estimate square roots for these numbers.

Estimate ! "" 204 to the nearest whole number.

• The largest perfect square less than 204 is 196.

• The smallest perfect square less than 204 is 225.

196 < 204 < 225 Write an inequality.

142 < 204 < 152 196 = 142 and 225 = 152.

! "" 142 < ! "" 204 < ! "" 152 Find the square root of each number.

14 < ! "" 204 < 15 Simplify.

So, ! "" 204 is between 14 and 15. Since 204 is closer to 196 than 225, the best whole number estimate for ! "" 204 is 14.

Estimate ! "" 79.3 to the nearest whole number.

• The largest perfect square less than 79.3 is 64.

• The smallest perfect square less than 79.3 is 81.

64 < 79.3 < 81 Write an inequality.

82 < 79.3 < 92 64 = 82 and 81 = 92.

! " 82 < ! "" 79.3 < ! " 92 Find the square root of each number.

8 < ! "" 79.3 < 9 Simplify.

So, ! "" 79.3 is between 8 and 9. Since 79.3 is closer to 81 than 64, the best whole number estimate for ! "" 79.3 is 9.

Exercises

Estimate to the nearest whole number.

1. ! " 8 2. ! "" 37 3. ! "" 14

4. ! "" 26 5. ! "" 62 6. ! "" 48

7. ! "" 103 8. ! "" 141 9. ! "" 14.3

10. ! "" 51.2 11. ! "" 82.7 12. ! """ 175.2

Example 1

Example 2

C


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5-3C

Skills PracticeEstimate Square Roots


1. ! " 5 2. ! "" 18 3. ! "" 10

4. ! "" 34 5. ! "" 53 6. ! "" 80

7. ! "" 69 1 # 2 8. ! "" 99 9. ! "" 120

10. ! "" 77 11. ! "" 171 12. ! "" 230

13. ! "" 147 14. ! "" 194 15. ! """ 290 3 # 7

16. ! "" 440 17. ! "" 578 18. ! "" 730

19. ! """ 1,010 20. ! """ 1,230 21. ! "" 8.42

22. ! "" 17.8 23. ! "" 11.5 24. ! "" 37.7

25. ! "" 23.8 26. ! "" 59.4 27. ! "" 97.3

28. ! """ 118.4 29. ! """ 84.35 30. ! """ 45.92


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5-3C

Homework PracticeEstimate Square Roots


1. ! "" 38 2. ! "" 53 3. ! "" 99 4. ! "" 227

5. ! "" 8.5 6. ! "" 35.1 7. ! "" 67.3 8. ! """ 103.6

9. ! "" 86.4 10. ! "" 45.2 11. ! "" 7 2 # 5 12. ! "" 27 3 #

8

Order from least to greatest.

13. 8, 10, ! "" 61 , ! "" 73 14. ! "" 45 , 9, 6, ! "" 63 15. ! "" 50 , 7, ! "" 44 , 5

ALGEBRA Estimate the solution of each equation to the nearest integer.

16. d2 = 61 17. z2 = 85 18. r2 = 3.7

19. GEOMETRY The radius of a cylinder with volume V and height

10 centimeters is approximately ! "" V # 30

. If a can that is 10 centimeters tall has a volume of 900 cubic centimeters, estimate its radius.

20. TRAVEL The formula s = ! "" 18d can be used to find the speed s of a car in miles per hour when the car needs d feet to come to a complete stop after slamming on the brakes. If it took a car 12 feet to come to a complete stop after slamming on the brakes, estimate the speed of the car.

GEOMETRY The formula for the area of a square is A = s2, where s is the length of a side. Estimate the length of a side for each square.

21. Area =

40 square inches

22.

Area = 97 square

feet



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5-3 Problem-Solving PracticeEstimate Square RootsC

1. GEOMETRY If the area of a square is 29 square inches, estimate the length of each side of the square to the nearest whole number.

2. DECORATING Miki has an square rug in her living room that has an area of 19 square yards. Estimate the length of a side of the rug to the nearest whole number.

3. GARDENING Ruby is planning to put a square garden with an area of 200 square feet in her back yard. Estimate the length of each side of the garden to the nearest whole number.

4. ALGEBRA Estimate the solution of c2 = 40 to the nearest integer.

5. ALGEBRA Estimate the solution of x2 = 138.2 to the nearest integer.

6. ARITHMETIC The geometric mean of two numbers a and b can be found by evaluating ! "" a · b . Estimate the geometric mean of 5 and 10 to the nearest whole number.

7. GEOMETRY The radius r of a certain circle is given by r = ! "" 71 . Estimate the radius of the circle to the nearest foot.

