Chapter 4Resource Masters
Copyright © by The McGraw-Hill Companies, Inc. All rights reserved.Printed in the United States of America. Permission is granted to reproduce thematerial contained herein on the condition that such material be reproduced onlyfor classroom use; be provided to students, teacher, and families without charge;and be used solely in conjunction with Glencoe Pre-Algebra. Any other reproduc-tion, for use or sale, is prohibited without prior written permission of the publisher.
Send all inquiries to:Glencoe/McGraw-Hill8787 Orion PlaceColumbus, OH 43240
ISBN: 0-07-827770-1
5 6 7 8 9 10 047 11 10 09 08 07 06 05 04 03 Pre-Algebra Chapter 4 Resource Masters
Consumable Workbooks
Many of the worksheets contained in the Chapter Resource Masters booklets are
available as consumable workbooks in both English and Spanish.
Study Guide and Intervention Workbook 0-07-827794-9
Study Guide and Intervention Workbook (Spanish) 0-07-827795-7
Skills Practice Workbook 0-07-827788-4
Skills Practice Workbook (Spanish) 0-07-827790-6
Practice Workbook 0-07-827789-2
Practice Workbook (Spanish) 0-07-827791-4
Answers for Workbooks The answers for Chapter 4 of these workbooks
can be found in the back of this Chapter Resource Masters booklet.
Spanish Assessment Masters Spanish versions of forms 2A and 2C
of the Chapter 4 Test are available in the Pre-Algebra Spanish Assessment
Masters (0-07-830412-1).
iii
Vocabulary Builder............................vii
Lesson 4-1Study Guide and Intervention ........................165Skills Practice.................................................166Practice ..........................................................167Reading to Learn Mathematics......................168Enrichment .....................................................169
Lesson 4-2Study Guide and Intervention ........................170Skills Practice.................................................171Practice ..........................................................172Reading to Learn Mathematics......................173Enrichment .....................................................174
Lesson 4-3Study Guide and Intervention ........................175Skills Practice.................................................176Practice ..........................................................177Reading to Learn Mathematics......................178Enrichment .....................................................179
Lesson 4-4Study Guide and Intervention ........................180Skills Practice.................................................181Practice ..........................................................182Reading to Learn Mathematics......................183Enrichment .....................................................184
Lesson 4-5Study Guide and Intervention ........................185Skills Practice.................................................186Practice ..........................................................187Reading to Learn Mathematics......................188Enrichment .....................................................189
Lesson 4-6Study Guide and Intervention ........................190Skills Practice.................................................191Practice ..........................................................192Reading to Learn Mathematics......................193Enrichment .....................................................194
Lesson 4-7Study Guide and Intervention ........................195Skills Practice.................................................196Practice ..........................................................197Reading to Learn Mathematics......................198Enrichment .....................................................199
Lesson 4-8Study Guide and Intervention ........................200Skills Practice.................................................201Practice ..........................................................202Reading to Learn Mathematics......................203Enrichment .....................................................204
Chapter 4 AssessmentChapter 4 Test, Form 1 ..........................205–206Chapter 4 Test, Form 2A ........................207–208Chapter 4 Test, Form 2B ........................209–210Chapter 4 Test, Form 2C........................211–212Chapter 4 Test, Form 2D........................213–214Chapter 4 Test, Form 3 ..........................215–216Chapter 4 Open-Ended Assessment .............217Chapter 4 Vocabulary Test/Review.................218Chapter 4 Quizzes 1 & 2................................219Chapter 4 Quizzes 3 & 4................................220Chapter 4 Mid-Chapter Test ...........................221Chapter 4 Cumulative Review........................222Chapter 4 Standardized Test Practice....223–224
Standardized Test Practice Student Recording Sheet ..............................................A1ANSWERS ................................................A2–A31
CONTENTS
iv
Teacher s Guide to Using theChapter 4 Resource Masters
The Fast File Chapter Resource system allows you to conveniently file the resources youuse most often. The Chapter 4 Resource Masters includes the core materials needed forChapter 4. These materials include worksheets, extensions, and assessment options. Theanswers for these pages appear at the back of this booklet.
All of the materials found in this booklet are included for viewing and printing in the Pre-Algebra TeacherWorks CD-ROM.
Vocabulary Builder Pages vii-viiiinclude a student study tool that presents up to twenty of the key vocabulary termsfrom the chapter. Students are to record definitions and/or examples for each term.You may suggest that students highlight or star the terms with which they are not familiar.
When to Use Give these pages to studentsbefore beginning Lesson 4-1. Encourage themto add these pages to their Pre-Algebra StudyNotebook. Remind them to add definitionsand examples as they complete each lesson.
Study Guide and InterventionEach lesson in Pre-Algebra addresses one ortwo objectives. There is one Study Guide andIntervention master for each lesson.
When to Use Use these masters as reteach-ing activities for students who need addi-tional reinforcement. These pages can alsobe used in conjunction with the StudentEdition as an instructional tool for studentswho have been absent.
Skills Practice There is one master foreach lesson. These provide computationalpractice at a basic level.
When to Use These masters can be usedwith students who have weaker mathematicsbackgrounds or need additional reinforcement.
Practice There is one master for each lesson. These problems more closely followthe structure of the Practice and Apply section of the Student Edition exercises.These exercises are of average difficulty.
When to Use These provide additionalpractice options or may be used as home-work for second day teaching of the lesson.
Reading to Learn MathematicsOne master is included for each lesson. Thefirst section of each master asks questionsabout the opening paragraph of the lesson inthe Student Edition. Additional questionsask students to interpret the context of andrelationships among terms in the lesson.Finally, students are asked to summarizewhat they have learned using various repre-sentation techniques.
When to Use This master can be used as a study tool when presenting the lesson or as an informal reading assessment after presenting the lesson. It is also a helpful tool for ELL (English Language Learner)students.
Enrichment There is one extension master for each lesson. These activities may extend the concepts in the lesson, offeran historical or multicultural look at theconcepts, or widen students’ perspectives on the mathematics they are learning.These are not written exclusively for honorsstudents, but are accessible for use with alllevels of students.
When to Use These may be used as extracredit, short-term projects, or as activitiesfor days when class periods are shortened.
v
Assessment OptionsThe assessment masters in the Chapter 4Resource Masters offer a wide range ofassessment tools for intermediate and finalassessment. The following lists describe eachassessment master and its intended use.
Chapter Assessment Chapter Tests• Form 1 contains multiple-choice questions
and is intended for use with basic levelstudents.
• Forms 2A and 2B contain multiple-choicequestions aimed at the average level student. These tests are similar in formatto offer comparable testing situations.
• Forms 2C and 2D are composed of free-response questions aimed at the averagelevel student. These tests are similar informat to offer comparable testing situa-tions. Grids with axes are provided forquestions assessing graphing skills.
• Form 3 is an advanced level test withfree-response questions. Grids withoutaxes are provided for questions assessinggraphing skills.
All of the above tests include a free-response Bonus question.
• The Open-Ended Assessment includesperformance assessment tasks that aresuitable for all students. A scoring rubric isincluded for evaluation guidelines. Sampleanswers are provided for assessment.
• A Vocabulary Test, suitable for all stu-dents, includes a list of the vocabularywords in the chapter and ten questionsassessing students’ knowledge of thoseterms. This can also be used in conjunc-tion with one of the chapter tests or as areview worksheet.
Intermediate Assessment• Four free-response quizzes are included
to offer assessment at appropriate inter-vals in the chapter.
• A Mid-Chapter Test provides an optionto assess the first half of the chapter. It iscomposed of both multiple-choice andfree-response questions.
Continuing Assessment• The Cumulative Review provides
students an opportunity to reinforce andretain skills as they proceed throughtheir study of Pre-Algebra. It can also be used as a test. This master includesfree-response questions.
• The Standardized Test Practice offerscontinuing review of pre-algebra conceptsin various formats, which may appear onthe standardized tests that they mayencounter. This practice includes multiple-choice, grid-in, and open-ended questions.Bubble-in and grid-in answer sections areprovided on the master.
Answers• Page A1 is an answer sheet for the
Standardized Test Practice questions that appear in the Student Edition onpages 196–197. This improves students’familiarity with the answer formats theymay encounter in test taking.
• The answers for the lesson-by-lesson masters are provided as reduced pageswith answers appearing in red.
• Full-size answer keys are provided for theassessment masters in this booklet.
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This is an alphabetical list of key vocabulary terms you will learn in Chapter 4.As you study this chapter, complete each term’s definition or description.Remember to add the page number where you found the term. Add these pages toyour Pre-Algebra Study Notebook to review vocabulary at the end of the chapter.
Reading to Learn MathematicsVocabulary Builder
NAME ______________________________________________ DATE ____________ PERIOD _____
44
Vocabulary FoundDefinition/Description/ExampleTerm on Page
algebraic fraction
base
composite number
divisible
expanded form
exponent
factor
factor tree
Vo
cab
ula
ry B
uild
er
viii
Reading to Learn MathematicsVocabulary Builder (continued)
NAME ______________________________________________ DATE ____________ PERIOD _____
44
Vocabulary FoundDefinition/Description/ExampleTerm on Page
greatest commonfactor (GCF)
monomial
power
prime factorization
prime number
standard form
scientific notation
simplest form
Venn Diagram
Use divisibility rules to determine whether each number is divisible by 2, 3, 5, 6, or 10.
1. 105 2. 600
3. 462 4. 197
List all the factors of each number.
5. 76 6. 42
7. 182 8. 80
Determine whether each expression is a monomial. Explain why or why not.
9. 13 10. x � y
11. 3(x � 1) 12. 5st
Determine whether 108 is divisible by 2, 3, 5, 6, or 10.
Number Divisible? Reason2 yes The ones digit is 8, and 8 is divisible by 2.
3 yes The sum of the digits is 9, and 9 is divisible by 3.
5 no The ones digit is 8, not 0 or 5.
6 yes 108 is divisible by 2 and by 3.
10 no The ones digit is not 0.
108 is divisible by 2, 3, and 6.
A monomial is a number, a variable, or a product of numbers and/or variables. So, 108 is a monomial. The expression 5q is also a monomial since it is the product of a number and a variable, 5 � q. However, 2x � 1 is not a monomial since it is the sum of two terms.
Finding Factors Two or more numbers that are multiplied to form a product are called factors. Anynumber is divisible by its factors. The following rules can be used to determine mentally whether a number is divisible by 2, 3, 5, 6, or 10.
A number is divisible by:
• 2 if the ones digit is divisible by 2.
• 3 if the sum of the digits is divisible by 3.
• 5 if the ones digit is 0 or 5.
• 6 if the number is divisible by 2 and by 3.
• 10 if the ones digit is 0.
ExampleExample
Study Guide and InterventionFactors and Monomials
NAME ______________________________________________ DATE ______________ PERIOD _____
4-14-1
© Glencoe/McGraw-Hill 165 Glencoe Pre-Algebra
ExercisesExercises
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© Glencoe/McGraw-Hill 166 Glencoe Pre-Algebra
Skills PracticeFactors and Monomials
NAME ______________________________________________ DATE ______________ PERIOD _____
4-14-1
Use divisibility rules to determine whether each number is divisible by 2, 3, 5, 6, or 10.
1. 100 2. 66
3. 88 4. 123
5. 240 6. 280
7. 255 8. 165
9. 318 10. 1000
List all the factors of each number.
11. 36 12. 29
13. 45 14. 81
15. 125 16. 117
17. 16 18. 63
Determine whether each expression is a monomial. Explain why or why not.
19. p 20. 73
21. 2 � n 22. h � w
23. 3(a � 6) 24. �3k
25. q � r 26. 4y � 6
27. 3(x � 3) 28. 6s � 4p
29. SEATING Can 132 graduates be seated in rows of 6 at the graduation ceremony?Explain.
30. SCHOOL SUPPLIES When Alex’s mother buys pencils for school, she divides themequally among Alex and his sister. Should she buy the pencils in packages of 15 or 30?Explain.
PracticeFactors and Monomials
NAME ______________________________________________ DATE ______________ PERIOD _____
4-14-1
© Glencoe/McGraw-Hill 167 Glencoe Pre-Algebra
Use divisibility rules to determine whether each number is divisible by 2, 3, 5, 6, or 10.
1. 476 2. 117
3. 426 4. 29
5. 735 6. 276
7. 1200 8. 2370
9. 700 10. 4200
List all the factors of each number.
11. 48 12. 24
13. 121 14. 82
15. 37 16. 196
17. 95 18. 110
19. 96 20. 200
Determine whether each expression is a monomial. Explain why or why not.
21. 82 22. 4(�m)
23. m 24. rv
25. 6(x � 6) 26. 8n � 8
27. (�12)(�8)x 28. w � �
29. 2� � 2w 30. 2s � t
NEWSPAPERS For Exercises 31 and 32, refer to the following information.
Brandon delivers newspapers in his neighborhood. On Sunday, he must deliver 112 papers.Since he rides his bike, he separates the papers into smaller stacks and delivers one stackat a time.
31. What size stacks can he make?
32. If Brandon can carry no more than 30 papers at a time and can return home to restockno more than 5 times, how can he organize the 112 papers?
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© Glencoe/McGraw-Hill 168 Glencoe Pre-Algebra
Reading to Learn MathematicsFactors and Monomials
NAME ______________________________________________ DATE ______________ PERIOD _____
4-14-1
How are side lengths of rectangles related to factors?
Do the activity at the top of page 148 in your textbook. Writeyour answers below.
a. Use grid paper to draw as many other rectangles as possible with an area of 36 square units. Label the length and width of each rectangle.
b. Did you draw a rectangle with a length of 5 units? Why or why not?
c. List all of the pairs of whole numbers whose product is 36. Compare this list to the lengths and widths of all the rectangles that have an area of 36 square units. What do you observe?
d. Predict the number of rectangles that can be drawn with an area of 64 square units. Explain how you can predict without actually drawingthem.
Write a definition and give an example of each new vocabulary word.
4. Is the expression 2x � 1 a monomial? Explain.
Helping You Remember5. Explain in your own words how to determine whether an expression is a monomial.
Pre-Activity
Reading the Lesson
Vocabulary Definition Example
1. factors
2. divisible
3. monomial
Enrichment
NAME ______________________________________________ DATE ______________ PERIOD _____
4-14-1
© Glencoe/McGraw-Hill 169 Glencoe Pre-Algebra
Determine whether 4032 is divisible by 7.
4032 Cross out the ones digit.
� 4 Subtract twice the value of the ones digit from the rest of the number.
399 If the difference is a number that you know is divisible by 7, stop. If not,
� 18
21 Since 21 is divisible by 7, 4032 is divisible by 7.
1. 266
4. 936
7. 2957
2. 4312
5. 13,293
8. 3124
3. 8976
6. 7085
9. 6545
Divisibility rule for 7
Divisibility rule for 11
Determine whether 5159 is divisible by 11.
Method 1
5159 Cross out the ones digit.
� 9 Subtract the value of the ones digit from the rest of the number.
506 If the difference is a number that you know is divisible by 11, stop. If not,
� 6
44 Since 44 is divisible by 11, 5159 is divisible by 11.
Method 2
5159
5 � 5 � 10 Add the odd-numbered digits (first and third).
1 � 9 � 10 Add the even-numbered digits (second and fourth).
0 Subtract the sums. If the difference is divisible by 11, the number is
Since 0 is divisible by 11, 5159 is divisible by 11.
Determine whether 62,382 is divisible by 11.
6 � 3 � 2 � 11 Add the odd-numbered digits.
2 � 8 � 10 Add the even-numbered digits.
1 Subtract the sums.
Since 1 is not divisible by 11, 62,382 is not divisible by 11.
Determine whether each number is divisible by 7 or 11.
Divisibility
X
X
X
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repeat.
divisible by 11.
repeat.
X
Write each expression using exponents.
a. 10 � 10 � 10 � 10 � 10
The base is 10. It is a factor 5 times, so the exponent is 5.
10 � 10 � 10 � 10 � 10 � 105
b. ( p � 2)( p � 2)(p � 2)
The base is p � 2. It is a factor 3 times, so the exponent is 3.
( p � 2)( p � 2)( p � 2) � ( p � 2)3
© Glencoe/McGraw-Hill 170 Glencoe Pre-Algebra
A number that is expressed using an exponent is called a power. The base is the number that is multiplied. The exponent tells how many times the base is used as a factor. So, 43 has a base of 4 and an exponent of 3, and 43 � 4 � 4 � 4 � 64.
Evaluate x2 � 4 if x � �6.
x2 � 4 = (�6)2 � 4 Replace x with �6.
= (�6)(�6) � 4 �6 is a factor 2 times.
= 36 � 4 Multiply.
= 32 Subtract.
Study Guide and InterventionPowers and Exponents
NAME ______________________________________________ DATE ______________ PERIOD _____
Example 1Example 1
4-24-2
Example 2Example 2
Write each expression using exponents.
1. 5 � 5 � 5 � 5 � 5 � 5 � 5 2. (–7)(–7)(–7)
3. d � d � d � d 4. x � x � y � y
5. (z – 4)(z – 4) 6. 3(–t)(–t)(–t)
Evaluate each expression if g � 3, h � �1, and m � 9.
7. g 5 8. 5g2
9. g2 � m 10. hm2
11. g3 � 2h 12. m � hg3
ExercisesExercises
Expressions involving powers are evaluated using order of operations. Powers are repeated multiplications. They are evaluated after any grouping symbols and before other multiplication or division operations.
Skills PracticePowers and Exponents
NAME ______________________________________________ DATE ______________ PERIOD _____
4-24-2
© Glencoe/McGraw-Hill 171 Glencoe Pre-Algebra
Write each expression using exponents.
1. 7 � 7 2. (�3)(�3)(�3)(�3)(�3)
3. 4 4. (k � k)(k � k)(k � k)
5. p � p � p � p � p � p 6. 3 � 3
7. (�a)(�a)(�a)(�a) 8. 6 � 6 � 6 � 6
9. 9 � 9 � 9 10. 4 � y � z � z � z
11. s � s � s � s � t � u � u 12. 5 � 5 � 5 � q � q
Express each number in expanded form.
13. 135 14. 8732
15. 1005 16. 989
Evaluate each expression if b � 8, c � 2, and d � �3.
17. 4c 18. c0
19. b3 20. c3 � 3c
21. 3c 22. c4
23. c2 � d 24. 2b2
25. b2 � c3 26. d2
27. d3 28. b2 � d3
29. b2d 30. (b � c)2
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© Glencoe/McGraw-Hill 172 Glencoe Pre-Algebra
PracticePowers and Exponents
NAME ______________________________________________ DATE ______________ PERIOD _____
4-24-2
Write each expression using exponents.
1. 11 � 11 � 11 2. 2 � 2 � 2 � 2 � 2 � 2 � 2 � 2
3. 5 4. (�4)(�4)
5. a � a � a � a 6. n � n � n � n � n
7. 4 � 4 � 4 8. (b � b)(b � b)(b � b)
9. (�v)(�v)(�v)(�v) 10. x � x � z � z � z
11. 2 � 2 � 2 � 2 � 2 � t � t 12. m � m � m � n � p � p
Express each number in expanded form.
13. 13 14. 1006
15. 17,629 16. 897
Evaluate each expression if x � 3, y � �2, and z � 4.
17. yx 18. 510
19. z2 20. x2
21. 9x 22. z2 � 22
23. y5 24. z2 � y4
25. x2 � y2 � z2 26. z2 � x2
FAMILY TREE For Exercises 27 and 28, refer to the following information.
When examining a family tree, the branches are many. You are generation “now.” One generationago, your 2 parents were born. Two generations ago your 4 grandparents were born.
27. How many great-grandparents were born three generations ago?
28. How many “great” grandparents were born ten generations ago?
Reading to Learn MathematicsPowers and Exponents
NAME ______________________________________________ DATE ______________ PERIOD _____
4-24-2
© Glencoe/McGraw-Hill 173 Glencoe Pre-Algebra
Why are exponents important in comparing computer data?Do the activity at the top of page 153 in your textbook. Write youranswers below.
a. Write 16 as a product of factors of 2. How many factors are there?
b. How many factors of 2 form the product 128?
c. One megabyte is 1024 kilobytes. How many factors of 2 form the product 1024?
Write a definition and give an example of each new vocabulary word or phrase.
6. Write each expression using exponents.
a. 4 � 4 � 4 � 4 b. x � x � x � y � y
c. (�2)(�2)(�2) d. 5 � r � r � m � m � m
7. The number (3 � 103) � (5 � 102) � (0 � 101) � (2 � 100) is written in
form, while 3502 is written in form.
Helping You Remember8. Explain how the terms base, power, and exponent are related. Provide an example.
Pre-Activity
Reading the Lesson
Less
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Vocabulary Definition Example
1. base
2. exponent
3. power
4. standardform
5. expandedform
© Glencoe/McGraw-Hill 174 Glencoe Pre-Algebra
Enrichment
NAME ______________________________________________ DATE ______________ PERIOD _____
4-24-2
ExponentsNumbers can be expressed in several ways. Some numbers are expressed as sums. Somenumbers are expressed as products of factors, while other numbers are expressed as powers.
Two ways to express 27 are 3 � 3 � 3 and 33.
The number 1 million can be expressed in the following ways.
1,000,000 1000 � 1000 100 � 100 � 100 102 � 102 � 102
1,000,0001 10002 1003 106
Write names for each number below using the given exponents.
1. 16; exponents: 2 and 4 2. 81; exponents: 2 and 4
3. 64; exponents: 2 and 6 4. 256; exponents: 2 and 8
5. 625; exponents: 2 and 4 6. 729; exponents: 2 and 6
7. 2401; exponents: 2 and 4 8. 4096; exponents: 2 and 12
9. 6561; exponents: 2 and 8 10. 390,625; exponents: 2 and 8
Numbers that can be named as powers with like bases can be multiplied by adding theexponents.
8 � 8 � 23 � 23
� 23�3
� 26
Write the product of each pair of factors in exponential form.
11. 9 � 9 12. 4 � 4
13. 16 � 8 14. 125 � 25
15. 27 � 9 16. 81 � 27
17. 49 � 49 18. 121 � 121
Find the prime factorization of 48.
Determine whether each number is prime or composite.
1. 27 2. 151
3. 77 4. 25
Write the prime factorization for each number. Use exponents for repeated factors.
5. 16 6. 45
7. 78 8. 70
Factor each monomial.
9. 6m3 10. –20xy2
11. a2b2c3 12. 25h
Example 2Example 2
A prime number is a whole number that has exactly two factors, 1 and itself. A composite number is a whole number that has more than two factors. Zero and 1 are neither prime nor composite.
Determine whether 29 is prime or composite.
Find the factors of 29.
29 � 1 � 29
The only factors of 29 are 1 and 29, therefore 29 is a prime number.
Any composite number can be written as a product of prime numbers. A factor tree can be used to findthe prime factorization.
Example 1Example 1
Study Guide and InterventionPrime Factorization
NAME ______________________________________________ DATE ______________ PERIOD _____
4-34-3
© Glencoe/McGraw-Hill 175 Glencoe Pre-Algebra
The prime factorization of 48 is 2 � 2 � 2 � 2 � 3 or 24 � 3.
In algebra, monomials can be factored as a product of prime numbers and variables with no exponentgreater than 1. So, 8x 2 factors as 2 � 2 � 2 � x � x.
48 is the number to be factored.
Find any pair of whole number factors of 48.
Continue to factor any number that is not prime.
The factor tree is complete when there is a row of prime numbers.2 � 2232
48
6 � 8
2 � 3 2 � 4
ExercisesExercises
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© Glencoe/McGraw-Hill 176 Glencoe Pre-Algebra
Skills PracticePrime Factorization
NAME ______________________________________________ DATE ______________ PERIOD _____
4-34-3
Determine whether each number is prime or composite.
1. 41 2. 29
3. 87 4. 36
5. 57 6. 61
7. 71 8. 103
9. 39 10. 91
11. 47 12. 67
Write the prime factorization of each number. Use exponents for repeated factors.
13. 20 14. 40
15. 32 16. 44
17. 90 18. 121
19. 46 20. 30
21. 65 22. 80
Factor each monomial.
23. 15t 24. 16r2
25. �11m2 26. �49y3
27. 21ab 28. �42xyz
29. 45j2k 30. 17u2v2
31. 27d4 32. �16cd2
PracticePrime Factorization
NAME ______________________________________________ DATE ______________ PERIOD _____
4-34-3
© Glencoe/McGraw-Hill 177 Glencoe Pre-Algebra
Determine whether each number is prime or composite.
1. 11 2. 63
3. 73 4. 75
5. 49 6. 69
7. 53 8. 83
Write the prime factorization of each number. Use exponents for repeated factors.
9. 33 10. 24
11. 72 12. 276
13. 85 14. 1024
15. 95 16. 200
17. 243 18. 735
Factor each monomial.
19. 35v 20. 49c2
21. �14b3 22. �81h2
23. 33wz 24. �56ghj
25. NUMBER THEORY Twin primes are a pair of consecutive odd primes, which differ by 2. For example, 3 and 5 are twin primes. Find the twin primes less than 100.(Hint: There are 8 pairs of twins less than 100.)
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© Glencoe/McGraw-Hill 178 Glencoe Pre-Algebra
Reading to Learn MathematicsPrime Factorization
NAME ______________________________________________ DATE ______________ PERIOD _____
4-34-3
How can models be used to determine whether numbers are prime?Do the activity at the top of page 159 in your textbook. Write youranswers below.
a. Use grid paper to draw as many different rectangular arrangements of2, 3, 4, 5, 6, 7, 8, and 9 squares as possible.
b. Which numbers of squares can be arranged in more than one way?
c. Which numbers of squares can only be arranged one way?
d. What do all rectangles that you listed in part c have in common?Explain.
Pre-Activity
Reading the Lesson
Write a definition and give an example of each new vocabulary word or phrase.
Helping You Remember
6. Composite is a word used in everyday English.
a. Find the definition of composite in the dictionary. Write the definition.
b. Explain how the English definition can help you remember how composite is used in mathematics.
Vocabulary Definition Example
1. compositenumber
2. factor
3. factor tree
4. primefactorization
5. primenumber
Enrichment
NAME ______________________________________________ DATE ______________ PERIOD _____
4-34-3
© Glencoe/McGraw-Hill 179 Glencoe Pre-Algebra
Prime PyramidA prime number is a whole number that has exactly two factors—itself and 1. The pyramid below is called a prime pyramid. Each row begins with 1 and ends with the numberof that row. So, row 2 begins with 1 and ends with 2, row 3 begins with 1 and ends with 3, andso on. In each row, the numbers from 1 to the row number are arranged such that the sum ofany two adjacent numbers is a prime number.
For example, look at row 4:
• It must contain the numbers 1, 2, 3, and 4.• It must begin with 1 and end with 4.• The sum of adjacent pairs must be a prime number:
1 � 2 � 3, 2 � 3 � 5, 3 � 4 � 7
1. Complete the pyramid by filling in the missing numbers.
2. Extend the pyramid to row 13.
3. Explain the patterns you see in the completed pyramid.
1 12
1 11
1 10
1 98567432
1 8
1 75
1 4 3 2 5 6
1 4 3 2 5
1 2 3 4
1 2 3
1
*
2
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4-3
© Glencoe/McGraw-Hill 180 Glencoe Pre-Algebra
The greatest number that is a factor of two or more numbers is the greatest common factor (GCF).Two ways to find the GCF are shown below.
Find the GCF of 24 and 32.
Method 1 List the factors.
factors of 24: 1, 2, 3, 4, 6, 8, 12, 24 Look for factors common to both lists, 1, 2, 4, and 8.
factors of 32: 1, 2, 4, 8, 16, 32
The greatest common factor of 24 and 32 is 8.
Method 2 Use prime factorization.
24 � 2 � 2 � 2 � 3 Find the common prime factors of 24 and 32.
32 � 2 � 2 � 2 � 2 � 2
Multiply the common prime factors. The greatest common factor of 24 and 32 is 2 � 2 � 2 or 8.
In algebra, greatest common factors are used to factor expressions.
Factor 5x � 10.
First, find the GCF of 5x and 10.
5x � 5 � x
10 � 2 � 5 The GCF is 5.
Now write each term as a product of the GCF and its remaining factors.
5x � 10 � 5(x) � 5(2)
� 5(x � 2) Distributive Property
So, 5x � 10 � 5(x � 2).
Study Guide and InterventionGreatest Common Factor (GCF)
NAME ______________________________________________ DATE ______________ PERIOD _____
Example 1Example 1
4-44-4
Example 2Example 2
Find the GCF of each set of numbers.
1. 30, 42 2. 15, 33 3. 44, 110 4. 16, 48
Factor each expression.
5. 4g � 16 6. 2d � 6 7. 8a � 24
8. f 2 � 2f 9. 6 � 3j 10. 16n2 � 40n
ExercisesExercises
Skills PracticeGreatest Common Factor (GCF)
NAME ______________________________________________ DATE ______________ PERIOD _____
4-44-4
© Glencoe/McGraw-Hill 181 Glencoe Pre-Algebra
Find the GCF of each set of numbers or monomials.
1. 15, 50 2. 24, 81
3. 18, 27 4. 36, 64
5. 88, 40 6. 54, 63
7. 11, 22 8. 14, 25
9. 20, 30 10. 16, 18
11. 64, 80 12. 16, 24
13. 30t, 40t2 14. 6, 9t
15. 16k2, 40k 16. 9m, 15n
17. 7pq, 8q 18. 18p, 45
Factor each expression.
19. 5b � 15 20. 7t � 49
21. 6w � 18 22. 100 � 50x
23. 7x � 7 24. 12n � 60
25. 24 � 8g 26. 50 � 5f
27. 3n � 24 28. 9� � 63
29. 6u � 36 30. 70 � 7c
31. 42 � 21x 32. 12y � 16
33. 6p � 12 34. 9r � 81
35. 6 � 8q 36. 21x � 33 Less
on
4-4
© Glencoe/McGraw-Hill 182 Glencoe Pre-Algebra
PracticeGreatest Common Factor (GCF)
NAME ______________________________________________ DATE ______________ PERIOD _____
4-44-4
Find the GCF of each set of numbers or monomials.
1. 9, 36 2. 42, 60
3. 16, 60 4. 29, 58
5. 18, 35 6. 90, 480
7. 80, 45 8. 700, 200
9. 17, 85 10. 24, 84, 168
11. 55, 105 12. 252, 126
13. 5p, 20p2 14. 28a, 49ab
15. 8b, 5c 16. 6a2, 18b2
17. 88s2t, 40st2 18. 42a2b, 60ab2
Factor each expression.
19. 10x � 40 20. 8v � 56
21. 9t � 9 22. 13m � 39
23. 90 � 45n 24. 15p � 60
25. 48 � 8r 26. 11z � 55
27. 18q � 54 28. 125 � 25h
29. 42a � 77 30. 30 � 45s
31. 50n � 30 32. 18 � 12d
33. 27m � 105 34. 65 � 39b
35. 21d � 63 36. 48 � 84m
37. SCHOOL TRIP Thirty-two seventh graders, 48 eighth graders, and 60 ninth graders aretaking a ski trip. In order to help students get better acquainted, students from eachgrade level are to ride each bus. What is the greatest number of buses that can be usedif students from each grade level are divided equally among the buses?
Reading to Learn MathematicsGreatest Common Factor (GCF)
NAME ______________________________________________ DATE ______________ PERIOD _____
4-44-4
© Glencoe/McGraw-Hill 183 Glencoe Pre-Algebra
How can a diagram be used to find the greatest common factor?
Do the activity at the top of page 164 in your textbook. Writeyour answers below.
a. Which numbers are in both circles?
b. Find the product of the numbers that are in both circles.
c. Is the product also a factor of 12 and 20?
d. Make a Venn diagram showing the prime factors of 16 and 28. Thenuse it to find the common factors of the numbers.
Pre-Activity
Reading the LessonWrite a definition and give an example of each new vocabulary word or phrase.
Helping You Remember3. Summarize in your own words how to find the greatest common factor of two numbers using
each method.
a. prime factorization
b. lists of factors
c. a Venn diagram
Vocabulary Definition Example
1. Venndiagram
2. greatestcommon factor
Less
on
4-4
© Glencoe/McGraw-Hill 184 Glencoe Pre-Algebra
Enrichment
NAME ______________________________________________ DATE ______________ PERIOD _____
4-44-4
GCFs by Successive DivisionAnother way to find the greatest common factor (GCF) of two numbers is to use successivedivision. This method works well for large numbers.
Find the GCF of 848 and 1325.
Step 1 Divide the smaller number into the greater number.
1 R477848 �1�3�2�5�
848477
Step 2 Divide the remainder into the divisor. Repeat this step until you get a remainder of 0.
1 R371 1 R106 3 R53 2 R0477 �8�4�8� 371�4�7�7� 106 �3�7�1� 53 �1�0�6�
477 371 318 106371 106 53 0
The last divisor is the GCF of the two original numbers. So the GCF of 848 and 1325 is 53.
Use the method above to find the GCF of each pair of numbers.
1. 187; 578 2. 161; 943
3. 215; 1849 4. 453; 484
5. 432; 588 6. 279; 403
7. 1325; 3498 8. 9840; 1751
9. 3484; 5963 10. 1802; 106
11. 45,787; 69,875 12. 35,811; 102,070
Simplify �2346aab2
2�.
�2346aab2
2� � Divide the numerator and denominator by the GCF, 2 � 2 � 3 � a.
� �2
3� b
� a� b
� or �23ba
2� Simplify.
2 � 2 � 2 � 3 � a � b � b���
2 � 2 � 3 � 3 � a � a
Simplify each fraction. If the fraction is already in simplest form, write simplified.
1. �1220� 2. �
1366� 3. �
17050
�
4. �165� 5. �
284� 6. �
38
�
7. �cc3� 8. �
rr
4
2� 9. �1241bb
�
10. �2246ww� 11. �
152st
� 12. �3dd2�
Example 2Example 2
A fraction is in simplest form when the GCF of the numerator and the denominator is 1. One way towrite a fraction in simplest form is to write the prime factorization of the numerator and the denominator.Then divide the numerator and denominator by the GCF.
Write �1284� in simplest form.
Write the prime factorization of the numerator and the denominator.
�1284� � �
22� 2
� 3� 2
� 3� 3
� Divide the numerator and denominator by the GCF, 2 � 3.
� �2
3� 2� or �
34
� Simplify.
Algebraic fractions can also be written in simplest form. Again, you can write the prime factorizationof the numerator and the denominator, then divide by the GCF.
Example 1Example 1
Study Guide and InterventionSimplifying Algebraic Fractions
NAME ______________________________________________ DATE ______________ PERIOD _____
4-54-5
© Glencoe/McGraw-Hill 185 Glencoe Pre-Algebra
ExercisesExercises
Less
on
4-5
1 1
1 1
/ // /
1 1 1 1
1 1 1 1
/ / / // / / /
© Glencoe/McGraw-Hill 186 Glencoe Pre-Algebra
Skills PracticeSimplifying Algebraic Fractions
NAME ______________________________________________ DATE ______________ PERIOD _____
4-54-5
Write each fraction in simplest form. If the fraction is already in simplest form,write simplified.
1. �1700� 2. �
1128� 3. �
3405�
4. �284� 5. �
46
� 6. �5663�
7. �1284� 8. �
479� 9. �
1339�
10. �2316� 11. �
3420� 12. �
346�
13. �4545� 14. �
144� 15. �
3468�
16. �8910� 17. �
255� 18. �
5764�
19. �2422� 20. �
178� 21. �
dd
3
4�
22. �yy3� 23. �
3� 24. �
ss
4
2�
25. �xy
2� 26. �
192aa
� 27. �186tt
�
28. �1244gg
� 29. �3450j
� 30. �210000pp2�
31. �17050nn3� 32. �
261kk
5
2� 33. �34ab�
34. �2146db
� 35. �284aa
� 36. �355tt
3
2�
PracticeSimplifying Algebraic Fractions
NAME ______________________________________________ DATE ______________ PERIOD _____
4-54-5
© Glencoe/McGraw-Hill 187 Glencoe Pre-Algebra
Write each fraction in simplest form. If the fraction is already in simplest form,write simplified.
1. �396� 2. �
160� 3. �
1597�
4. �2214� 5. �
369� 6. �
18050
�
7. �1762� 8. �
13326
� 9. �4752�
10. �4962� 11. �
3555� 12. �
6840�
13. �5670� 14. �
1576� 15. �
3633�
16. �3640� 17. �
2542� 18. �
19068
�
19. �4458� 20. �
1249� 21. �
xx
3
7�
22. �mm
4
5� 23. �aa
7
4� 24. �uu
5�
25. �2214yy
� 26. �144qq
2
2� 27. �1158xx
2
2�
28. �16236cc
� 29. �11211vv
2� 30. �
4429b2�
31. �ee
2
3
ff
2� 32. �
mp3
2� 33. �
120aa
3b5b
4�
34. SKI RESORT A local ski resort is open for business 13 weeks in the winter. Write a
fraction in simplest form that represents the fraction of a year the resort is open.
Less
on
4-5
© Glencoe/McGraw-Hill 188 Glencoe Pre-Algebra
Reading to Learn MathematicsSimplifying Algebraic Fractions
NAME ______________________________________________ DATE ______________ PERIOD _____
4-54-5
How are simplified fractions useful in representing measurements?Do the activity at the top of page 169 in your textbook. Writeyour answers below.
a. Are the three fractions equivalent? Explain your reasoning.
b. Which figure is divided into the least number of parts?
c. Which fraction would you say is written in simplest form? Why?
Pre-Activity
Reading the Lesson
Write a definition and give an example of each new vocabulary phrase.
3. Use a Venn diagram to explain how to simplify �1485�.
Vocabulary Definition Example
1. simplestform
2. algebraicfraction
Helping You Remember
4. Explain the similarities and differences between simplifying a numerical fraction andsimplifying an algebraic fraction.
Enrichment
NAME ______________________________________________ DATE ______________ PERIOD _____
4-54-5
© Glencoe/McGraw-Hill 189 Glencoe Pre-Algebra
Matching Equivalent Fractions
Cut out the pieces below and match the edges so that equivalent fractions meet.The pieces form a 4 � 6 rectangle. The outer edges of the rectangle formed willhave no fractions.
17
38
1242
210
59
1535
324910
58
1628
45501112
47
820
40451213 89
37
515
321
12
5256310911
1521
4044
19
3577
713
18
49
1848
218
810
2733
4991
110
2454
220112
45
68
2133
78
13
318
57
915
4448711
15
1456
511
34
46
933
14
28
1628
27
25
35401314
67
2545
336
23
620
35
36
4852311
2032
30351011
Less
on
4-5
Find �((�
�
88))
4
2�. Express your answer using exponents.
�((�
�
88))
4
2� � (�8)4� 2 The common base is �8.
� (�8)2 Subtract the exponents.
© Glencoe/McGraw-Hill 190 Glencoe Pre-Algebra
When multiplying powers with the same base, add the exponents.
Symbols Example
am � an � am +n 42 � 45 � 42 + 5 or 47
When dividing powers with the same base, subtract the exponents.
Symbols Example
�aa
m
n� � am � n, where a � 0 �55
6
2� � 56 � 2 or 54
Find 2a2(3a). Express your answer using exponents.
2a2(3a) � (2 � 3)(a2 � a) Use the Commutative and Associative Properties.
� (6)(a2 � 1) The common base is a.
� 6a3 Add the exponents.
Find each product or quotient. Express your answer using exponents.
1. 47 � 46 2. v5 � v4 3. ( f 3)( f 9)
4. 225 � 225 5. 7h(5h3) 6. �10x2(7x3)
7. �77
5
2� 8. �11
8
6� 9. �((�
�
1122))
3
3�
10. 38 � 33 11. �cc
2
1
0
3� 12. �((�
�
pp
))
1
1
8
2�
13. �7u6(�6u5) 14. �22ww
3� 15. �5m3(4m6)
16. the product of two cubed and two squared
17. the quotient of six to the eighth power and six squared
Study Guide and InterventionMultiplying and Dividing Monomials
NAME ______________________________________________ DATE ______________ PERIOD _____
Example 1Example 1
4-64-6
Example 2Example 2
ExercisesExercises
Skills PracticeMultiplying and Dividing Monomials
NAME ______________________________________________ DATE ______________ PERIOD _____
4-64-6
© Glencoe/McGraw-Hill 191 Glencoe Pre-Algebra
Find each product or quotient. Express your answer using exponents.
1. 23 � 25 2. 102 � 107
3. 14 � 1 4. 63 � 63
5. (�3)2(�3)3 6. (�9)2(�9)2
7. a2 � a3 8. n8 � n3
9. ( p4)( p4) 10. (z6)(z7)
11. (6b3)(3b4) 12. (�v)3(�v)7
13. 11a2 � 3a6 14. 10t2 � 4t10
15. (8c2)(9c) 16. (4f 8)(5 f 6)
17. �55
1
2
0� 18. �
1100
6
2�
19. �77
9
6� 20. �1122
8
3�
21. �110000
9
8� 22. �(�
�
22)3
�
23. �rr
8
7� 24. �zz
1
8
0�
25. �qq
8
4� 26. �gg
1
8
2�
27. �((�
�
yy))
7
2� 28. �((�
�
zz))
1
5
2�
29. the product of two squared and two to the sixth power
30. the quotient of ten to the seventh power and ten cubed
31. the product of y squared and y cubed
32. the quotient of a to the twentieth power and a to the tenth power
Less
on
4-6
© Glencoe/McGraw-Hill 192 Glencoe Pre-Algebra
PracticeMultiplying and Dividing Monomials
NAME ______________________________________________ DATE ______________ PERIOD _____
4-64-6
Find each product or quotient. Express your answer using exponents.