8. GEOMETRY In a triangle whose base and height are equal, the base b is given by the formula b = ! "" 2A , where A is the area of the triangle. Estimate to the nearest whole number the base of this triangle if the area is 17 square meters.


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5-3C

Enrich

Heron’s FormulaA formula named after Heron of Alexandria can be used to find the area of a triangle if you know the lengths of the sides.

Step 1 Step 2

Find s, the semi-perimeter. For a triangle with Substitute s, a, b, and c intosides a, b, and c, the semi-perimeter is: Heron’s Formula to find the area, A.

S = a + b + c ! 2 . A = " ######### s(s - a)(s - b)(s - c)

Estimate the area of each triangle by counting squares. Then use Heron’s Formula to compute a more exact area. Give each answer to the nearest tenth of a unit.

1.

6

6

6

2.

910

9

3.

10

8

6

Estimated area: Estimated area: Estimated area: Computed area: Computed area: Computed area:

4.

7

7

7

5.

8

8

3

6.

5

9

7

Estimated area: Estimated area: Estimated area: Computed area: Computed area: Computed area:

7. Why would it be foolish to use Heron’s Formula to find the area of a right triangle?


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5-3 TI-73 ActivityEstimate Square RootsC

A graphing calculator can be used to estimate square roots. On the TI-73, you can use the square root function. To use the square root function, you must first press .

Estimate ! "" 32 . Round to the nearest tenth.

Enter: [ ! "" ] 32 ) 5.656854249

So, ! "" 32 # 5.7.

Estimate ! "" 130 . Round to the nearest tenth.

Enter: [ ! "" ] 130 ) 11.40175425

So, ! "" 130 # 11.4.

Exercises

Estimate each square root. Round to the nearest tenth.

1. ! "" 15 2. ! "" 13 3. ! "" 20

4. ! "" 24 5. ! " 7 6. ! " 2

7. ! "" 37 8. ! "" 50 9. ! " 5

10. ! "" 120 11. ! "" 140 12. ! "" 11

13. ! " 8 14. ! "" 21 15. ! "" 17

16. ! "" 19 17. ! "" 221 18. ! "" 315

Example 1

Example 2


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5-3D

ReteachCompare Real Numbers

Numbers may be classified by identifying to which of the following sets they belong.

Whole Numbers 0, 1, 2, 3, 4, … Integers …, -2, -1, 0, 1, 2, …

Rational Numbers numbers that can be expressed in the form a ! b , where a and b are

integers and b " 0

Irrational Numbers numbers that cannot be expressed in the form a ! b , where a and b

are integers and b " 0

Name all sets of numbers to which each real number belongs.

5 whole number, integer, rational number

0.666… Decimals that terminate or repeat are rational numbers, since they can be expressed as fractions. 0.666… = 2 !

3

- # $$ 25 Since - # $$ 25 = -5, it is an integer and a rational number.

# $$ 11 # $$ 11 % 3.31662479… Since the decimal does not terminate or repeat, it is an irrational number.

Replace with <, >, or = to make 2 1 ! 4 " # 5 a true statement.

Write each number as a decimal.

2 1 ! 4 = 2.25

# $ 5 % 2.236067…

Since 2.25 is greater than 2.236067…, 2 1 ! 4 > # $ 5 .

Exercises


1. 30 2. -11

3. 5 4 ! 7 4. # $$ 21

5. 0 6. - # $ 9

7. 6 ! 3 8. - # $$ 101

Replace each with <, >, or = to make a true statement.

9. 2.7 # $ 7 10. # $$ 11 3 1 ! 2 11. 4 1 !

6 # $$ 17 12. 3. ! 8 # $$ 15

Examples

Example 5

To compare real numbers, write each number as a decimal and then compare the decimal values.


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5-3D

Skills PracticeCompare Real Numbers


1. 12 2. -15

3. 1 1 ! 2 4. 3.18

5. 8 ! 4 6. 9. ! 3

7. -2 7 ! 9 8. " ## 25

9. " # 3 10. - " ## 64

11. - " ## 12 12. " ## 13


13. 1.7 " # 3 14. " # 6 2 1 ! 2

15. 4 2 ! 5 " ## 19 16. 4. ! 8 " ## 24

17. 6 1 ! 6 " ## 38 18. " ## 55 7.4 ! 2

19. 2.1 " ## 4.41 20. 2. ! 7 " ## 7.7

Order each set of numbers from least to greatest. Verify your answer by graphing on a number line.

21. 1.84, " # 5 , 5 ! 2 , 2.3, " # 3

1.5 2 2.5

22. –3.01, –2.95, –2.9, –3.1, –3.5

-3.5 -3 -2.5


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5-3D

Homework PracticeCompare Real Numbers

Name all sets of numbers to which the real number belongs.