1. 42 � 43 2. 98 � 96
3. 74 � 72 4. 132 � 134
5. (�8)5(�8)3 6. (�21)9(�21)5
7. t9 � t3 8. h4 � h13
9. (m6)(m6) 10. (u11)(u10)
11. (�r)7(�r)20 12. (�w)(�w)9
13. 4d5 � 8d6 14. 7j50 � 6j50
15. �5b9 � 6b2 16. 121 � 122
17. �66
1
3
1� 18. �
1155
3
2�
19. �99
9
7� 20. �1188
4
4�
21. �((�
�
77))
6
5� 22. �9955
2
1
1
8�
23. �vv
3
2
0
0� 24. �nn
1
1
9
1�
25. the product of five cubed and five to the fourth power
26. the quotient of eighteen to the ninth power and eighteen squared
27. the product of z cubed and z cubed
28. the quotient of x to the fifth power and x cubed
29. SOUND Decibels are units used to measure sound. The softest sound that can be heardis rated as 0 decibels (or a relative loudness of 1). Ordinary conversation is rated atabout 60 decibels (or a relative loudness of 106). A rock concert is rated at about 120decibels (or a relative loudness of 1012). How many times greater is the relative loudness of a rock concert than the relative loudness of ordinary conversation?
Reading to Learn MathematicsMultiplying and Dividing Monomials
NAME ______________________________________________ DATE ______________ PERIOD _____
4-64-6
© Glencoe/McGraw-Hill 193 Glencoe Pre-Algebra
How are powers of monomials useful in comparing earthquake magnitudes?Do the activity at the top of page 175 in your textbook. Writeyour answers below.
a. Examine the exponents of the factors and the exponents of the productsin the last column. What do you observe?
b. Make a conjecture about a rule for determining the exponent of theproduct when you multiply powers with the same base. Test your ruleby multiplying 22 � 24 using a calculator.
Pre-Activity
Reading the Lesson
1. When multiplying powers with like bases, the exponents.
2. When dividing powers with like bases, the exponents.
3. Write a division expression whose quotient is 72.
4. Write a multiplication expression whose product is v5.
5. Find each product.
a. 4 � 43 b. y7 � y5
c. (�2x2)(5x2) d. �3r2 � r
6. Find each quotient.
a. �77
4
2� b. �vv
9
3�
c. �66
7
6� d. �ab
2b2
2�
Helping You Remember
7. Explain how dividing powers is related to simplifying fractions. Provide an example aspart of your explanation.
Less
on
4-6
© Glencoe/McGraw-Hill 194 Glencoe Pre-Algebra
Enrichment
NAME ______________________________________________ DATE ______________ PERIOD _____
4-64-6
Dividing Powers with Different BasesSome powers with different bases can be divided. First, you must be able to write both aspowers of the same base. An example is shown below.
�28
5
2� � �(223
5
)2� To find the power of a power, multiply the exponents.
� �22
5
6�
� 2�1 or �12
�
This method could not have been used to divide �29
5
2�, since 9 cannot be written as a power of2 using integers.
Simplify each fraction using the method shown above. Express the solution without exponents.
1. �82
2
2� 2. �1863
4� 3. �
93
3
3�
4. �8314
4� 5. �
831
9
2� 6. �3126
4
4�
7. �122553
2� 8. �
261
6
62� 9. �11000
6
03�
10. �6845
3� 11. �
2974
5� 12. �
34735
3�
Study Guide and InterventionNegative Exponents
NAME ______________________________________________ DATE ______________ PERIOD _____
4-74-7
© Glencoe/McGraw-Hill 195 Glencoe Pre-Algebra
Extending the pattern below shows that 4–1 = �14
� or �411� .
42 � 16� 4
41 � 4� 4
40 � 1� 4
4�1 � �14
�
This suggests the following definition.
a�n = �a1n�, for a � 0 and any integer n.
Write each expression using a positive exponent.
a. 3�4 b. y�2
3�4 � �314� y�2 � �
y12�
We can evaluate algebraic expressions with negative exponents using the definition of
negative exponents.
Write each expression using a positive exponent.
1. 6�4 2. (�7)�8 3. b�6 4. n�1
Write each fraction as an expression using a negative exponent other than �1.
5. �212� 6. �
1134� 7. �
215� 8. �
419�
Evaluate each expression if m � �4, n � 1, and p � 6.
9. p�2 10. m�3 11. (np) –1 12. 3m
Evaluate b�2 if b� 3.
b�2 � 3�2 Replace b with 3.
� �312� Definition of negative exponent
� �19
� Find 32.
Example 1Example 1
Example 2Example 2
ExercisesExercises
Less
on
4-7
© Glencoe/McGraw-Hill 196 Glencoe Pre-Algebra
Skills PracticeNegative Exponents
NAME ______________________________________________ DATE ______________ PERIOD _____
4-74-7
Write each expression using a positive exponent.
1. 3�4 2. 8�7 3. 10�4
4. (�2)�6 5. (�40)�3 6. (�17)�12
7. n�10 8. b�8 9. q�5
10. m�4 11. v�11 12. p�2
Write each fraction as an expression using a negative exponent other than �1.
13. �812� 14. �
1105� 15. �
213�
16. �617� 17. �
1174� 18. �
2112�
19. �317� 20. �
912� 21. �
312�
22. �1121� 23. �
215� 24. �
316�
Evaluate each expression if x � 1, y � 2, and z � �3.
25. y�z 26. z�2 27. x�8
28. y�5 29. z�3 30. y�1
31. z�4 32. 5 z 33. x�99
34. 1z 35. 4z 36. yz
PracticeNegative Exponents
NAME ______________________________________________ DATE ______________ PERIOD _____
4-74-7
© Glencoe/McGraw-Hill 197 Glencoe Pre-Algebra
Write each expression using a positive exponent.
1. 7�8 2. 10�6 3. 23�1
4. (�5)�2 5. (�18)�10 6. m�99
7. (�1)�12 8. c�6 9. p�5
10. g�17 11. 5z�4 12. 3t�1
Write each fraction as an expression using a negative exponent.
13. �2110� 14. �
2193� 15. �
414�
16. �319� 17. �
8117� 18. �
m1
4�
19. �x13� 20. �
a12� 21. �
419�
22. �18
� 23. �1144� 24. �
1169�
Evaluate each expression if x � 3, y � �2, and z � 4.
25. x�4 26. y�2 27. y�5
28. z�4 29. 5y 30. 10y
31. 3z�1 32. zy 33. (xz)�2
34. HAIR Hair grows at a rate of �614� inch per day. Write this number using negative
exponents.
Less
on
4-7
© Glencoe/McGraw-Hill 198 Glencoe Pre-Algebra
Reading to Learn MathematicsNegative Exponents
NAME ______________________________________________ DATE ______________ PERIOD _____
4-74-7
How do negative exponents represent repeated division?
Do the activity at the top of page 181 in your textbook. Writeyour answers below.
a. Describe the pattern of powers in the first column. Continue the pattern by writing the next two powers in the table.
b. Describe the pattern of values in the second column. Then completethe second column.
c. Verify that the powers you wrote in part a are equal to the valuesthat you found in part b.
d. Determine how 3�1 should be defined.
Pre-Activity
Reading the Lesson
1. Explain the value of 5�3 using a pattern.
2. Using what you know about the Quotient of Powers rule, fill in the missing number.
5�3 = �5?5�
Helping You Remember
3. Are �x2 and x�2 equivalent? Explain.
Power Value
51 5
50 1
5�1 �15
�
5�2 �215�
5�3
Enrichment
NAME ______________________________________________ DATE ______________ PERIOD _____
4-74-7
© Glencoe/McGraw-Hill 199 Glencoe Pre-Algebra
Proving Definitions of ExponentsRecall the rules for multiplying and dividing powers with the same base. Use these rules,along with other properties you have learned, to justify each definition. Abbreviations forsome properties you may wish to use are listed below.
Associative Property of Multiplication (APM) Additive Identity Property (AIP)Multiplicative Identity Property (MIP) Inverse Property of Addition (IPA)Inverse Property of Multiplication (IPM)
Write the reason for each statement.
1. Prove: a0 � 1
StatementLet m be an integer, and let a be any nonzero number.
am � a0 � am � 0
am � a0 � am
�a1m� � (am � a0) � �
a1m� � am
��a1m� � am� � a0 � �
a1m� � am
1 � a0 � 1
a0 � 1
2. Prove: a–n � �a1n�
StatementLet n be an integer, and let a be any nonzero number.
a–n � an � a–n � n
a–n � an � a0
a–n � an � 1
(a–n � an) � �a1n� � 1 � �
a1n�
a–n � �an � �a1n�� � 1 � �
a1n�
a–n � 1 � 1 � �a1n�
a–n � �a1n�
Reasona. Given
b.
c.
d.
e.
f.
g.
Reason
a. Given
b.
c.
d.
e.
f.
g.
h.
Less
on
4-7
© Glencoe/McGraw-Hill 200 Glencoe Pre-Algebra
When you deal with very large numbers like 5,000,000 or very small numbers like 0.0005, it is difficult tokeep track of place value. Numbers such as these can be written in scientific notation. A number isexpressed in scientific notation when it is written as a product of a factor and a power of 10. The factormust be greater than or equal to 1 and less than 10.
By definition, a number in scientific notation is written as a � 10n, where 1 a 10 and n is an integer.
Express each number in scientific notation.
a. 62,000,00
To write in scientific notation, place the decimal point after the first nonzero digit, then find the power of 10.
62,000,000 � 6.2 � 107 The decimal point moves 7 places. The power of 10 is 7.
b. 0.00025
0.00025 � 2.5 � 10�4 Place the decimal point after the first nonzero digit. The power of 10 is �4.
1. 4.12 � 106 2. 5.8 � 102 3. 9.01 � 10�3
4. 6.72 � 10�7 5. 8.72 � 104 6. 4.44 � 10�5
Express each number in scientific notation.
7. 12,000,000,000 8. 5000 9. 0.00475
10. 0.00007463 11. 235,000 12. 0.000377
Choose the greater number in each pair.
13. 4.9 � 104, 9.9 � 10�4 14. 2.004 � 103, 2.005 � 10–2
15. 3.2 � 102, 700 16. 0.002, 3.6 � 10�4
Study Guide and InterventionScientific Notation
NAME ______________________________________________ DATE ______________ PERIOD _____
4-84-8
Express each number in standard form.
a. 6.32 � 105
6.32 � 105 � 632,000 Move the decimal point 5 places to the right.
b. 7.8 � 10�6
7.8 � 10�6 � 0.0000078 Move the decimal point 6 places to the left.
Example 1Example 1
Example 2Example 2
ExercisesExercises
Skills PracticeScientific Notation
NAME ______________________________________________ DATE ______________ PERIOD _____
4-84-8
© Glencoe/McGraw-Hill 201 Glencoe Pre-Algebra
Express each number in standard form.
1. 1.5 � 103 2. 4.01 � 104
3. 6.78 � 102 4. 5.925 � 106
5. 7.0 � 108 6. 9.99 � 107
7. 3.0005 � 105 8. 2.54 � 105
9. 1.75 � 104 10. 1.2 � 10�6
11. 7.0 � 10�1 12. 6.3 � 10�3
13. 5.83 � 10�2 14. 8.075 � 10�4
15. 1.1 � 10�5 16. 7.3458 � 107
Express each number in scientific notation.
17. 1,000,000 18. 17,400
19. 500 20. 803,000
21. 0.00027 22. 5300
23. 18 24. 0.125
25. 17,000,000,000 26. 0.01
27. 21,800 28. 2,450,000
29. 0.0054 30. 0.000099
31. 8,888,800 32. 0.00912
Choose the greater number in each pair.
33. 8.8 � 103, 9.1 � 10�4 34. 5.01 � 102, 5.02 � 10�1
35. 6.4 � 103, 900 36. 1.9 � 10�2, 0.02
37. 2.2 � 10�3, 2.1 � 102 38. 8.4 � 102, 839
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on
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© Glencoe/McGraw-Hill 202 Glencoe Pre-Algebra
PracticeScientific Notation
NAME ______________________________________________ DATE ______________ PERIOD _____
4-84-8
Express each number in standard form.
1. 2.4 � 104 2. 9.0 � 103
3. 4.385 � 107 4. 1.03 � 108
5. 3.05 � 102 6. 5.11 � 1010
7. 6.000032 � 106 8. 1.0 � 101
9. 8.75 � 105 10. 8.49 � 10�2
11. 7.1 � 10�6 12. 1.0 � 10�3
13. 4.39 � 10�7 14. 1.25 � 10�4
Express each number in scientific notation.
15. 40,000 16. 16
17. 876,000,000 18. 4500
19. 151 20. 0.00037
21. 83,000,000 22. 919,100
23. 5,000,000,000,000 24. 0.13
25. 0.0000007 26. 0.0067
NIAGARA FALLS For Exercises 27 and 28, use the following information.
Every minute, 840,000,000,000 drops of water flow over Niagara Falls.
27. Write this number in scientific notation.
28. How many drops flow over the falls in a day?
Reading to Learn MathematicsScientific Notation
NAME ______________________________________________ DATE ______________ PERIOD _____
4-84-8
© Glencoe/McGraw-Hill 203 Glencoe Pre-Algebra
Why is scientific notation an important tool in comparingreal-world data?
Do the activity at the top of page 186 in your textbook. Writeyour answers below.
a. Write the track length in millimeters.
b. Write the track width in millimeters. (1 micron � 0.001 millimeter)
Pre-Activity
Reading the LessonWrite a definition and give an example of the new vocabulary phrase.
2. To multiply by a power of 10, move the decimal point to the if the exponent is positive.
3. Which is larger, �2.1 × 104 or �2.1 × 10�4? Explain.
Helping You Remember4. Explain how to express each number in scientific notation.
a. a number greater than 1
b. a number less than one
c. the number 1
Vocabulary Definition Example
slope intercept form See students’ work.1.
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© Glencoe/McGraw-Hill 204 Glencoe Pre-Algebra
Enrichment
NAME ______________________________________________ DATE ______________ PERIOD _____
4-84-8
Scientific NotationIt is sometimes necessary to multiply and divide very large or very small numbers usingscientific notation.
To multiply numbers in scientific notation, use the following rule.
For any numbers a and b, and any numbers c and d,(c � 10a)(d � 10b) � (c � d) � 10a � b.
For any numbers a and b, and any numbers c and d, (d � 0)(c � 10a) � (d � 10b) � (c � d) � 10a � b.
(3.0 � 104)(�5.0 � 10�2) � [3.0 � (�5.0)] � 104 � (�2)
� �15.0 � 102
� �1.5 � 103 or �1500
(24 � 10�4) � (1.5 � 102) � (24 � 1.5) � 10�4 � 2
� 16 � 10�6
� 1.6 � 10�5 or 0.000016
Multiply or divide. Express each product or quotient in scientific notation.
1. (2.7 � 109) � (3.1 � 102) 2. (6.1 � 10�2) � (1.3 � 105)
3. (5.4 � 10�3) � (1.8 � 102) 4. (6.9 � 10�3) � (3.0 � 10�8)
5. (1.1 � 10�5) � (9.9 � 10�1) 6. (4.0 � 100) � (1.0 � 10�2)
Solve. Write your answers in standard form.
To divide numbers in scientific notation, use the following rule.
7. The distance from Earth to the Moon is about 2.0 � 105 miles. The distance from Earth to the Sun is about 9.3 � 107 miles. How many times farther is it to the Sun than to the Moon?
8. If each of the 3.0 � 104 people employed by Sunny Motors earned 4.0 � 104 dollars lastyear, how much money did the company pay out to its employees?
Example 1Example 1
Example 2Example 2
Chapter 4 Test, Form 1
NAME DATE PERIOD
SCORE
© Glencoe/McGraw-Hill 205 Glencoe Pre-Algebra
Ass
essm
ent
Write the letter for the correct answer in the blank at the right of each question.
1. Use divisibility rules to determine which number is a factor of 39.A. 3 B. 5 C. 6 D. 10 1.
2. Determine which expression is not a monomial.A. 36 B. x C. x � 5 D. 4k 2.
3. Write (6)(6)(6) using exponents.A. 36 B. 66 C. 13 D. 63 3.
4. Evaluate k3 if k � 2.A. 2 B. 6 C. 8 D. 27 4.
5. Write the prime factorization of 18.A. 2 � 9 B. 2 � 3 � 3 C. 2 � 2 � 3 D. 3 � 6 5.
6. Factor 35x2y completely.A. 5 � 7 � x � x � y B. 35 � 1 � x � x � yC. 5 � 7 � x � y � y D. 5 � 7 � x2 � y 6.
For Questions 7 and 8, find the GCF of each set of numbers or monomials.
7. 12, 20A. 2 B. 32 C. 240 D. 4 7.
8. 6, 8A. 2 B. 3 C. 4 D. 48 8.
9. Factor 3b � 12.A. 12(3b � 1) B. 4(b � 3) C. 3(b � 4) D. b(3 � 12) 9.
For Questions 10 and 11, write each fraction in simplest form.
10. �1221�
A. �67� B. �
73� C. �
47� D. �
12� 10.
11. �1255aa�
A. �35� B. �
35a� C. �
35aa�
D. �53a�
11.
14
© Glencoe/McGraw-Hill 206 Glencoe Pre-Algebra
Chapter 4 Test, Form 1 (continued)
12. Eight inches is what part of 1 foot?
A. �29� B. �
23� C. �2
25�
D. �12� 12.
Find each product.
13. 45 � 42
A. 167 B. 452 C. 410 D. 47 13.
14. m4 � mA. 4m B. m5 C. m4 D. 2m4 14.
For Questions 15 and 16, find each quotient.
15. �55
4
2�
A. 12 B. 56 C. 52 D. 58 15.
16. �tt3�
A. 13 B. t3 C. t4 D. t2 16.
17. Write �315� using a negative exponent.
A. 3�5 B. 5�3 C. �35 D. �53 17.
18. Evaluate y�2 if y � 4.
A. ��18� B. �1
16�
C. 8 D. ��116�
18.
19. The speed of light is 300,000,000 meters per second. Express this number in scientific notation.A. 300 � 103 B. 30.0 � 108 C. 3.0 � 108 D. 0.03 � 107 19.
20. Choose the true statement.A. 3.1 � 105 2.7 � 105 B. 1.8 � 10�1 � 1.1 � 101
C. 5.4 � 104 � 3.7 � 107 D. 3.7 � 10�4 3.4 � 10�1 20.
Bonus Dyenitha is renting tables for her wedding reception. B:She can choose from tables that seat 6 or 8. If she is expecting 176 people and wants the same number of people at each table, which size table should she order?
NAME DATE PERIOD
14
Chapter 4 Test, Form 2A
NAME DATE PERIOD
SCORE
© Glencoe/McGraw-Hill 207 Glencoe Pre-Algebra
Ass
essm
ent
Write the letter for the correct answer in the blank at the right of each question.
1. Use divisibility rules to determine which number is a factor of 126.A. 6 B. 10 C. 4 D. 5 1.
2. Determine which expression is not a monomial.A. �16xyz B. 7(x � y) C. 12 D. u 2.
3. Write ( y)( y)( y)( y)( y) using exponents.A. 5 � y B. 5y C. 5y D. y5 3.
4. Evaluate 3m2 if m � 5.A. 21 B. 30 C. 75 D. 225 4.
5. Write the prime factorization of 36.A. 2 � 2 � 9 B. 2 � 2 � 2 � 2 C. 6 � 6 D. 2 � 2 � 3 � 3 5.
6. Factor 20x2y completely.A. 2 � 2 � 5 � x � x � y B. 2 � 25x2yC. 4 � 5 � x � x � y D. 2 � 10 � x � x � y 6.
7. Find the GCF of 14x and 35x2.A. 490x B. 7x C. 490x2 D. 14x3 7.
8. Factor 5 � 10y.A. 5y(1 � 2) B. 5(1 � 10y) C. 5(1 � 2y) D. 5y(1 � 2y) 8.
For Questions 9 and 10, write each fraction in simplest form.
9. �3757�
A. �151�
B. �12� C. �
1343�
D. �13� 9.
10. �2500aabb2
�
A. �2a
5b
� B. �52ab� C. �
25aabb2
� D. �25b� 10.
11. Twenty centimeters is what part of a meter?
A. �15� B. �5
10�
C. �110�
D. �49� 11.
14
© Glencoe/McGraw-Hill 208 Glencoe Pre-Algebra
Chapter 4 Test, Form 2A (continued)
Find each product.
12. 84 � 84
A. 164 B. 88 C. 816 D. 648 12.
13. 4x2 � 5x4
A. 9x6 B. 9x8 C. 20x6 D. 20x8 13.
For Questions 14 and 15, find each quotient.
14. �mm
5�
A. m4 B. 5 C. m6 D. 15 14.
15. �44
6
2�
A. 14 B. 43 C. 44 D. 48 15.
16. Write �516� as an expression using a negative exponent.
A. �5�6 B. 5�6 C. 6�5 D. �6 16.
For Questions 17 and 18, evaluate each expression if a � �2 and b � 3.
17. b�3
A. �217�
B. �9 C. ��217�
D. 27 17.
18. 2a
A. ��12� B. �
14� C. ��
14� D. 4 18.
19. A red blood cell is about 7.5 � 10�4 centimeter long. Express this number in standard form.A. 0.0075 B. 0.07500 C. 7500 D. 0.00075 19.
20. Choose the number that is greater than 2.7 � 104.A. 26,000 B. 1.4 � 105 C. 3.1 � 10�6 D. 2.5 � 104 20.
Bonus Order the planets in B:the table at the right fromleast to greatest diameter.
NAME DATE PERIOD
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Planet Diameter (mi)
VenusUranusNeptuneEarth
7.52 � 103
3.18 � 104
3.08 � 104
7.93 � 103
Chapter 4 Test, Form 2B
NAME DATE PERIOD
SCORE
© Glencoe/McGraw-Hill 209 Glencoe Pre-Algebra
Ass
essm
ent
Write the letter for the correct answer in the blank at the right of each question.
1. Use divisibility rules to determine which number is a factor of 105.A. 4 B. 5 C. 6 D. 10 1.
2. Determine which expression is not a monomial.A. 3(x � y) B. �13xyz C. 16 D. w 2.
3. Write (m)(m)(m)(m) using exponents.A. 4 � m B. 4m C. 4m D. m4 3.
4. Evaluate 3m3 if m � 6.A. 54 B. 648 C. 216 D. 5832 4.
5. Write the prime factorization of 24.A. 2 � 2 � 3 B. 2 � 2 � 2 � 3 C. 4 � 6 D. 2 � 2 � 3 � 3 5.
6. Factor 28xy2 completely.A. 4 � 7 � x � y � y B. 2 � 2 � 7 � x � y � yC. 1 � 28 � x � y � y D. 2 � 2 � 7 � x � y2 6.
7. Find the GCF of 16x and 64x2.A. 4x B. 8x2 C. 16x D. 32x3 7.
8. Factor 12 � 6y.A. 12(1 � 6y) B. 6(2 � y) C. 2(6 � y) D. 6(3y) 8.
For Questions 9 and 10, write each fraction in simplest form.
9. �3666�
A. �12� B. �
1383�
C. D. �1222�
9.
10. �3500aa2
�
A. �35� B. �
35a� C. �
3500a
� D. �65� 10.
11. Twenty-five centimeters is what part of a meter?
A. �14� B. �
2356�
C. �12� D. �4
10�
11.
6�11
14
© Glencoe/McGraw-Hill 210 Glencoe Pre-Algebra
Chapter 4 Test, Form 2B (continued)
Find each product.
12. 103 � 106
A. 1009 B. 1036 C. 1018 D. 109 12.
13. (3y2)(6y3)A. 18y5 B. 9y4 C. 18y6 D. 9y5 13.
For Questions 14 and 15, find each quotient.
14. �xx6�
A. 16 B. 6 C. x7 D. x5 14.
15. �((��
33))4
1�
A. 13 B. 4 C. (�3)3 D. (�3)5 15.
16. Write �417� as an expression using a negative exponent.
A. �4�7 B. 4�7 C. �74 D. �7 16.
For Questions 17 and 18, evaluate each expression if s � �2 and t � 3.
17. t�2
A. ��32� B. �9 C. �
19� D. ��
19� 17.
18. 4�1
A. �14� B. �4 C. ��
14� D. 4 18.
19. Bacteria are among the smallest living things. Some of the largest bacteria measure 7.87 � 10�5 inch across. Express this number in standard form.A. 0.0000787 B. 787,000 C. 0.00787 D. 0.000787 19.
20. Choose the number that is less than 3.4 � 10�4.A. 2.1 � 106 B. 2.1 � 102 C. 43,000 D. 5.4 � 10�6 20.
Bonus Order the planets in the table at the right from least to greatest diameter.
NAME DATE PERIOD
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B:Planet Diameter (mi)
SaturnUranusJupiterNeptune
1.21 � 105
5.11 � 104
1.43 � 105
4.95 � 104
Chapter 4 Test, Form 2C
© Glencoe/McGraw-Hill 211 Glencoe Pre-Algebra
Use divisibility rules to determine whether each number is divisible by 2, 3, 5, 6, or 10.
1. 1000 1.
2. 324 2.
For Questions 3 and 4, determine whether each expression is a monomial.
3. 10m(n � 8) 3.
4. 9b(7e) 4.
5. Write (2)(2)(2)(2)(2) using exponents. 5.
6. Evaluate 7a3 if a � 2. 6.
Write the prime factorization of each number or monomial.
7. 88 7.
8. 42a3x 8.
For Questions 9 and 10, find the GCF of each set of numbers or monomials.
9. 20, 36, 48 9.
10. 20y, 30y2 10.
11. Factor 18b � 9. 11.
For Questions 12 and 13, write each fraction in simplest form.
12. �1605�
12.
13. �2450rr2
� 13.
14. Forty minutes is what part of an hour? 14.
NAME DATE PERIOD
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Ass
essm
ent
© Glencoe/McGraw-Hill 212 Glencoe Pre-Algebra
Chapter 4 Test, Form 2C (continued)
Find each product. Express using exponents.
15. m4 � m3 15.
16. (11x3y)(5y) 16.
Find each quotient. Express using exponents.
17. �((��
44))7
2� 17.
18. �bb11� 18.
For Questions 19 and 20, write each expression using a negative exponent other than �1.
19. �319� 19.
20. 0.0001 20.
21. Evaluate b�3 if b � 5. 21.
22. Express 5.09 � 10�4 in standard form. 22.
23. Scientists have discovered that many mammals can expect 23.to live for 1.5 billion heartbeats. Write this number in scientific notation.
24. The distance between Saturn and Mars is 7.53 � 108 miles. 24.The distance between Mars and Mercury is 8.37 � 107 miles.Is Mars closer to Saturn or to Mercury?
25. David is packing bundles of 3-inch-by-5-inch cards, face up, 25.into a square box. He places the cards side-by-side so that there is no wasted space. Find the smallest possible measure for the edge of the box.
Bonus Write the prime factorization of 1188. Use exponents B:for repeated factors.
NAME DATE PERIOD
14
Chapter 4 Test, Form 2D
© Glencoe/McGraw-Hill 213 Glencoe Pre-Algebra
Use divisibility rules to determine whether each number is divisible by 2, 3, 5, 6, or 10.
1. 98 1.
2. 360 2.
For Questions 3 and 4, determine whether each expression is a monomial.
3. 9k(p � 3) 3.
4. 6a(4c) 4.
5. Write (3)(3)(3)(3)(3)(3) using exponents. 5.
6. Evaluate 5y4 if y � 3. 6.
Write the prime factorization of each number or monomial.
7. 68 7.
8. 32p2x 8.
For Questions 9 and 10, find the GCF of each set of numbers or monomials.
9. 20, 30, 45 9.
10. 16x2, 18x 10.
11. Factor 20a � 4. 11.
Write each fraction in simplest form.
12. �1684�
12.
13. �1260aa2
� 13.
14. Forty-five seconds is what part of a minute? 14.
NAME DATE PERIOD
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Ass
essm
ent
© Glencoe/McGraw-Hill 214 Glencoe Pre-Algebra
Chapter 4 Test, Form 2D (continued)
Find each product. Express using exponents.
15. d3 � d2 15.
16. (12x2y)(3y) 16.
Find each quotient. Express using exponents.
17. �((��
33))5
2� 17.
18. �aa16� 18.
For Questions 19 and 20, write each expression using a negative exponent other than �1.
19. �417� 19.
20. 0.001 20.
21. Evaluate a�4 if a � 2. 21.
22. Express 4.68 � 10�4 in standard form. 22.
23. The number of different hands possible in the game of 23.bridge is about 635 billion. Write this number in scientific notation.
24. The number of neurons in the neocortex of the human 24.brain is 3.0 � 1010. The neocortex of a gorilla contains 7.5 � 108 neurons. Which mammal has more neurons?
25. Scott is packing bundles of 4-inch-by-6-inch cards, face up, 25.into a square box. He places the cards side-by-side so that there is no wasted space. Find the smallest possible measure for the edge of the box.
Bonus Write the prime factorization of 1584. Use exponents B:for repeated factors.
NAME DATE PERIOD
14
Chapter 4 Test, Form 3
© Glencoe/McGraw-Hill 215 Glencoe Pre-Algebra
Use divisibility rules to determine whether each number is divisible by 2, 3, 5, 6, or 10.
1. 6036 1.
2. 12,420 2.
3. 22,523 3.
Determine whether each expression is a monomial.Explain why or why not.
4. 6a � 2b 4.
5. (�12x)(13y) 5.
6. 14k(2p � 3) 6.
7. �29c2d(3c � 4)
7.Write each expression using exponents.
8. 3 � 3 � x � x � y 8.
9. 7 � (2 � d) � (2 � d) 9.
Evaluate each expression if a � �2, b � 3, and c � 5.
10. 4a4 � 3b 10.
11. 2(3a � c)4 11.
Determine whether each number is prime or composite.
12. 211 12.
13. 57 13.
Factor each number or monomial completely.
14. 99 14.
15. �45qr2s3 15.
Find the GCF of each set of numbers or monomials.
16. 26, 65, 91 16.
17. 12a2c2, 30a3b2 17.
NAME DATE PERIOD
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Ass
essm
ent
© Glencoe/McGraw-Hill 216 Glencoe Pre-Algebra
Chapter 4 Test, Form 3 (continued)
Factor each expression.
18. x2 � 3x 18.
19. 18 � 6y 19.
For Questions 20 and 21, write each fraction in simplest form. If already in simplest form, write simplified.
20. �19516�
20.
21. �6293ppq2
� 21.
22. Forty-four feet is what part of a mile? 22.
23. Twelve inches is what part of one yard? 23.
For Questions 24–27, find each product or quotient.Express using exponents.
24. (�5x3)(3x2) 24.
25. (4s4t)(st2) 25.
26. �ba5
3
ab2� 26.
27. ��3mm4
����m12
2�� 27.
28. Write �316�
using a negative exponent other than �1. 28.
Evaluate each expression if a � 4, b � �3, and c � �1.
29. (b3)(3c) 29.
30. (bc)�5 30.
31. Express 1.057 � 10�4 in standard form. 31.
32. To find how many seconds it takes light to travel from the 32.Sun to Earth, divide the total distance, 93,000,000 miles,by the distance light travels in one second, 186,000 miles.Write the result in scientific notation.
33. Order 3.13 � 10�4, 0.0313, 3.03 � 10�4, 0.00303, and
33.
3.0 � 10�4 from least to greatest.
Bonus Write all of the prime numbers between 1 and 50.B:
NAME DATE PERIOD
14
Chapter 4 Open-Ended Assessment
© Glencoe/McGraw-Hill 217 Glencoe Pre-Algebra
Demonstrate your knowledge by giving a clear, concise solutionto each problem. Be sure to include all relevant drawings andjustify your answers. You may show your solution in more thanone way or investigate beyond the requirements of the problem.
1. Methods of finding prime numbers have intriguedmathematicians for years. For example, Goldbach’s conjecturestates that every even number greater than 2 can be written asthe sum of two prime numbers. Choose three even numbersgreater than 20 and less than 100. Write each as the sum of two prime numbers.
2. Write an argument or counterexample to support your answers tothe following questions.a. Are all numbers that are divisible by 9 also divisible by 3?b. Are all numbers that are divisible by 3 also divisible by 9?
3. The Warrior High School band and drill team have been invitedto participate in the Thanksgiving parade. There are 210 membersin the band and 40 members in the drill team.a. Can the band march in a rectangular formation having
8 band members in each row? Why or why not? If not, give anexample of a rectangular formation that would be possible.
b. Twenty-six drill team members and 140 band members turnedin their permission slips for the Thanksgiving trip. Explainhow to tell whether a greater fractional part of the drill teamor band turned in their slips. Find which part is greater.
4. To understand mathematics, you must understand the languageor symbols used.a. Explain the difference in the meanings of 3a and a3.b. Explain the difference in the meanings of �3b and b�3.
NAME DATE PERIOD
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Ass
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© Glencoe/McGraw-Hill 218 Glencoe Pre-Algebra
Chapter 4 Vocabulary Test/Review
Write the letter of the term that best matches each statement or phrase.
1. shows relationships among sets of numbers or objects using overlapping circles in a rectangle
2. a whole number with exactly two factors,1 and itself
3. tells how many times a number is used as a factor
4. GCF of the numerator and denominator is 1
5. (1 � 103) � (3 � 102) � (2 � 101) � (7 � 100)
6. language that uses a base two system of numbers
7. a number, variable, or product of numbers and/or variables
8. the greatest number that is a factor of two or more numbers
9. a number that is expressed using an exponent
In your own words—Define each term.
10. scientific notation
11. base
12. algebraic fraction
algebraic fractionbasebase twobinarydivisible
expanded formexponentfactor greatest common factor (GCF) monomial
power prime number scientific notation simplest form Venn diagram
NAME DATE PERIOD
SCORE 14
a. power
b. binary
c. exponent
d. Venn diagram
e. monomial
f. GCF
g. expanded form
h. simplest form
i. prime number
Chapter 4 Quiz (Lessons 4—1 and 4—2)
14
© Glencoe/McGraw-Hill 219 Glencoe Pre-Algebra
Use divisibility rules to determine whether each number is divisible by 2, 3, 5, 6, or 10.
1. 52 2. 90
3. 711 4. 435
Determine whether each expression is a monomial.Explain why or why not.
5. xy 6. x � y
Write each expression using exponents.
7. x � x � x 7.
8. (9 � 9 � 9) � (9 � 9) 8.
Evaluate each expression if x � 2, a � 3, and b � 2.
9. 12x4 9.
10. a2b3 10.
NAME DATE PERIOD
SCORE
Chapter 4 Quiz (Lessons 4—3 and 4—4)
Write the prime factorization of each number.Use exponents for repeated factors.
1. 27 2. 63 3. 112
Factor each monomial completely.
4. 21b 4.
5. 30x2y 5.
Find the GCF of each set of numbers or monomials.
6. 12, 16 6.
7. 120, 130, 140 7.
8. 28x4, 35x6 8.
Factor each expression.
9. 4a � 14 9.
10. 30 � 5y 10.
NAME DATE PERIOD
SCORE 14
Ass
essm
ent
1.
2.
3.
4.
5.
6.
1.
2.
3.
© Glencoe/McGraw-Hill 220 Glencoe Pre-Algebra
Write each fraction in simplest form. If the fraction is already in simplest form, write simplified.
1. �2346�
2. �3402�
3. �2480xx6
�
4. Fifteen centimeters is what part of a meter?
5. Standardized Test Practice Which fraction is �5r2r�
written in simplest form?
A. �51r�
B. 5r C. �5r
� D. �5r�
Find each product or quotient. Express using exponents.
6. 36 � 34 7. x4 � x7 � x2 8. (2x3)(5x2)
9. �((��
44))6
6� 10. �xx1
9
3�
Chapter 4 Quiz
Write each expression using a positive exponent.
1. 7�5 2. y�3
Evaluate each expression if x � 3 and y � 2.
3. x�4 4. (xy)�3
Write each number in scientific notation.
5. 0.0000001602 6. 200,000,000 7. 3,000,000,000
Choose the greater number in each pair.
8. 4.62 � 10�3, 0.000462 9. 3.06 � 10�5, 3.60 � 10�5
10. The table at the right shows the masses of three subatomic particles. Write the names of the particles in order from least to greatest mass.
NAME DATE PERIOD
SCORE
Chapter 4 Quiz 14
NAME DATE PERIOD
SCORE
14
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
Particle Mass
electron 9.10 � 10�31
neutron 1.68 � 10�27
proton 1.67 � 10�27
Chapter 4 Mid-Chapter Test (Lessons 4—1 through 4—8)
© Glencoe/McGraw-Hill 221 Glencoe Pre-Algebra
Write the letter for the correct answer in the blank at the right of each question.
1. Determine which expression is not a monomial.A. 9(a � b) B. �29x2yz C. 112 D. k 1.
2. Evaluate 2x2 if x � 3.A. 36 B. 8 C. 18 D. 12 2.
3. Write the prime factorization of 84.A. 2 � 2 � 21 B. 2 � 2 � 2 � 3 � 7C. 4 � 21 D. 2 � 2 � 3 � 7 3.
4. Factor 21b � 35.A. 3(7b � 5) B. 7(3b � 5)C. b(21 � 35) D. 7(3 � 5b) 4.
5. Factor 15 � 45y.A. 15 � 3y B. 15(1 � 3y)C. 60y D. y(15 � 45) 5.
6. Choose the number that is prime.A. 21 B. 55 C. 37 D. 49 6.
7. Determine whether 252 is divisible by 2, 3, 5, 6, or 10. 7.
8. Write (3)(3)(3)(2)(2) using exponents. 8.
Factor each number or monomial completely.
9. 66 9.
10. 24x3y2 10.
In Questions 11 and 12, find the GCF of each set of numbers or monomials.
11. 32, 56 11.
12. 20x2, 8x 12.
13. Amy raised chickens for her 4-H project. The chickens have 13.laid 70 eggs. Does Amy have enough eggs to completely fill cartons that contain 12 eggs each? Why or why not?
Part II
Part I
NAME DATE PERIOD
SCORE 14
Ass
essm
ent
© Glencoe/McGraw-Hill 222 Glencoe Pre-Algebra
Chapter 4 Cumulative Review (Chapters 1—4)
1. State the domain and range of the relation 1.{(2.3, 4), (5, 3.2), (4.6, 3.3)}. (Lesson 1–6)
2. Determine whether a scatter plot of the outside temperatures 2.and the corresponding air conditioning bills might show a positive, negative, or no relationship. Explain your answer.(Lesson 1–7)
For Questions 3 and 4, simplify each expression. 3.
3. �7m � (�15m) 4. 9x � (�23x) 4.(Lesson 3–2) (Lesson 3–2)
5. Evaluate ��34kh
� if h � 6 and k � �2. (Lesson 2–3) 5.
6. Find the average (mean) of �24, 16, 21, 9, �12. (Lesson 2–5) 6.
7. Name the quadrant in which the graph of (6, �5) lies. 7.(Lesson 2–6)
8. When you divide a number by �9, the result is 18. 8.Write and solve an equation to find the number. (Lesson 3–4)
Graph the solution of each equation on a number line.
9. 2 � y � �1 (Lesson 3–3) 9.
10. �4 � z � 2 (Lesson 3–3) 10.
For Questions 11 and 12, evaluate each expression if n � 6 and r � 4.
11. n4 (Lesson 4–2) 11.
12. 15r3 (Lesson 4–2) 12.
13. Write the prime factorization of 78. (Lesson 4–3) 13.
14. Factor 810abc3 completely. (Lesson 1–6) 14.
For Questions 15 and 16, find the GCF of each set of numbers or monomials.
15. 45, 75, 90 (Lesson 4–4) 15.
16. 36r2s4t, 81r3t2 (Lesson 4–2) 16.
17. Factor 18 � 42y. (Lesson 4–4) 17.
�1�2�3�4 0 1 2 43
�1�2�3�4 0 1 2 43
NAME DATE PERIOD
14
Standardized Test Practice (Chapters 1—4)
© Glencoe/McGraw-Hill 223 Glencoe Pre-Algebra
1. Lenora wants to buy a CD player that costs $169 (including tax). She has $108 in the bank, she earns $30 babysitting, $18 taking care of a neighbor’s pet, and receives $12 in allowance. How much more money does she need to buy the CD player? (Lesson 1–1)
A. $7 B. $1 C. $4 D. none 1.
2. Simplify 7(k � 5) � 9k. (Lesson 1–4)
E. 16k � 35 F. 10k � 12 G. 16k � 5 H. 17k � 5 2.
3. Simplify (�4)(3a)(�4b). (Lesson 2–4)
A. 48ab B. �48ab C. �11ab D. �5ab 3.
4. Evaluate the expression �xy� if x � �60 and y � 5. (Lesson 2–5)
E. 12 F. �12 G. �112�
H. ��112�
4.