1. -9 2. ! "" 144 3. ! "" 35 4. 8 # 11

5. 9.55 6. 5. # 3 7. 20 # 5 8. - ! "" 44


9. ! " 8 2.7 10. ! "" 15 3.9 11. 5 2 # 5 ! "" 30

12. 2 3 # 10

! "" 5.29 13. ! "" 9.8 3. # 1 14. 8. # 2 8 2 # 9

Order each set of numbers from least to greatest. Verify your answer by graphing on a number line.

15. ! "" 10 , ! " 8 , 2.75, 2. # 8 16. 5.01, 5.0 # 1 , 5. ## 01 , ! "" 26 17. - ! "" 12 , ! "" 13 , -3.5, 3.5

2.7 2.8 2.9 3 3.1 3.2 5 5.1 -4 -3 -2 -1 0 1 2 3 4

18. ALGEBRA The geometric mean of two numbers a and b is ! "" ab . Find the geometric mean of 32 and 50.

19. ART The area of a square painting is 600 square inches. To the nearest hundredth inch, what is the perimeter of the painting?



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5-3D

Problem-Solving PracticeCompare Real Numbers

1. GEOMETRY If the area of a square is 33 square inches, estimate the length of a side of the square to the nearest tenth of an inch.

2. GARDENING Hal has a square garden in his back yard with an area of 210 square feet. Estimate the length of a side of the garden to the nearest tenth of a foot.

3. ALGEBRA Estimate the solution of a2 = 21 to the nearest tenth.

4. ALGEBRA Estimate the solution of b2 = 67.5 to the nearest tenth.

5. ARITHMETIC The geometric mean of two numbers a and b can be found by evaluating ! "" a · b . Estimate the geometric mean of 4 and 11 to the nearest tenth.

6. ELECTRICITY In a certain electrical circuit, the voltage V across a 20 ohm resistor is given by the formula V = ! "" 20P , where P is the power dissipated in the resistor, in watts. Estimate to the nearest tenth the voltage across the resistor if the power P is 4 watts.

7. GEOMETRY The length s of a side of a cube is related to the surface area A of

the cube by the formula s = ! " A # 6 . If the

surface area is 27 square inches, what is the length of a side of the cube to the nearest tenth of an inch?

8. PETS Alicia and Didia are comparing the weights of their pet dogs. Alicia reports that her dog weighs 11 1 #

5

pounds, while Didia says that her dog weighs ! "" 125 pounds. Whose dog weighs more?


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5-3D

Enrich

The Closure PropertyThe Real Number System contains properties that can help you solve problems. These include the Commutative Property, the Distributive Property, and the Associative Property.

Another property of real numbers is the Closure Property.

Is the set of whole numbers closed under addition?

Yes. The sum cannot contain a decimal part because none of the addends has a decimal part. Also, the sum cannot be negative because none of the addends is negative. The sum must be a whole number.

Answer each of the following questions about the Closure Property in the real number system. If the answer is yes, explain how you know. If the answer is no, give a counterexample to show.

1. Is the set of rational numbers closed under multiplication?

2. Is the set of whole numbers closed under subtraction?

3. Is the set of integers closed under division?

4. Is the set of rational numbers closed under division?

5. Is the set of irrational numbers closed under subtraction?

6. Is the set of integers closed under subtraction?

Closure Property of Real Numbers

A set of numbers is closed under a particular operation if performing the operation on any number in the set results in a number in the set.


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1. Give an example that supports the statement that any nonzero number raised to the zero power is 1.

2. Give a real world example where scientific notation is used. Why do you think it is used in this situation?

3. Briefly explain why the square root of a real number less than zero is not real.



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ISBN: 978-0-07-892304-3MHID: 0-07-892304-2

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Contents Chapter 0 Start Smart Chapter 1 Rational Numbers and Percent Chapter 2 Expressions and Functions Chapter 3 Linear Functions and Systems of Equations Chapter 4 Equations and Inequalities Chapter 5 Operations on Real Numbers Chapter 6 Angles and Lines Chapter 7 Similar Triangles and the Pythagorean Theorem Chapter 8 Data Analysis Chapter 9 Units of Measure Chapter 10 Measurement: Area and Volume Chapter 11 Properties and Multi-Step Equations and InequalitiesChapter 12 Nonlinear Functions and Polynomials

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CChapter 5 hapter 5 RResource Mastersesource Masters · • Contains multiple-choice questions • 2A-2B On-level students • Contains both multiple-choice and free-response questions