5. Name the ordered pair for the point A graphed on the coordinate plane at the right. (Lesson 2–6)
A. (3, 4) B. (3, �4)C. (�4, 3) D. (�4, �3) 5.
6. In the school cafeteria, an apple costs a cents and a carton of milk costs 40 cents. Which expression represents the total cost of an apple and 2 cartons of milk for s students? (Lesson 3–1)
E. a � 80 � s F. s(a � 80) G. sa � 80 H. s(a � 40) 6.
7. Wanda drove for w hours on a trip. Her husband drove 3 hours more than Wanda. Which expression represents the total time they spent driving? (Lesson 3–2)
A. w � 6 B. 2w � 3 C. w � 6 D. 2w � 3 7.
8. If z � 5 � �9, find the numerical value of �2z � 5. (Lesson 3–3)
E. 23 F. �33 G. �13 H. 3 8.
9. Nine more than eight times a number is �47. Translate this sentence into an equation. (Lesson 3–6)
A. 8n � 9 � �47 B. 9n � 8 � �47C. 8(n � 9) � �47. D. 9(8 � n) � �47 9.
10. The formula d � rt can be rewritten as (Lesson 3–7)
E. dr � t. F. r � �dt�. G. t � �
dr�. H. r � dt. 10. HGFE
DCBA
HGFE
DCBA
HGFE
DCBA
HGFE
DCBA
HGFE
DCBA
NAME DATE PERIOD
44
Ass
essm
ent
Part 1: Multiple Choice
Instructions: Fill in the appropriate oval for the best answer.
y
xO
A
© Glencoe/McGraw-Hill 224 Glencoe Pre-Algebra
Standardized Test Practice (continued)
11. Choose the expression that is not a monomial. (Lesson 4–1)
A. (4k2)(9m) B. �3r
4�s
s� C. �29xy2 D. �
173c2� 11.
12. Write (a)(a)(a)(b)(b) using exponents. (Lesson 4–2)
E. a3b2 F. a�3b�2 G. 3a2b H. 3a32b2 12.
13. Write the prime factorization of 42. (Lesson 4–3)
A. 1 � 42 B. 2 � 21 C. 2 � 3 � 7 D. 6 � 7 13.
14. Find the GCF of 36 and 54. (Lesson 4–4)
E. 2 F. 3 G. 6 H. 18 14.
15. Simplify �11162cc4
2�. (Lesson 4–5)
A. �71c42
� B. �1c42� C. �c
72� D. 7c4 15.
16. Which is �1144�
written with a negative exponent? (Lesson 4–7)
E. 14�2 F. �1122� G. 144�2 H. 12�2 16.
17. The end zone of a football field is 30 feet wide 17. 18.and 160 feet long. What is its area in square feet? (Lesson 3–7)
18. What is the least 3-digit number that is divisible by 4 and 7? (Lesson 4–1)
19. Order the integers {32, �18, 2, 7, 0, �5, �11} from least to 19.greatest. (Lesson 2–1)
20. Saturn is about 799,800,000 miles from Earth. Write this 20.number in scientific notation. (Lesson 4–8)
Part 3: Short Response
Instructions: Write your answer in the blank at the right of each question.
0 0 0
.. ./ /
.
99 9 987654321
87654321
87654321
87654321
0 0 0
.. ./ /
.
99 9 987654321
87654321
87654321
87654321
HGFE
DCBA
HGFE
DCBA
HGFE
DCBA
NAME DATE PERIOD
44
NAME DATE PERIOD
Part 2: Grid In
Instructions: Enter your answer by writing each digit of the answer in a column boxand then shading in the appropriate oval that corresponds to that entry.
Standardized Test PracticeStudent Record Sheet (Use with pages 196—197 of the Student Edition.)
© Glencoe/McGraw-Hill A1 Glencoe Pre-Algebra
NAME DATE PERIOD
44
An
swer
s
Select the best answer from the choices given and fill in the corresponding oval.
1 4 7 10
2 5 8 11
3 6 9
Solve the problem and write your answer in the blank.
For Questions 15, 17, and 19, also enter your answer by writing each number orsymbol in a box. Then fill in the corresponding oval for that number or symbol.
12 15 17 19
13
14
15 (grid in)
16
17 (grid in)
18
19 (grid in)
20
21
Record your answers for Question 22 on the back of this paper.
0 0 0
.. ./ /
.
99 9 987654321
87654321
87654321
87654321
0 0 0
.. ./ /
.
99 9 987654321
87654321
87654321
87654321
0 0 0
.. ./ /
.
99 9 987654321
87654321
87654321
87654321
DCBADCBADCBA
DCBADCBADCBADCBA
DCBADCBADCBADCBA
Part 2 Short Response/Grid InPart 2 Short Response/Grid In
Part 1 Multiple ChoicePart 1 Multiple Choice
Part 3 Extended ResponsePart 3 Extended Response
©G
lenc
oe/M
cGra
w-H
ill16
6G
lenc
oe P
re-A
lgeb
ra
Skil
ls P
ract
ice
Fact
ors
an
d M
on
om
ials
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
__P
ER
IOD
____
_
4-1
4-1
Use
div
isib
ilit
y ru
les
to d
eter
min
e w
het
her
eac
h n
um
ber
is
div
isib
le b
y 2,
3,5,
6,or
10.
1.10
02,
5,10
2.66
2,3,
6
3.88
24.
123
3
5.24
02,
3,5,
6,10
6.28
02,
5,10
7.25
53,
58.
165
3,5
9.31
82,
3,6
10.
1000
2,5,
10
Lis
t al
l th
e fa
ctor
s of
eac
h n
um
ber
.
11.
361,
2,3,
4,6,
9,12
,18,
3612
.29
1,29
13.
451,
3,5,
9,15
,45
14.
811,
3,9,
27,8
1
15.
125
1,5,
25,1
2516
.11
71,
3,9,
13,3
9,11
7
17.
161,
2,4,
8,16
18.
631,
3,7,
9,21
,63
Det
erm
ine
wh
eth
er e
ach
exp
ress
ion
is
a m
onom
ial.
Exp
lain
wh
y or
wh
y n
ot.
19.
pye
s;a
vari
able
20.
73ye
s;a
nu
mb
er
21.
2 �
nn
o;
sum
of
two
ter
ms
22.
h�
wn
o;
dif
fere
nce
of
two
ter
ms
23.
3(a
�6)
no
;su
m o
f tw
o t
erm
s24
.�
3kye
s;p
rod
uct
of
a n
um
ber
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d
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riab
le25
.q
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no
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m o
f tw
o t
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s26
.4y
�6
no
;d
iffe
ren
ce o
f tw
o t
erm
s
27.
3(x
�3)
no;d
iffer
ence
oftw
ote
rms
28.
6s�
4pye
s;p
rod
uct
of
nu
mb
ers
and
var
iab
les
29.
SEA
TIN
GC
an 1
32 g
radu
ates
be
seat
ed i
n r
ows
of 6
at
the
grad
uat
ion
cer
emon
y?E
xpla
in.
Yes.
Sin
ce 1
32 is
div
isib
le b
y 2
and
by
3,it
is d
ivis
ible
by
6.S
o t
he
gra
du
ates
can
be
seat
ed in
ro
ws
of
6 w
ith
no
ext
ra p
eop
le o
rem
pty
ch
airs
.
30.
SCH
OO
L SU
PPLI
ESW
hen
Ale
x’s
mot
her
bu
ys p
enci
ls f
or s
choo
l,sh
e di
vide
s th
emeq
ual
ly a
mon
g A
lex
and
his
sis
ter.
Sh
ould
sh
e bu
y th
e pe
nci
ls i
n p
acka
ges
of 1
5 or
30?
Exp
lain
.S
he
sho
uld
bu
y p
acka
ges
of
30 s
ince
30
is d
ivis
ible
by
2,bu
t 15
is n
ot.
Use
div
isib
ilit
y ru
les
to d
eter
min
e w
het
her
eac
h n
um
ber
is d
ivis
ible
by
2,3,
5,6,
or 1
0.
1.10
53,
52.
600
2,3,
5,6,
10
3.46
22,
3,6
4.19
7n
on
e
Lis
t al
l th
e fa
ctor
s of
eac
h n
um
ber
.
5.76
1,2,
4,19
,38,
766.
421,
2,3,
6,7,
14,2
1,42
7.18
21,
2,7,
13,1
4,26
,91,
182
8.80
1,2,
4,5,
8,10
,16,
20,4
0,80
Det
erm
ine
wh
eth
er e
ach
exp
ress
ion
is
a m
onom
ial.
Exp
lain
wh
y or
wh
y n
ot.
9.13
yes;
a n
um
ber
10.
x�
yn
o;
sum
of
two
ter
ms
11.
3(x
�1)
no
;12
.5s
tye
s;p
rod
uct
of
a n
um
ber
dif
fere
nce
of
two
ter
ms
and
var
iab
les
Det
erm
ine
wh
eth
er 1
08 i
s d
ivis
ible
by
2,3,
5,6,
or 1
0.
Nu
mb
erD
ivis
ible
?R
easo
n2
yes
Th
e on
es d
igit
is
8,an
d 8
is d
ivis
ible
by
2.
3ye
sT
he
sum
of
the
digi
ts i
s 9,
and
9 is
div
isib
le b
y 3.
5n
oT
he
ones
dig
it i
s 8,
not
0 o
r 5.
6ye
s10
8 is
div
isib
le b
y 2
and
by 3
.
10n
oT
he
ones
dig
it i
s n
ot 0
.
108
is d
ivis
ible
by
2,3,
and
6.
A m
onom
ial
is a
num
ber,
a va
riab
le,o
r a
prod
uct
of n
umbe
rs a
nd/o
r va
riab
les.
So,
108
is
a m
onom
ial.
The
exp
ress
ion
5qis
als
o a
mon
omia
l si
nce
it i
s th
e pr
oduc
t of
a n
umbe
r an
d a
vari
able
,5�
q.H
owev
er,2
x�
1 is
not
a m
onom
ial
sinc
e it
is
the
sum
of
two
term
s.
Fin
din
g F
acto
rsTw
o or
mor
e nu
mbe
rs t
hat
are
mul
tiplie
d to
form
a p
rodu
ct a
re c
alle
d fa
ctor
s.A
nynu
mbe
r is
div
isib
le b
y its
fact
ors.
The
follo
win
g ru
les
can
be u
sed
to d
eter
min
e m
enta
lly w
heth
er a
nu
mbe
r is
div
isib
le b
y 2,
3,
5, 6
, or
10.
A n
umbe
r is
div
isib
le b
y:
•2
if th
e on
es d
igit
is d
ivis
ible
by
2.
•3
if th
e su
m o
f th
e di
gits
is d
ivis
ible
by
3.
•5
if th
e on
es d
igit
is 0
or
5.
•6
if th
e nu
mbe
r is
div
isib
le b
y 2
and
by 3
.
•10
if t
he o
nes
digi
t is
0.
Exam
ple
Exam
pleS
tudy G
uid
e a
nd I
nte
rven
tion
Fact
ors
an
d M
on
om
ials
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
__P
ER
IOD
____
_
4-1
4-1
©G
lenc
oe/M
cGra
w-H
ill16
5G
lenc
oe P
re-A
lgeb
ra
Exer
cises
Exer
cises
Lesson 4-1
© Glencoe/McGraw-Hill A2 Glencoe Pre-Algebra
Answers (Lesson 4-1)
©G
lenc
oe/M
cGra
w-H
ill16
8G
lenc
oe P
re-A
lgeb
ra
Readin
g t
o L
earn
Math
em
ati
csFa
cto
rs a
nd
Mo
no
mia
ls
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
__P
ER
IOD
____
_
4-1
4-1
Ho
w a
re s
ide
len
gth
s o
f re
ctan
gle
s re
late
d t
o f
acto
rs?
Do
the
acti
vity
at
the
top
of
pag
e 14
8 in
you
r te
xtb
ook
.Wri
teyo
ur
answ
ers
bel
ow.
a.U
se g
rid
pape
r to
dra
w a
s m
any
oth
er r
ecta
ngl
es a
s po
ssib
le w
ith
an
ar
ea o
f 36
squ
are
un
its.
Lab
el t
he
len
gth
an
d w
idth
of
each
rec
tan
gle.
Stu
den
ts s
ho
uld
dra
w r
ecta
ng
les
wit
h d
imen
sio
ns
1 �
36,
2 �
18,3
�12
,an
d 6
�6.
b.D
id y
ou d
raw
a r
ecta
ngl
e w
ith
a l
engt
h o
f 5
un
its?
Wh
y or
wh
y n
ot?
No
,if
the
len
gth
wer
e 5,
ther
e is
no
wh
ole
nu
mb
er w
idth
th
at w
ou
ld g
ive
an a
rea
of
36.
c.L
ist
all
of t
he
pair
s of
wh
ole
nu
mbe
rs w
hos
e pr
odu
ct i
s 36
.Com
pare
th
is l
ist
to t
he
len
gth
s an
d w
idth
s of
all
th
e re
ctan
gles
th
at h
ave
an
area
of
36 s
quar
e u
nit
s.W
hat
do
you
obs
erve
?1
and
36,
2 an
d 1
8,3
and
12,
4 an
d 9
,6 a
nd
6;
they
are
th
e sa
me.
d.P
redi
ct t
he
nu
mbe
r of
rec
tan
gles
th
at c
an b
e dr
awn
wit
h a
n a
rea
of
64 s
quar
e u
nit
s.E
xpla
in h
ow y
ou c
an p
redi
ct w
ith
out
actu
ally
dra
win
gth
em.
4 re
ctan
gle
s;fi
nd
th
e fa
cto
r p
airs
wh
ose
pro
du
ct is
64
:1
�64
,2 �
32,4
�16
,8 �
8.
Wri
te a
def
init
ion
an
d g
ive
an e
xam
ple
of
each
new
voc
abu
lary
wor
d.
4.Is
th
e ex
pres
sion
2x
�1
a m
onom
ial?
Exp
lain
.2x
�1
is n
ot
a m
on
om
ial
bec
ause
it is
th
e d
iffe
ren
ce o
f tw
o t
erm
s.
Hel
pin
g Y
ou
Rem
emb
er5.
Exp
lain
in
you
r ow
n w
ords
how
to
dete
rmin
e w
het
her
an
exp
ress
ion
is
a m
onom
ial.
Sam
ple
an
swer
:A
mo
no
mia
l is
a n
um
ber
,a v
aria
ble
,or
the
pro
du
ct o
fn
um
ber
s an
d/o
r va
riab
les.
Pre-
Act
ivit
y
Rea
din
g t
he
Less
on
1–3.
See
stu
den
ts’w
ork
.
Voca
bula
ryD
efin
itio
nE
xam
ple
1.fa
ctor
s
2.di
visi
ble
3.m
onom
ial
Pra
ctic
eFa
cto
rs a
nd
Mo
no
mia
ls
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
__P
ER
IOD
____
_
4-1
4-1
©G
lenc
oe/M
cGra
w-H
ill16
7G
lenc
oe P
re-A
lgeb
ra
Use
div
isib
ilit
y ru
les
to d
eter
min
e w
het
her
eac
h n
um
ber
is
div
isib
le b
y 2,
3,5,
6,or
10.
1.47
62
2.11
73
3.42
62,
3,6
4.29
no
ne
5.73
53,
56.
276
2,3,
6
7.12
002,
3,5,
6,10
8.23
702,
3,5,
6,10
9.70
02,
5,10
10.
4200
2,3,
5,6,
10
Lis
t al
l th
e fa
ctor
s of
eac
h n
um
ber
.
11.
481,
2,3,
4,6,
8,12
,16,
24,4
812
.24
1,2,
3,4,
6,8,
12,2
4
13.
121
1,11
,121
14.
821,
2,41
,82
15.
371,
3716
.19
61,
2,4,
7,14
,28,
49,9
8,19
6
17.
951,
5,19
,95
18.
110
1,2,
5,10
,11,
22,5
5,11
0
19.
961,
2,3,
4,6,
8,12
,16,
24,
20.
200
1,2,
4,5,
8,10
,20,
25,
32,4
8,96
40,5
0,10
0,20
0
Det
erm
ine
wh
eth
er e
ach
exp
ress
ion
is
a m
onom
ial.
Exp
lain
wh
y or
wh
y n
ot.
21.
82ye
s;a
nu
mb
er22
.4(
�m
)ye
s;p
rod
uct
of
a n
um
ber
an
d a
var
iab
lle23
.m
yes;
a va
riab
le24
.rv
yes;
pro
du
ct o
f va
riab
les
25.
6(x
�6)
no;d
iffer
ence
of t
wo
term
s26
.8n
�8
no
;d
iffe
ren
ce o
f tw
o t
erm
s
27.
(�12
)(�
8)x
yes;
pro
du
ct o
f 28
.w
��
yes;
pro
du
ct o
f va
riab
les
nu
mb
ers
and
a v
aria
ble
29.
2��
2wn
o;
sum
of
two
ter
ms
30.
2s�
tn
o;
dif
fere
nce
of
two
ter
ms
NEW
SPA
PER
SF
or E
xerc
ises
31
and
32,
refe
r to
th
e fo
llow
ing
info
rmat
ion
.
Bra
ndo
n d
eliv
ers
new
spap
ers
in h
is n
eigh
borh
ood.
On
Su
nda
y,h
e m
ust
del
iver
112
pap
ers.
Sin
ce h
e ri
des
his
bik
e,h
e se
para
tes
the
pape
rs i
nto
sm
alle
r st
acks
an
d de
live
rs o
ne
stac
kat
a t
ime.
31.
Wh
at s
ize
stac
ks c
an h
e m
ake?
2 st
acks
of
56 (
or
56 s
tack
s o
f 2)
,4 s
tack
s o
f28
(o
r 28
sta
cks
of
4),7
sta
cks
of
16 (
or
16 s
tack
s o
f 7)
,8 s
tack
s o
f 14
(or
14 s
tack
s o
f 8)
32.
If B
ran
don
can
car
ry n
o m
ore
than
30
pape
rs a
t a
tim
e an
d ca
n r
etu
rn h
ome
to r
esto
ckn
o m
ore
than
5 t
imes
,how
can
he
orga
niz
e th
e 11
2 pa
pers
?4
stac
ks o
f 28
pap
ers
Lesson 4-1
© Glencoe/McGraw-Hill A3 Glencoe Pre-Algebra
Answers (Lesson 4-1)
An
swer
s
Wri
te e
ach
exp
ress
ion
usi
ng
exp
onen
ts.
a.10
�10
�10
�10
�10
Th
e ba
se i
s 10
.It
is a
fac
tor
5 ti
mes
,so
the
expo
nen
t is
5.
10�10
�10
�10
�10
�10
5
b.(
p�
2)(p
�2)
(p�
2)
Th
e ba
se i
s p
�2.
It i
s a
fact
or 3
tim
es,s
o th
e ex
pon
ent
is 3
.
(p�
2)(p
�2)
(p�
2) �
(p�
2)3
©G
lenc
oe/M
cGra
w-H
ill17
0G
lenc
oe P
re-A
lgeb
ra
A n
umbe
r th
at is
exp
ress
ed u
sing
an
expo
nent
is c
alle
d a
pow
er.T
he b
ase
is th
e nu
mbe
r th
at is
m
ultip
lied.
The
exp
on
ent
tells
how
man
y tim
es th
e ba
se is
use
d as
a fa
ctor
.So,
43
has
a ba
se o
f 4
and
an e
xpon
ent o
f 3, a
nd 4
3 �
4�
4�
4 �
64.
Eva
luat
e x2
�4
if x
��
6.
x2�
4 =
(�6)
2 �
4R
epla
ce x
with
�6.
= (�
6)(�
6) �
4�
6 is
a fa
ctor
2 t
imes
.
= 36
�4
Mul
tiply
.
= 32
Sub
trac
t.
Stu
dy G
uid
e a
nd I
nte
rven
tion
Pow
ers
and
Exp
on
ents
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
__P
ER
IOD
____
_
Exam
ple1
Exam
ple1
4-2
4-2
Exam
ple2
Exam
ple2
Wri
te e
ach
exp
ress
ion
usi
ng
exp
onen
ts.
1.5
�5
�5
�5
�5
�5
�5
572.
(–7)
(–7)
(–7)
(�7)
3
3.d
�d
�d
�d
d4
4.x
�x
�y
�y
x2 y
2
5.(z
– 4)
(z–
4)(z
�4)
26.
3(–t
)(–t
)(–t
)3(
�t)
3
Eva
luat
e ea
ch e
xpre
ssio
n i
f g
�3,
h�
�1,
and
m�
9.
7.g
524
38.
5g2
45
9.g2
�m
010
.h
m2
�81
11.
g3�
2h25
12.
m�
hg3
�18
Exer
cises
Exer
cises
Exp
ress
ions
invo
lvin
g po
wer
s ar
e ev
alua
ted
usin
g or
der
of o
pera
tions
.Pow
ers
are
repe
ated
m
ultip
licat
ions
.The
y ar
e ev
alua
ted
afte
r an
y gr
oupi
ng s
ymbo
ls a
nd b
efor
e ot
her
mul
tiplic
atio
n or
di
visi
on o
pera
tions
.
En
rich
men
t
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
__P
ER
IOD
____
_
4-1
4-1
©G
lenc
oe/M
cGra
w-H
ill16
9G
lenc
oe P
re-A
lgeb
ra
Det
erm
ine
wh
eth
er 4
032
is d
ivis
ible
by
7.
4032
Cro
ss o
ut
the
ones
dig
it.
�4
Su
btr
act
twic
e th
e va
lue
of t
he
ones
dig
it f
rom
th
e re
st o
f th
e n
um
ber
.
399
If t
he
dif
fere
nce
is
a n
um
ber
that
you
kn
ow i
s d
ivis
ible
by
7,st
op.I
f n
ot,
�18 21
Sin
ce 2
1 is
div
isib
le b
y 7,
4032
is
divi
sibl
e by
7.
1.26
67
4.93
6n
eith
er
7.29
57n
eith
er
2.43
127
and
11
5.13
,293
7
8.31
2411
3.89
7611
6.70
85n
eith
er
9.65
457
and
11
Div
isib
ilit
y ru
le f
or 7
Div
isib
ilit
y ru
le f
or 1
1
Det
erm
ine
wh
eth
er 5
159
is d
ivis
ible
by
11.
Met
hod
1
5159
Cro
ss o
ut
the
ones
dig
it.
�9
Su
btr
act
the
valu
e of
th
e on
es d
igit
fro
m t
he
rest
of
the
nu
mb
er.
506
If t
he
dif
fere
nce
is
a n
um
ber
that
you
kn
ow i
s d
ivis
ible
by
11,s
top
.If
not
,
�6
44S
ince
44
is d
ivis
ible
by
11,5
159
is d
ivis
ible
by
11.
Met
hod
2
5159
5�
5�
10A
dd
th
e od
d-n
um
ber
ed d
igit
s (f
irst
an
d t
hir
d).
1�
9�
10A
dd
th
e ev
en-n
um
ber
ed d
igit
s (s
econ
d a
nd
fou
rth
).
0S
ub
trac
t th
e su
ms.
If t
he
dif
fere
nce
is
div
isib
le b
y 11
,th
e n
um
ber
is
Sin
ce 0
is
divi
sibl
e by
11,
5159
is
divi
sibl
e by
11.
Det
erm
ine
wh
eth
er 6
2,38
2 is
div
isib
le b
y 11
.
6�
3�
2 �
11A
dd
th
e od
d-n
um
ber
ed d
igit
s.
2�
8�
10A
dd
th
e ev
en-n
um
ber
ed d
igit
s.
1S
ub
trac
t th
e su
ms.
Sin
ce 1
is
not
div
isib
le b
y 11
,62,
382
is n
ot d
ivis
ible
by
11.
Det
erm
ine
wh
eth
er e
ach
nu
mb
er i
s d
ivis
ible
by
7 or
11.
Div
isib
ility
XX X
Lesson 4-1
rep
eat.
div
isib
le b
y 11
.
rep
eat.
X
© Glencoe/McGraw-Hill A4 Glencoe Pre-Algebra
Answers (Lessons 4-1 and 4-2)
©G
lenc
oe/M
cGra
w-H
ill17
2G
lenc
oe P
re-A
lgeb
ra
Pra
ctic
eP
ower
s an
d E
xpo
nen
ts
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
__P
ER
IOD
____
_
4-2
4-2
Wri
te e
ach
exp
ress
ion
usi
ng
exp
onen
ts.
1.11
�11
�11
113
2.2
�2
�2
�2
�2
�2
�2
�2
28
3.5
514.
(�4)
(�4)
(�4)
2
5.a
�a
�a
�a
a4
6.n
�n
�n
�n
�n
n5
7.4
�4
�4
438.
(b �
b)(b
�b)
(b �
b)b
6
9.(�
v)(�
v)(�
v)(�
v)(�
v)4
10.
x �
x �
z �
z �
zx
2z
3
11.
2 �
2 �
2 �
2 �
2 �
t �
t2
5 t2
12.
m �
m �
m �
n �
p �
pm
3 np
2
Exp
ress
eac
h n
um
ber
in
exp
and
ed f
orm
.
13.
13
14.
1006
(1�
101)
�(3
�10
0)
(1 �
103)
�(0
�10
2)
�(0
�10
1 ) �
(6 �
100 )
15.
17,6
29(1
�10
4)
�(7
�10
3)
�16
.89
7(8
�10
2)
�(9
�10
1 ) �
(6 �
102)
�(2
�10
1)
�(9
�10
0)
(7 �
100)
Eva
luat
e ea
ch e
xpre
ssio
n i
f x
�3,
y�
�2,
and
z�
4.
17.
yx�
818
.51
01
19.
z216
20.
x29
21.
9x72
922
.z2
�22
64
23.
y5�
3224
.z2
�y4
0
25.
x2�
y2�
z229
26.
z2�
x27
FAM
ILY
TR
EEF
or E
xerc
ises
27
and
28,
refe
r to
th
e fo
llow
ing
info
rmat
ion
.
Whe
n ex
amin
ing
a fa
mily
tre
e,th
e br
anch
es a
re m
any.
You
are
gene
rati
on “
now
.”O
ne g
ener
atio
nag
o,yo
ur 2
par
ents
wer
e bo
rn.T
wo
gene
rati
ons
ago
your
4 g
rand
pare
nts
wer
e bo
rn.
27.
How
man
y gr
eat-
gran
dpar
ents
wer
e bo
rn t
hre
e ge
ner
atio
ns
ago?
23
or
8
28.
How
man
y “g
reat
”gr
andp
aren
ts w
ere
born
ten
gen
erat
ion
s ag
o?21
0o
r 10
24
Skil
ls P
ract
ice
Pow
ers
and
Exp
on
ents
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
__P
ER
IOD
____
_
4-2
4-2
©G
lenc
oe/M
cGra
w-H
ill17
1G
lenc
oe P
re-A
lgeb
ra
Wri
te e
ach
exp
ress
ion
usi
ng
exp
onen
ts.
1.7
�7
722.
(�3)
(�3)
(�3)
(�3)
(�3)
(–3)
5
3.4
414.
(k �
k)(k
�k)
(k �
k)k
6
5.p
�p
�p
�p
�p
�p
p6
6.3
�3
32
7.(�
a)(�
a)(�
a)(�
a)(–
a)4
8.6
�6
�6
�6
64
9.9
�9
�9
9310
.4
�y
�z
�z
�z
4yz
3
11.
s �
s �
s �
s �
t �
u �
us
4 tu
212
.5
�5
�5
�q
�q
53q
2
Exp
ress
eac
h n
um
ber
in
exp
and
ed f
orm
.
13.
135
14.
8732
(1 �
102)
�(3
�10
1)
�(5
�10
0)
(8 �
103)
�(7
�10
2)
�
(3 �
101)
�(2
�10
0)
15.
1005
16.
989
(1 �
103)�
(0 �
102
) �
(9 �
102
) �
(8 �
101 )
�(9
�10
0)
(0 �
101 )
�(5
�10
0)
Eva
luat
e ea
ch e
xpre
ssio
n i
f b
�8,
c�
2,an
d d
��
3.
17.
4c16
18.
c01
19.
b351
220
.c3
�3c
72
21.
3c9
22.
c416
23.
c2�
d1
24.
2b2
128
25.
b2�
c372
26.
d2
9
27.
d3
–27
28.
b2�
d3
37
29.
b2d
–192
30.
(b�
c)2
36
Lesson 4-2
© Glencoe/McGraw-Hill A5 Glencoe Pre-Algebra
Answers (Lesson 4-2)
An
swer
s
©G
lenc
oe/M
cGra
w-H
ill17
4G
lenc
oe P
re-A
lgeb
ra
En
rich
men
t
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
__P
ER
IOD
____
_
4-2
4-2
Exp
on
ents
Num
bers
can
be
expr
esse
d in
sev
eral
way
s.S
ome
num
bers
are
exp
ress
ed a
s su
ms.
Som
enu
mbe
rs a
re e
xpre
ssed
as
prod
ucts
of
fact
ors,
whi
le o
ther
num
bers
are
exp
ress
ed a
s po
wer
s.
Tw
o w
ays
to e
xpre
ss 2
7 ar
e 3
�3
�3
and
33.
Th
e n
um
ber
1 m
illi
on c
an b
e ex
pres
sed
in t
he
foll
owin
g w
ays.
1,00
0,00
010
00�
1000
100
�10
0�
100
102
�10
2�
102
1,00
0,00
0110
002
1003
106
Wri
te n
ames
for
eac
h n
um
ber
bel
ow u
sin
g th
e gi
ven
exp
onen
ts.
1.16
;exp
onen
ts:2
an
d 4
42,2
42.
81;e
xpon
ents
:2 a
nd
492
,34
3.64
;exp
onen
ts:2
an
d 6
82,2
64.
256;
expo
nen
ts:2
an
d 8
162 ,
28
5.62
5;ex
pon
ents
:2 a
nd
4 25
2 ,54
6.72
9;ex
pon
ents
:2 a
nd
627
2 ,36
7.24
01;e
xpon
ents
:2 a
nd
449
2 ,74
8.40
96;e
xpon
ents
:2 a
nd
1264
2 ,21
2
9.65
61;e
xpon
ents
:2 a
nd
881
2 ,38
10.
390,
625;
expo
nen
ts:2
an
d 8
6252
,58
Nu
mbe
rs t
hat
can
be
nam
ed a
s po
wer
s w
ith
lik
e ba
ses
can
be
mu
ltip
lied
by
addi
ng
the
expo
nen
ts.
8�
8�
23 �
23
�23
�3
�26
Wri
te t
he
pro
du
ct o
f ea
ch p
air
of f
acto
rs i
n e
xpon
enti
al f
orm
.
11.
9�
932
�32
�34
12.
4�
42
2�
22�
24
13.
16�
824
�23
�27
14.
125
�25
53�
52�
55
15.
27�
933
�32
�35
16.
81�
2734
�33
�37
17.
49�
4972
�72
�74
18.
121
�12
111
2�
112
�11
4
Readin
g t
o L
earn
Math
em
ati
csP
ower
s an
d E
xpo
nen
ts
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
__P
ER
IOD
____
_
4-2
4-2
©G
lenc
oe/M
cGra
w-H
ill17
3G
lenc
oe P
re-A
lgeb
ra
Why
are
exp
onen
ts im
port
ant
in c
ompa
ring
com
pute
r da
ta?
Do
the
acti
vity
at
the
top
of
pag
e 15
3 in
you
r te
xtb
ook
.Wri
te y
our
answ
ers
bel
ow.
a.W
rite
16
as a
pro
duct
of
fact
ors
of 2
.How
man
y fa
ctor
s ar
e th
ere?
2 �
2 �
2 �
2;4
fact
ors
b.
How
man
y fa
ctor
s of
2 f
orm
th
e pr
odu
ct 1
28?
7 fa
cto
rs
c.O
ne
meg
abyt
e is
102
4 ki
loby
tes.
How
man
y fa
ctor
s of
2 f
orm
th
e pr
odu
ct 1
024?
10 f
acto
rs
Wri
te a
def
init
ion
an
d g
ive
an e
xam
ple
of
each
new
voc
abu
lary
wor
d o
r p
hra
se.
6.W
rite
eac
h e
xpre
ssio
n u
sin
g ex
pon
ents
.
a.4
�4
�4
�4
44b
.x
�x
�x
�y
�y
x3 y
2
c.(�
2)(�
2)(�
2)(�
2)3
d.
5�
r�
r�
m�
m�
m5r
2 m3
7.T
he n
umbe
r (3
�10
3 ) �
(5 �
102 )
�(0
�10
1 ) �
(2 �
100 )
is w
ritt
en i
n
ex
pan
ded
form
,wh
ile
3502
is
wri
tten
in
st
and
ard
form
.
Hel
pin
g Y
ou
Rem
emb
er8.
Exp
lain
how
th
e te
rms
base
,pow
er,a
nd
expo
nen
tar
e re
late
d.P
rovi
de a
n e
xam
ple.
Sam
ple
an
swer
:A p
ower
is a
n e
xpre
ssio
n w
ith t
wo
par
ts—
a b
ase
and
an
exp
on
ent.
Fo
r ex
amp
le,t
he
pow
er 2
3h
as a
bas
e o
f 2
and
an
exp
on
ent
of
3.
Pre-
Act
ivit
y
Rea
din
g t
he
Less
on
1–5.
See
stu
den
ts’w
ork
.
Lesson 4-2
Voca
bula
ryD
efin
itio
nE
xam
ple
1.ba
se
2.ex
pon
ent
3.po
wer
4.st
anda
rdfo
rm
5.ex
pan
ded
form
© Glencoe/McGraw-Hill A6 Glencoe Pre-Algebra
Answers (Lesson 4-2)
©G
lenc
oe/M
cGra
w-H
ill17
6G
lenc
oe P
re-A
lgeb
ra
Skil
ls P
ract
ice
Pri
me
Fact
ori
zati
on
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
__P
ER
IOD
____
_
4-3
4-3
Det
erm
ine
wh
eth
er e
ach
nu
mb
er i
s p
rim
e or
com
pos
ite.
1.41
pri
me
2.29
pri
me
3.87
com
po
site
4.36
com
po
site
5.57
com
po
site
6.61
pri
me
7.71
pri
me
8.10
3p
rim
e
9.39
com
po
site
10.
91co
mp
osi
te
11.
47p
rim
e12
.67
pri
me
Wri
te t
he
pri
me
fact
oriz
atio
n o
f ea
ch n
um
ber
.Use
exp
onen
ts f
or r
epea
ted
fac
tors
.
13.
2022
�5
14.
402
3�
5
15.
322
516
.44
22
�11
17.
902
�3
2�
518
.12
111
2
19.
462
�23
20.
302
�3
�5
21.
655
�13
22.
802
4�
5
Fac
tor
each
mon
omia
l.
23.
15t
3 �
5 �
t24
.16
r22
�2
�2
�2
�r
�r
25.
�11
m2
�1
�11
�m
�m
26.
�49
y3�
1 �
7 �
7 �
y�
y�
y
27.
21ab
3 �
7 �
a�
b28
.�
42xy
z�
1 �
2 �
3 �
7 �
x�
y�
z
29.
45j2
k3
�3
�5
�j�
j�k
30.
17u
2 v2
17 �
u�
u�
v�
v
31.
27d
43
�3
�3
�d
�d
�d
�d
32.
�16
cd2
�1
�2
�2
�2
�2
�c
�d
�d
Fin
d t
he
pri
me
fact
oriz
atio
n o
f 48
.
Det
erm
ine
wh
eth
er e
ach
nu
mb
er i
s p
rim
eor
com
pos
ite.
1.27
com
po
site
2.15
1p
rim
e
3.77
com
po
site
4.25
com
po
site
Wri
te t
he
pri
me
fact
oriz
atio
n f
or e
ach
nu
mb
er.U
se e
xpon
ents
for
rep
eate
d f
acto
rs.
5.16
246.
4532
�5
7.78
2 �
3 �
138.
702
�5
�7
Fac
tor
each
mon
omia
l.
9.6m
32
�3
�m
�m
�m
10.
–20x
y2�
1 �
2 �
2 �
5 �
x�
y�
y
11.
a2b2
c3a
�a
�b
�b
�c
�c
�c
12.
25h
5 �
5 �
h
Exam
ple2
Exam
ple2
A p
rim
e n
um
ber
is a
who
le n
umbe
r th
at h
as e
xact
ly t
wo
fact
ors,
1 a
nd it
self.
A c
om
po
site
nu
mb
eris
a
who
le n
umbe
r th
at h
as m
ore
than
tw
o fa
ctor
s.Z
ero
and
1 ar
e ne
ither
prim
e no
r co
mpo
site
.
Det
erm
ine
wh
eth
er 2
9 is
pri
me
or c
omp
osit
e.
Fin
d th
e fa
ctor
s of
29.
29 �
1�
29
Th
e on
ly f
acto
rs o
f 29
are
1 a
nd
29,t
her
efor
e 29
is
a pr
ime
nu
mbe
r.
Any
com
posi
te n
umbe
r ca
n be
writ
ten
as a
pro
duct
of
prim
e nu
mbe
rs.A
fact
or t
ree
can
be u
sed
to f
ind
the
prim
e fa
ctor
izat
ion.
Exam
ple1
Exam
ple1S
tudy G
uid
e a
nd I
nte
rven
tion
Pri
me
Fact
ori
zati
on
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
__P
ER
IOD
____
_
4-3
4-3
©G
lenc
oe/M
cGra
w-H
ill17
5G
lenc
oe P
re-A
lgeb
ra
Th
e pr
ime
fact
oriz
atio
n o
f 48
is
2�
2 �
2 �
2 �
3 or
24
�3.
In a
lgeb
ra,
mon
omia
ls c
an b
e fa
ctor
ed a
s a
prod
uct
of p
rime
num
bers
and
var
iabl
es w
ith n
o ex
pone
ntgr
eate
r th
an 1
.So,
8x
2fa
ctor
s as
2 �
2 �
2 �
x�
x.
48 is
the
num
ber
to b
e fa
ctor
ed.
Fin
d an
y pa
ir of
who
le n
umbe
r fa
ctor
s of
48.
Con
tinue
to
fact
or a
ny n
umbe
r th
at is
not
prim
e.
The
fact
or t
ree
is c
ompl
ete
whe
n th
ere
is a
row
of
prim
e nu
mbe
rs.
2 �
22
32
48 6 �
8
2 �
32
� 4
Exer
cises
Exer
cises
Lesson 4-3
© Glencoe/McGraw-Hill A7 Glencoe Pre-Algebra
Answers (Lesson 4-3)
An
swer
s
©G
lenc
oe/M
cGra
w-H
ill17
8G
lenc
oe P
re-A
lgeb
ra
Readin
g t
o L
earn
Math
em
ati
csP
rim
e Fa
cto
riza
tio
n
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
__P
ER
IOD
____
_
4-3
4-3
Ho
w c
an m
od
els
be
use
d t
o d
eter
min
e w
het
her
nu
mb
ers
are
pri
me?
Do
the
acti
vity
at
the
top
of
pag
e 15
9 in
you
r te
xtb
ook
.Wri
te y
our
answ
ers
bel
ow.
a.U
se g
rid
pape
r to
dra
w a
s m
any
diff
eren
t re
ctan
gula
r ar
ran
gem
ents
of
2,3,
4,5,
6,7,
8,an
d 9
squ
ares
as
poss
ible
. See
stu
den
ts’a
nsw
ers.
b.W
hic
h n
um
bers
of
squ
ares
can
be
arra
nge
d in
mor
e th
an o
ne
way
? 4,
6,8,
9
c.W
hic
h n
um
bers
of
squ
ares
can
on
ly b
e ar
ran
ged
one
way
?2,
3,5,
7
d.W
hat
do
all
rect
angl
es t
hat
you
lis
ted
in p
art
ch
ave
in c
omm
on?
Exp
lain
.T
hey
all
hav
e a
wid
th o
f 1
bec
ause
no
oth
er p
air
of
fact
ors
can
be
fou
nd
.
Pre-
Act
ivit
y
Rea
din
g t
he
Less
on
1–
5.S
ee s
tud
ents
’wo
rk.
Wri
te a
def
init
ion
an
d g
ive
an e
xam
ple
of
each
new
voc
abu
lary
wor
d o
r p
hra
se.
Hel
pin
g Y
ou
Rem
emb
er
6.C
ompo
site
is
a w
ord
use
d in
eve
ryda
y E
ngl
ish
.
a.F
ind
the
defi
nit
ion
of
com
posi
tein
th
e di
ctio
nar
y.W
rite
th
e de
fin
itio
n.
mad
e u
p o
f d
isti
nct
par
ts
b.
Exp
lain
how
th
e E
ngl
ish
def
init
ion
can
hel
p yo
u r
emem
ber
how
com
posi
te i
s u
sed
in m
ath
emat
ics.
Sam
ple
an
swer
:C
om
po
site
nu
mb
ers
are
mad
e u
p o
f m
any
dis
tin
ct p
arts
,or
fact
ors
.
Voca
bula
ryD
efin
itio
nE
xam
ple
1.co
mpo
site
nu
mbe
r
2.fa
ctor
3.fa
ctor
tre
e
4.pr
ime
fact
oriz
atio
n
5.pr
ime
nu
mbe
r
Pra
ctic
eP
rim
e Fa
cto
riza
tio
n
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
__P
ER
IOD
____
_
4-3
4-3
©G
lenc
oe/M
cGra
w-H
ill17
7G
lenc
oe P
re-A
lgeb
ra
Det
erm
ine
wh
eth
er e
ach
nu
mb
er i
s p
rim
e or
com
pos
ite.
1.11
pri
me
2.63
com
po
site
3.73
pri
me
4.75
com
po
site
5.49
com
po
site
6.69
com
po
site
7.53
pri
me
8.83
pri
me
Wri
te t
he
pri
me
fact
oriz
atio
n o
f ea
ch n
um
ber
.Use
exp
onen
ts f
or r
epea
ted
fac
tors
.
9.33
3 �
1110
.24
23�
3
11.
7223
�32
12.
276
22�
3 �
23
13.
855
�17
14.
1024
210
15.
955
�19
16.
200
23
�5
2
17.
243
35
18.
735
3 �
5 �
72
Fac
tor
each
mon
omia
l.
19.
35v
5 �
7 �
v20
.49
c27
�7
�c
�c
21.
�14
b3�
1 �
2 �
7 �
b�
b�
b22
.�
81h
2�
1 �
3 �
3 �
3 �
3 �
h�
h
23.
33w
z3
�11
�w
�z
24.
�56
ghj
�1
�2
�2
�2
�7
�g
�h
�j
25.
NU
MB
ERTH
EORY
Tw
in p
rim
esar
e a
pair
of
con
secu
tive
odd
pri
mes
,wh
ich
dif
fer
by 2
.For
exa
mpl
e,3
and
5 ar
e tw
in p
rim
es.F
ind
the
twin
pri
mes
les
s th
an 1
00.
(Hin
t:T
her
e ar
e 8
pair
s of
tw
ins
less
th
an 1
00.)
3,5;
5,7;
11,1
3;17
,19;
29,3
1;41
,43;
59,6
1;71
,73
Lesson 4-3
© Glencoe/McGraw-Hill A8 Glencoe Pre-Algebra
Answers (Lesson 4-3)
©G
lenc
oe/M
cGra
w-H
ill18
0G
lenc
oe P
re-A
lgeb
ra
The
gre
ates
t nu
mbe
r th
at is
a fa
ctor
of
two
or m
ore
num
bers
is t
he g
reat
est
com
mo
n f
acto
r (G
CF
).Tw
o w
ays
to f
ind
the
GC
F a
re s
how
n be
low
.
Fin
d t
he
GC
F o
f 24
an
d 3
2.
Met
ho
d 1
Lis
t th
e fa
ctor
s.
fact
ors
of 2
4:1,
2,3,
4,6,
8,12
,24
Look
for
fact
ors
com
mon
to
both
list
s, 1
, 2,
4,
and
8.
fact
ors
of 3
2:1,
2,4,
8,16
,32
Th
e gr
eate
st c
omm
on f
acto
r of
24
and
32 i
s 8.
Met
ho
d 2
U
se p
rim
e fa
ctor
izat
ion
.
24 �
2 �
2 �
2 �
3F
ind
the
com
mon
prim
e fa
ctor
s of
24
and
32.
32 �
2 �
2 �
2 �
2 �
2
Mul
tipl
y th
e co
mm
on p
rim
e fa
ctor
s.T
he g
reat
est
com
mon
fac
tor
of 2
4 an
d 32
is
2 �
2 �
2or
8.
In a
lgeb
ra,
grea
test
com
mon
fact
ors
are
used
to
fact
or e
xpre
ssio
ns.
Fac
tor
5x�
10.
Fir
st,f
ind
the
GC
F o
f 5x
and
10.
5x�
5�
x
10 �
2 �
5T
he G
CF
is 5
.
Now
wri
te e
ach
ter
m a
s a
prod
uct
of
the
GC
F a
nd
its
rem
ain
ing
fact
ors.
5x�
10 �
5(x)
�5(
2)
�5(
x�
2)
Dis
trib
utiv
e P
rope
rty
So,
5x �
10 �
5(x
�2)
.
Stu
dy G
uid
e a
nd I
nte
rven
tion
Gre
ates
t C
om
mo
n F
acto
r (G
CF
)
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
__P
ER
IOD
____
_
Exam
ple1
Exam
ple1
4-4
4-4
Exam
ple2
Exam
ple2
Fin
d t
he
GC
F o
f ea
ch s
et o
f n
um
ber
s.
1.30
,42
62.
15,3
33
3.44
,110
224.
16,4
816
Fac
tor
each
exp
ress
ion
.
5.4g
�16
4(g
�4)
6.2d
�6
2(d
�3)
7.8a
�24
8(a
�3)
8.f2
�2f
f(f
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9.6
�3j
3(2
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10.
16n2
�40
n8n
(2n
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Exer
cises
Exer
cises
En
rich
men
t
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
__P
ER
IOD
____
_
4-3
4-3
©G
lenc
oe/M
cGra
w-H
ill17
9G
lenc
oe P
re-A
lgeb
ra
Pri
me
Pyr
amid
A p
rim
e n
um
ber
is a
who
le n
umbe
r th
at h
as e
xact
ly t
wo
fact
ors—
itse
lf a
nd 1
.The
py
ram
id b
elow
is c
alle
d a
prim
e py
ram
id.E
ach
row
beg
ins
wit
h 1
and
ends
wit
h th
e nu
mbe
rof
tha
t ro
w.S
o,ro
w 2
beg
ins
wit
h 1
and
ends
wit
h 2,
row
3 b
egin
s w
ith
1 an
d en
ds w
ith
3,an
dso
on.
In e
ach
row
,the
num
bers
from
1 t
o th
e ro
w n
umbe
r ar
e ar
rang
ed s
uch
that
the
sum
of
any
two
adja
cent
num
bers
is a
pri
me
num
ber.
For
exa
mpl
e,lo
ok a
t ro
w 4
:
• It
mu
st c
onta
in t
he
nu
mbe
rs 1
,2,3
,an
d 4.
• It
mu
st b
egin
wit
h 1
an
d en
d w
ith
4.
• T
he
sum
of
adja
cen
t pa
irs
mu
st b
e a
prim
e n
um
ber:
1�
2�
3,2
�3
�5,
3�
4�
7
1.C
ompl
ete
the
pyra
mid
by
fill
ing
in t
he
mis
sin
g n
um
bers
.
2.E
xten
d th
e py
ram
id t
o ro
w 1
3.
1,4,
3,2,
5,6,
7,10
,9,8
,11,
12,1
3
3.E
xpla
in t
he
patt
ern
s yo
u s
ee i
n t
he
com
plet
ed p
yram
id.
Sam
ple
an
swer
:E
ach
ro
w a
lter
nat
es o
dd
an
d e
ven
nu
mb
ers.
Mu
ltip
les
of
3 fo
rm d
iag
on
als
that
are
co
nst
ant.
112
43
25
67
109
811
111
43
25
67
109
8
110
23
47
65
89
19
85
67
43
2
18
23
47
65
17
54
32
6
14
32
56
14
32
5
12
34
12
3
1
*
2
Lesson 4-3
© Glencoe/McGraw-Hill A9 Glencoe Pre-Algebra
Answers (Lessons 4-3 and 4-4)
An
swer
s
©G
lenc
oe/M
cGra
w-H
ill18
2G
lenc
oe P
re-A
lgeb
ra
Pra
ctic
eG
reat
est
Co
mm
on
Fac
tor
(GC
F)
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
__P
ER
IOD
____
_
4-4
4-4
Fin
d t
he
GC
F o
f ea
ch s
et o
f n
um
ber
s or
mon
omia
ls.
1.9,
369
2.42
,60
6
3.16
,60
44.
29,5
829
5.18
,35
16.
90,4
8030
7.80
,45
58.
700,
200
100
9.17
,85
1710
.24
,84,
168
12
11.
55,1
055
12.
252,
126
126
13.
5p,2
0p2
5p14
.28
a,49
ab7a
15.
8b,5
c1
16.
6a2 ,
18b2
6
17.
88s2
t,40
st2
8st
18.
42a2
b,60
ab2
6ab
Fac
tor
each
exp
ress
ion
.
19.
10x
�40
10(x
�4)
20.
8v�
568(
v�
7)
21.
9t�
99
(t�
1)22
.13
m�
3913
(m�
3)
23.
90 �
45n
45(2
�n
)24
.15
p�
6015
(p�
4)
25.
48 �
8r8
(6 �
r)26
.11
z�
5511
(z�
5)
27.
18q
�54
18(q
�3)
28.
125
�25
h25
(5 �
h)
29.
42a
�77
7(6a
�11
)30
.30
�45
s15
(2 �
3s)
31.
50n
�30
10(5
n�
3)32
.18
�12
d6(
3 �
2d
)
33.
27m
�10
53(
9m�
35)
34.
65 �
39b
13(5
�3b
)
35.
21d
�63
7(3d
�9)
36.
48 �
84m
12(4
�7m
)
37.
SCH
OO
L TR
IPT
hir
ty-t
wo
seve
nth
gra
ders
,48
eigh
th g
rade
rs,a
nd
60 n
inth
gra
ders
are
taki
ng
a sk
i tr
ip.I
n o
rder
to
hel
p st
ude
nts
get
bet
ter
acqu
ain
ted,
stu
den
ts f
rom
eac
hgr
ade
leve
l ar
e to
rid
e ea
ch b
us.
Wh
at i
s th
e gr
eate
st n
um
ber
of b
use
s th
at c
an b
e u
sed
if s
tude
nts
fro
m e
ach
gra
de l
evel
are
div
ided
equ
ally
am
ong
the
buse
s?4
buse
s
Skil
ls P
ract
ice
Gre
ates
t C
om
mo
n F
acto
r (G
CF
)
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
__P
ER
IOD
____
_
4-4
4-4
©G
lenc
oe/M
cGra
w-H
ill18
1G
lenc
oe P
re-A
lgeb
ra
Fin
d t
he
GC
F o
f ea
ch s
et o
f n
um
ber
s or
mon
omia
ls.
1.15
,50
52.
24,8
13
3.18
,27
94.
36,6
44
5.88
,40
86.
54,6
39
7.11
,22
118.
14,2
51
9.20
,30
1010
.16
,18
2
11.
64,8
016
12.
16,2
48
13.
30t,
40t2
10t
14.
6,9t
3
15.
16k2
,40k
8k16
.9m
,15n
3
17.
7pq,
8qq
18.
18p,
459
Fac
tor
each
exp
ress
ion
.
19.
5b�
155(
b�
3)20
.7t
�49
7(t
�7)
21.
6w�
186(
w�
3)22
.10
0 �
50x
50(2
�x)
23.
7x�
77(
x�
1)24
.12
n�
6012
(n�
5)
25.
24 �
8g8(
3 �
g)
26.
50 �
5f5(
10 �
f)
27.
3n�
243(
n�
8)28
.9
��
639(
l�7)
29.
6u�
366(
u�
6)30
.70
�7c
7(10
�c
)
31.
42 �
21x
21(2
�x
)32
.12
y�
164(
3y�
4)
33.
6p�
126(
p�
2)34
.9r
�81
9(r
�9)
35.
6 �
8q2(
3 �
4q)
36.
21x
�33
3(7x
�11
)
Lesson 4-4
© Glencoe/McGraw-Hill A10 Glencoe Pre-Algebra
Answers (Lesson 4-4)
©G
lenc
oe/M
cGra
w-H
ill18
4G
lenc
oe P
re-A
lgeb
ra
En
rich
men
t
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
__P
ER
IOD
____
_
4-4
4-4
GC
Fs
by S
ucc
essi
ve D
ivis
ion
An
oth
er w
ay t
o fi
nd
the
grea
test
com
mon
fac
tor
(GC
F)
of t
wo
nu
mbe
rs i
s to
use
su
cces
sive
div
isio
n.T
his
met
hod
wor
ks w
ell
for
larg
e n
um
bers
.
Fin
d th
e G
CF
of
848
and
1325
.
Ste
p 1
Div
ide
the
smal
ler
nu
mbe
r in
to t
he
grea
ter
nu
mbe
r.
1 R
477
848
�1 �3�2�5� 848
477
Ste
p 2
Div
ide
the
rem
aind
er in
to t
he d
ivis
or.R
epea
t th
is s
tep
unti
l you
get
a r
emai
nder
of 0
.
1 R
371
1 R
106
3 R
532
R0
477
�8 �4�8�
371
�4�7�7�
106
�3�7�1�
53�1�
0�6�47
737
131
810
637
1
106
53
0
Th
e la
st d
ivis
or i
s th
e G
CF
of
the
two
orig
inal
nu
mbe
rs.S
o th
e G
CF
of
848
and
1325
is
53.
Use
th
e m
eth
od a
bov
e to
fin
d t
he
GC
F o
f ea
ch p
air
of n
um
ber
s.
1.18
7;57
817
2.16
1;94
323
3.21
5;18
4943
4.45
3;48
41
5.43
2;58
812
6.27
9;40
331
7.13
25;3
498
538.
9840
;175
11
9.34
84;5
963
6710
.18
02;1
0610
6
11.
45,7
87;6
9,87
51
12.
35,8
11;1
02,0
7017
3
Readin
g t
o L
earn
Math
em
ati
csG
reat
est
Co
mm
on
Fac
tor
(GC
F)
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
__P
ER
IOD
____
_
4-4
4-4
©G
lenc
oe/M
cGra
w-H
ill18
3G
lenc
oe P
re-A
lgeb
ra
Ho
w c
an a
dia
gra
m b
e u
sed
to
fin
d t
he
gre
ates
t co
mm
on
fac
tor?
Do
the
acti
vity
at
the
top
of
pag
e 16
4 in
you
r te
xtb
ook
.Wri
teyo
ur
answ
ers
bel
ow.
a.W
hic
h n
um
bers
are
in
bot
h c
ircl
es?
2,2
b.F
ind
the
prod
uct
of
the
nu
mbe
rs t
hat
are
in
bot
h c
ircl
es.
4
c.Is
th
e pr
odu
ct a
lso
a fa
ctor
of
12 a
nd
20?
yes
d.M
ake
a V
enn
dia
gram
sh
owin
g th
e pr
ime
fact
ors
of 1
6 an
d 28
.Th
enu
se i
t to
fin
d th
e co
mm
on f
acto
rs o
f th
e n
um
bers
.2,
2
Pre-
Act
ivit
y
Rea
din
g t
he
Less
on
1–2.
See
stu
den
ts’w
ork
.W
rite
a d
efin
itio
n a
nd
giv
e an
exa
mp
le o
f ea
ch n
ew v
ocab
ula
ry w
ord
or
ph
rase
.
Prim
e Fa
cto
rso
f 28
Prim
e Fa
cto
rso
f 16
2 22 2
16 �
2 •
2 •
2 •
228
� 2
• 2
• 7
7
Hel
pin
g Y
ou
Rem
emb
er3.
Sum
mar
ize
in y
our
own
wor
ds h
ow t
o fi
nd t
he g
reat
est
com
mon
fact
or o
f tw
o nu
mbe
rs u
sing
each
met
hod.
a.pr
ime
fact
oriz
atio
nW
rite
th
e p
rim
e fa
cto
riza
tio
n o
f ea
ch n
um
ber
,th
enlo
ok
for
the
pri
me
fact
ors
co
mm
on
to
bo
th n
um
ber
s.T
he
gre
ates
t co
mm
on
fac
tor
is t
he
pro
du
ct o
f th
e co
mm
on
fac
tors
.
b.
list
s of
fac
tors
Mak
e a
list
of
all f
acto
rs o
f b
oth
nu
mb
ers.
Th
e la
rges
t fa
cto
r th
at is
in b
oth
list
s is
th
e g
reat
est
com
mo
n f
acto
r.
c.a
Ven
n d
iagr
amM
ake
a V
enn
dia
gra
m w
ith
th
e p
rim
e fa
cto
rs o
f ea
ch n
um
ber
in a
cir
cle.
Th
e G
CF
is t
he
pro
du
ct o
f th
e fa
cto
rs in
th
e ov
erla
pp
ing
sec
tio
n o
f th
e d
iag
ram
.
Voca
bula
ryD
efin
itio
nE
xam
ple
1.V
enn
diag
ram
2.gr
eate
stco
mm
on f
acto
r
Lesson 4-4
© Glencoe/McGraw-Hill A11 Glencoe Pre-Algebra
Answers (Lesson 4-4)
An
swer
s
©G
lenc
oe/M
cGra
w-H
ill18
6G
lenc
oe P
re-A
lgeb
ra
Skil
ls P
ract
ice
Sim
plif
yin
g A
lgeb
raic
Fra
ctio
ns
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
__P
ER
IOD
____
_
4-5
4-5
Wri
te e
ach
fra
ctio
n i
n s
imp
lest
for
m.I
f th
e fr
acti
on i
s al
read
y in
sim
ple
st f
orm
,w
rite
sim
pli
fied
.
1.�1 70 0�
�1 7�2.
�1 12 8��2 3�
3.�3 40 5�
�2 3�
4.� 28 4�
�1 3�5.
�4 6��2 3�
6.�5 66 3�
�8 9�
7.�1 28 4�
�3 4�8.
� 47 9��1 7�
9.�1 33 9�
�1 3�
10.
�2 31 6�� 17 2�
11.
�3 42 0��4 5�
12.
� 34 6��1 9�
13.
�4 54 5��4 5�
14.
� 14 4��2 7�
15.
�3 46 8��3 4�
16.
�8 91 0�� 19 0�
17.
� 25 5��1 5�
18.
�5 76 4��2 38 7�
19.
�2 42 2��1 21 1�
20.
� 17 8�si
mp
lifie
d21
.�d d
3 4�� d1 �
22.
� yy 3�� y1 2�
23.
�q q3 �q
224
.�s s4 2�
s2
25.
�x y2 �si
mp
lifie
d26
.� 19 2a a�
�3 4�27
.� 18 6t t�
�1 2�
28.
�1 24 4g g�� 17 2�
29.
�3 45 0j ��7 8j �
30.
� 21 00 00 pp 2�
� 21 p�
31.
� 17 05 0n n3
�� 4n3
2�32
.� 26 1k k5 2
��2k 7
3 �33
.�3 4a b�
sim
plif
ied
34.
� 21 46 db ��2 3b d�
35.
� 28 4a a��1 3�
36.
� 35 5t t3 2�
� 7t �
Sim
pli
fy �2 34 6a ab 22
�.
�2 34 6a ab 22�
�D
ivid
e th
e nu
mer
ator
and
den
omin
ator
by
the
GC
F, 2
�2
�3
�a.
��2
3�b �
a�b
�or
�2 3b a2 �S
impl
ify.
2 �
2 �
2 �
3 �
a�
b�
b�
��
2 �
2 �
3 �
3 �
a�
a
Sim
pli
fy e
ach
fra
ctio
n.I
f th
e fr
acti
on i
s al
read
y in
sim
ple
st f
orm
,wri
te s
imp
lifi
ed.
1.�1 22 0�
�3 5�2.
�1 36 6��4 9�
3.� 17 05 0�
�3 4�
4.� 16 5�
�2 5�5.
� 28 4��1 3�
6.�3 8�
sim
plif
ied
7.� cc 3�
� c1 2�8.
�r r4 2�r2
9.�1 24 1b b�
�2 3�
10.
�2 24 6w w��1 12 3�
11.
� 15 2s t�si
mp
lifie
d12
.� 3d d
2�� 31 d�
Exam
ple2
Exam
ple2
A f
ract
ion
is in
sim
ple
st f
orm
whe
n th
e G
CF
of
the
num
erat
or a
nd t
he d
enom
inat
or is
1.O
ne w
ay t
ow
rite
a fr
actio
n in
sim
ples
t fo
rm is
to
writ
e th
e pr
ime
fact
oriz
atio
n of
the
num
erat
or a
nd t
he d
enom
inat
or.
The
n di
vide
the
num
erat
or a
nd d
enom
inat
or b
y th
e G
CF.
Wri
te �1 28 4�
in s
imp
lest
for
m.
Wri
te t
he
prim
e fa
ctor
izat
ion
of
the
nu
mer
ator
an
d th
e de
nom
inat
or.
�1 28 4��� 2
2 �2�
3 �2�
3 �3
�D
ivid
e th
e nu
mer
ator
and
den
omin
ator
by
the
GC
F, 2
�3.
�� 2
3 �2�
or �3 4�
Sim
plify
.
Alg
ebra
ic f
ract
ions
can
als
o be
writ
ten
in s
impl
est
form
.Aga
in,
you
can
writ
e th
e pr
ime
fact
oriz
atio
nof
the
num
erat
or a
nd t
he d
enom
inat
or,
then
div
ide
by t
he G
CF.
Exam
ple1
Exam
ple1S
tudy G
uid
e a
nd I
nte
rven
tion
Sim
plif
yin
g A
lgeb
raic
Fra
ctio
ns
NA
ME
____
____
____
____
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4-5
4-5
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w-H
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5G
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Exer
cises
Exer
cises
Lesson 4-5
11
11
//
//
11
1
1
11
1
1
/ /
/ /
/ /
//
© Glencoe/McGraw-Hill A12 Glencoe Pre-Algebra
Answers (Lesson 4-5)
©G
lenc
oe/M
cGra
w-H
ill18
8G
lenc
oe P
re-A
lgeb
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Readin
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Math
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ati
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imp
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4-5
4-5
Ho
w a
re s
imp
lifie
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ract
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s u
sefu
l in
rep
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nti
ng
m
easu
rem
ents
?D
o th
e ac
tivi
ty a
t th
e to
p o
f p
age
169
in y
our
text
boo
k.W
rite
you
r an
swer
s b
elow
.
a.A
re t
he
thre
e fr
acti
ons
equ
ival
ent?
Exp
lain
you
r re
ason
ing.
Yes;
the
sam
e p
ort
ion
of
each
cir
cle
is s
had
ed.
b.W
hic
h f
igu
re i
s di
vide
d in
to t
he
leas
t n
um
ber
of p
arts
? th
e th
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fig
ure
c.W
hic
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ract
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wou
ld y
ou s
ay i
s w
ritt
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n s
impl
est
form
? W
hy?
�1 4� ;T
he
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art
is n
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div
ided
into
sm
alle
r p
arts
.
Pre-
Act
ivit
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Rea
din
g t
he
Less
on
1–2.
See
stu
den
ts’w
ork
.
Wri
te a
def
init
ion
an
d g
ive
an e
xam
ple
of
each
new
voc
abu
lary
ph
rase
.
3.U
se a
Ven
n d
iagr
am t
o ex
plai
n h
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o si
mpl
ify
�1 48 5�.
Th
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e
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3,an
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.Th
e p
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e fa
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f th
e d
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ato
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e 3,
3,an
d 5
.Th
e G
CF
of
the
nu
mb
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9,is
sh
ow
n b
y th
e in
ters
ecti
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.So
,th
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mp
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ract
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is �2 5� .
Voca
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1.si
mpl
est
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frac
tion
Hel
pin
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ou
Rem
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4.E
xpla
in t
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sim
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itie
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s be
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a n
um
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an
dsi
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lgeb
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Bo
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s ar
e si
mp
lifie
d b
y d
ivid
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th
en
um
erat
or
and
den
om
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or
by t
he
GC
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he
on
ly d
iffe
ren
ce is
th
at a
nal
geb
raic
fra
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n h
as v
aria
ble
s as
fac
tors
in t
he
nu
mer
ato
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r th
ed
eno
min
ato
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1845
332
5
3�3
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____
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ER
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4-5
4-5
©G
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w-H
ill18
7G
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Wri
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fra
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n s
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lest
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Lesson 4-5
© Glencoe/McGraw-Hill A13 Glencoe Pre-Algebra
Answers (Lesson 4-5)
An
swer
s
Fin
d �( (� �
8 8) )4 2�
.Exp
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am�
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7
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2a2 (
3a)
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(a2
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Use
the
Com
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6)(a
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.
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r q
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g ex
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.
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9
16.
the
prod
uct
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two
cube
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o sq
uar
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17.
the
quot
ien
t of
six
to
the
eigh
th p
ower
an
d si
x sq
uar
ed66
Stu
dy G
uid
e a
nd I
nte
rven
tion
Mu
ltip
lyin
g a
nd
Div
idin
g M
on
om
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NA
ME
____
____
____
____
____
____
____
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____
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ATE
____
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ER
IOD
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Exam
ple1
Exam
ple1
4-6
4-6
Exam
ple2
Exam
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Exer
cises
Exer
cises
En
rich
men
t
NA
ME
____
____
____
____
____
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____
____
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ATE
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ER
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4-5
4-5
©G
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w-H
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9G
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Mat
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Fra
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Cu
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t fr
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17
3 812 42
210
5 915 35
324
910
5 816 28
4550
1112
4 78 20
4045
1213
89
3 75 15
321
1 2
5256
310
911
15 21
4044
19
35 777 13
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218
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112
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2133
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1 3
318
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4448
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15
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16
28
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336
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3035
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1815
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14
58
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4
2420
313
7
222
2117
9
161
12
34
56
78
910
1112
1314
1516
1718
1920
2122
2324
612
Lesson 4-5
© Glencoe/McGraw-Hill A14 Glencoe Pre-Algebra
Answers (Lessons 4-5 and 4-6)
©G
lenc
oe/M
cGra
w-H
ill19
2G
lenc
oe P
re-A
lgeb
ra
Pra
ctic
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ult
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an
d D
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Mo
no
mia
ls
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
__P
ER
IOD
____
_
4-6
4-6
Fin
d e
ach
pro
du
ct o
r q
uot
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t.E
xpre
ss y
our
answ
er u
sin
g ex
pon
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.
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45
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132
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413
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1 19 1�n
8
25.t
he
prod
uct
of
five
cu
bed
and
five
to
the
fou
rth
pow
er57
26.t
he
quot
ien
t of
eig
hte
en t
o th
e n
inth
pow
er a
nd
eigh
teen
squ
ared
187
27.t
he
prod
uct
of
zcu
bed
and
zcu
bed
z6
28.t
he
quot
ien
t of
xto
th
e fi
fth
pow
er a
nd
xcu
bed
x2
29.
SOU
ND
Dec
ibel
s ar
e u
nit
s u
sed
to m
easu
re s
oun
d.T
he
soft
est
sou
nd
that
can
be
hea
rdis
rat
ed a
s 0
deci
bels
(or
a r
elat
ive
lou
dnes
s of
1).
Ord
inar
y co
nve
rsat
ion
is
rate
d at
abou
t 60
dec
ibel
s (o
r a
rela
tive
lou
dnes
s of
106
).A
roc
k co
nce
rt i
s ra
ted
at a
bou
t 12
0de
cibe
ls (
or a
rel
ativ
e lo
udn
ess
of
1012
).H
ow m
any
tim
es g
reat
er i
s th
e re
lati
ve
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s of
a r
ock
con
cert
th
an t
he
rela
tive
lou
dnes
s of
ord
inar
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nve
rsat
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?10
6o
r1,
000,
000
tim
es
Skil
ls P
ract
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Mu
ltip
lyin
g a
nd
Div
idin
g M
on
om
ials
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
__P
ER
IOD
____
_
4-6
4-6
©G
lenc
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cGra
w-H
ill19
1G
lenc
oe P
re-A
lgeb
ra
Fin
d e
ach
pro
du
ct o
r q
uot
ien
t.E
xpre
ss y
our
answ
er u
sin
g ex
pon
ents
.
1.23
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282.
102
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9
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19.
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20.
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5
21.
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22.
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23.
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or
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426
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g4
27.
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(�y
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(�z)
7
29.t
he
prod
uct
of
two
squ
ared
an
d tw
o to
th
e si
xth
pow
er28
30.t
he
quot
ien
t of
ten
to
the
seve
nth
pow
er a
nd
ten
cu
bed
104
31.t
he
prod
uct
of
ysq
uar
ed a
nd
ycu
bed
y5
32.t
he
quot
ien
t of
ato
th
e tw
enti
eth
pow
er a
nd
ato
th
e te
nth
pow
era1
0
Lesson 4-6
© Glencoe/McGraw-Hill A15 Glencoe Pre-Algebra
Answers (Lesson 4-6)
An
swer
s
©G
lenc
oe/M
cGra
w-H
ill19
4G
lenc
oe P
re-A
lgeb
ra
En
rich
men
t
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
__P
ER
IOD
____
_
4-6
4-6
Div
idin
g P
ower
s w
ith
Dif
fere
nt
Bas
esS
ome
pow
ers
wit
h d
iffe
ren
t ba
ses
can
be
divi
ded.
Fir
st,y
ou m
ust
be
able
to
wri
te b
oth
as
pow
ers
of t
he
sam
e ba
se.A
n e
xam
ple
is s
how
n b
elow
.
�2 85 2��
� (22 35 )2�
To f
ind
the
pow
er o
f a
pow
er,
mul
tiply
the
exp
onen
ts.
��2 25 6�
�2�
1 or
�1 2�
Th
is m
eth
od c
ould
not
hav
e be
en u
sed
to d
ivid
e �2 95 2�
,sin
ce 9
can
not
be
wri
tten
as
a po
wer
of
2 u
sin
g in
tege
rs.
Sim
pli
fy e
ach
fra
ctio
n u
sin
g th
e m
eth
od s
how
n a
bov
e.E
xpre
ss t
he
solu
tion
w
ith
out
exp
onen
ts.
1.�8 22 2�
162.
�1 86 34 �12
83.
�9 33 3�27
4.�8 31 44 �
531,
441
5.� 83 19 2�
36.
�3 12 64 4�16
7.�1 22 55 32
�1
8.� 26 16 62
�1
9.� 11 00 06 03
�0.
001
10.
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11.
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2401
Readin
g t
o L
earn
Math
em
ati
csM
ult
iply
ing
an
d D
ivid
ing
Mo
no
mia
ls
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
__P
ER
IOD
____
_
4-6
4-6
©G
lenc
oe/M
cGra
w-H
ill19
3G
lenc
oe P
re-A
lgeb
ra
Ho
w a
re p
ow
ers
of
mo
no
mia
ls u
sefu
l in
co
mp
arin
g
eart
hq
uak
e m
agn
itu
des
?D
o th
e ac
tivi
ty a
t th
e to
p o
f p
age
175
in y
our
text
boo
k.W
rite
you
r an
swer
s b
elow
.
a.E
xam
ine
the
expo
nent
s of
the
fac
tors
and
the
exp
onen
ts o
f th
e pr
oduc
tsin
the
last
col
umn.
Wha
t do
you
obs
erve
? T
he
exp
on
ents
of
the
fact
ors
are
add
ed t
o g
et t
he
exp
on
ent
of
the
pro
du
ct.
b.M
ake
a co
nje
ctu
reab
out
a ru
le f
or d
eter
min
ing
the
expo
nen
t of
th
epr
odu
ct w
hen
you
mu
ltip
ly p
ower
s w
ith
th
e sa
me
base
.Tes
t yo
ur
rule
by m
ult
iply
ing
22�
24u
sin
g a
calc
ula
tor.
Sam
ple
an
swer
:A
dd
th
eex
po
nen
ts.
Pre-
Act
ivit
y
Rea
din
g t
he
Less
on
1.W
hen
mu
ltip
lyin
g po
wer
s w
ith
lik
e ba
ses,
add
the
expo
nen
ts.
2.W
hen
div
idin
g po
wer
s w
ith
lik
e ba
ses,
sub
trac
tth
e ex
pon
ents
.
3.W
rite
a d
ivis
ion
exp
ress
ion
wh
ose
quot
ien
t is
72 .
Sam
ple
an
swer
:�7 73 �
4.W
rite
a m
ult
ipli
cati
on e
xpre
ssio
n w
hos
e pr
odu
ct i
s v5
.S
amp
le a
nsw
er:
v2
�v
3
5.F
ind
each
pro
duct
.
a.4
�43
44b
.y7
�y5
y12
c.(�
2x2)(
5x2)
�10
x4
d.
�3r
2�
r�
3r3
6.F
ind
each
qu
otie
nt.
a.�7 74 2�
72b
.�v v9 3�
v6
c.�6 67 6�
61
or
6d
.�a b2 b 2
2�
a2
Hel
pin
g Y
ou
Rem
emb
er
7.E
xpla
in h
ow d
ivid
ing
pow
ers
is r
elat
ed t
o si
mpl
ifyi
ng
frac
tion
s.P
rovi
de a
n e
xam
ple
aspa
rt o
f yo
ur
expl
anat
ion
.S
amp
le a
nsw
er:W
hen
div
idin
g p
ow
ers
wit
h li
keb
ases
,su
btr
acti
ng
th
e ex
po
nen
ts is
eq
uiv
alen
t to
sim
plif
yin
g f
ract
ion
s.F
or
exam
ple
,34
�32
�34
�2
or
32by
th
e Q
uo
tien
t o
f P
ow
ers
rule
.
Wh
en s
imp
lifyi
ng
�3 34 2�,d
ivid
e th
e n
um
erat
or
and
den
om
inat
or
by t
he
GC
F,32
,to
get
�3 12 � o
r 32
.
Lesson 4-6
© Glencoe/McGraw-Hill A16 Glencoe Pre-Algebra
Answers (Lesson 4-6)
©G
lenc
oe/M
cGra
w-H
ill19
6G
lenc
oe P
re-A
lgeb
ra
Skil
ls P
ract
ice
Neg
ativ
e E
xpo
nen
ts
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
____
PE
RIO
D__
___
4-7
4-7
Wri
te e
ach
exp
ress
ion
usi
ng
a p
osit
ive
exp
onen
t.
1.3�
4� 31 4�
2.8�
7� 81 7�
3.10
�4
� 11 04�
4.(�
2)�
6� (�
1 2)6
�5.
(�40
)�3
� (�41 0)
3�
6.(�
17)�
12� (�
11 7)12
�
7.n
�10
� n1 10�8.
b�8
� b1 8�9.
q�5
� q1 5�
10.
m�
4� m1 4�
11.
v�11
� v1 11�12
.p�
2� p1 2�
Wri
te e
ach
fra
ctio
n a
s an
exp
ress
ion
usi
ng
a n
egat
ive
exp
onen
t ot
her
th
an �
1.
13.
� 81 2�8�
214
.� 11 05�
10�
515
.� 21 3�
2�3
16.
� 61 7�6�
717
.� 11 74�
17�
418
.� 21 12�
21�
2
19.
� 31 7�3�
720
.� 91 2�
9�2
21.
� 31 2�3�
2
22.
� 11 21�11
�2
23.
� 21 5�5�
224
.� 31 6�
6�2
Eva
luat
e ea
ch e
xpre
ssio
n i
f x
�1,
y�
2,an
d z
��
3.
25.
y�z
�1 8�26
.z�
2�1 9�
27.
x�8
1
28.
y�5
� 31 2�29
.z�
3�
� 21 7�30
.y�
1�1 2�
31.
z�4
� 81 1�32
.5
z� 11 25�
33.
x�99
1
34.
1z1
35.
4z� 61 4�
36.
yz�1 8�
Stu
dy G
uid
e a
nd I
nte
rven
tion
Neg
ativ
e E
xpo
nen
ts
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
__P
ER
IOD
____
_
4-7
4-7
©G
lenc
oe/M
cGra
w-H
ill19
5G
lenc
oe P
re-A
lgeb
ra
Ext
endi
ng t
he p
atte
rn b
elow
sho
ws
that
4–1
= �1 4�
or � 41 1�
.
42�
16�
441
�4
�4
40�
1�
4
4�1
��1 4�
Thi
s su
gges
ts t
he fo
llow
ing
defin
ition
.
a�n
= � a1 n�
, fo
r a
�0
and
any
inte
ger
n.
Wri
te e
ach
exp
ress
ion
usi
ng
a p
osit
ive
exp
onen
t.
a.3�
4b
.y�
2
3�4
�� 31 4�
y�2
�� y1 2�
We
can
eval
uate
alg
ebra
ic e
xpre
ssio
ns w
ith n
egat
ive
expo
nent
s us
ing
the
defin
ition
of
nega
tive
expo
nent
s.
Wri
te e
ach
exp
ress
ion
usi
ng
a p
osit
ive
exp
onen
t.
1.6�
4� 61 4�
2.(�
7)�
8� (�
1 7)8
�3.
b�6
� b1 6�4.
n�
1� n1 1�
or
� n1 �
Wri
te e
ach
fra
ctio
n a
s an
exp
ress
ion
usi
ng
a n
egat
ive
exp
onen
t ot
her
th
an�
1.
5.� 21 2�
2�2
6.� 11 34�
13�
47.
� 21 5�5�
28.
� 41 9�7�
2
Eva
luat
e ea
ch e
xpre
ssio
n i
f m
��
4,n
�1,
and
p�
6.
9.p�
2� 31 6�
10.
m�
3�
� 61 4�11
.(n
p) –
1�1 6�
12.
3m� 81 1�
Eva
luat
e b�
2 if
b�
3.
b�2
�3�
2R
epla
ce b
with
3.
�� 31 2�
Def
initi
on o
f ne
gativ
e ex
pone
nt
��1 9�
Fin
d 32
.
Exam
ple
1Ex
ampl
e 1
Exam
ple
2Ex
ampl
e 2
Exer
cises
Exer
cises
Lesson 4-7
© Glencoe/McGraw-Hill A17 Glencoe Pre-Algebra
Answers (Lesson 4-7)
An
swer
s
©G
lenc
oe/M
cGra
w-H
ill19
8G
lenc
oe P
re-A
lgeb
ra
Readin
g t
o L
earn
Math
em
ati
csN
egat
ive
Exp
on
ents
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
__P
ER
IOD
____
_
4-7
4-7
Ho
w d
o n
egat
ive
exp
on
ents
rep
rese
nt
rep
eate
d d
ivis
ion
?
Do
the
acti
vity
at
the
top
of
pag
e 18
1 in
you
r te
xtb
ook
.Wri
teyo
ur
answ
ers
bel
ow.
a.D
escr
ibe
the
patt
ern
of
pow
ers
in t
he
firs
t co
lum
n.C
onti
nu
e th
e pa
tter
n b
y w
riti
ng
the
nex
t tw
o po
wer
s in
th
e ta
ble.
Th
e ex
po
-n
ents
dec
reas
e by
1;
20,2
–1.
b.
Des
crib
e th
e pa
tter
n o
f va
lues
in
th
e se
con
d co
lum
n.T
hen
com
plet
eth
e se
con
d co
lum
n.
Eac
h n
um
ber
is d
ivid
ed b
y 2;
1,�1 2� .
c.V
erif
y th
at t
he
pow
ers
you
wro
te i
n p
art
aar
e eq
ual
to
the
valu
esth
at y
ou f
oun
d in
par
t b
. See
stu
den
ts’w
ork
.
d.
Det
erm
ine
how
3�
1 sh
ould
be
defi
ned
.�1 3�
Pre-
Act
ivit
y
Rea
din
g t
he
Less
on
1.E
xpla
in t
he
valu
e of
5�
3u
sin
g a
patt
ern
.S
amp
le a
nsw
er:T
he
valu
e in
eac
h
row
is t
he
pre
vio
us
valu
e d
ivid
ed b
y 5,
so 5
�3
is � 11 25�
.
2.U
sin
g w
hat
you
kn
ow a
bou
t th
e Q
uot
ien
t of
Pow
ers
rule
,fil
l in
th
e m
issi
ng
nu
mbe
r.
5�3
= � 5? 5�
5–3�
�5 52 5�
Hel
pin
g Y
ou
Rem
emb
er
3.A
re �
x2an
d x�
2 eq
uiv
alen
t? E
xpla
in.
Sam
ple
an
swer
:N
o;
let
x=
3.�
32=
�9,
but
3�2
�� 31 2�
��1 9� .
Po
wer
Val
ue
515
501
5�1
�1 5�
5�2
� 21 5�
5�3
� 11 25�
Pra
ctic
eN
egat
ive
Exp
on
ents
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
__P
ER
IOD
____
_
4-7
4-7
©G
lenc
oe/M
cGra
w-H
ill19
7G
lenc
oe P
re-A
lgeb
ra
Wri
te e
ach
exp
ress
ion
usi
ng
a p
osit
ive
exp
onen
t.
1.7�
8� 71 8�
2.10
�6
� 11 06�3.
23�
1� 21 31�
4.(�
5)�
2� (�
1 5)2
�5.
(�18
)�10
� (�11 8)
10�
6.m
�99
� m1 99�
7.(�
1)�
12� (�
11 )12�
8.c�
6� c1 6�
9.p�
5� p1 5�
10.
g�17
� g1 17�11
.5z
�4
5 �� z1 4��
12.
3t�
13 ��1 t� �
Wri
te e
ach
fra
ctio
n a
s an
exp
ress
ion
usi
ng
a n
egat
ive
exp
onen
t.
13.
� 21 10�2�
1014
.� 21 93�
29�
315
.� 41 4�
4�4
16.
� 31 9�39
�1
17.
� 81 17�81
�7
18.
� m1 4�m
�4
19.
� x1 3�x
�3
20.
� a1 2�a�
221
.� 41 9�
7�2
22.
�1 8�2�
323
.� 11 44�
12�
224
.� 11 69�
13�
2
Eva
luat
e ea
ch e
xpre
ssio
n i
f x
�3,
y�
�2,
and
z �
4.
25.
x�4
� 81 1�26
.y�
2�1 4�
27.
y�5
�� 31 2�
28.
z�4
� 21 56�29
.5
y� 21 5�
30.
10y
� 11 00�
31.
3z�
1�3 4�
32.
zy� 11 6�
33.
(xz)
�2
� 11 44�
34.H
AIR
Hai
r gr
ows
at a
rat
e of
� 61 4�in
ch p
er d
ay.W
rite
th
is n
um
ber
usi
ng
neg
ativ
e
expo
nen
ts.
8�2
or
4�3
or
2�6
Lesson 4-7
© Glencoe/McGraw-Hill A18 Glencoe Pre-Algebra
Answers (Lesson 4-7)
©G
lenc
oe/M
cGra
w-H
ill20
0G
lenc
oe P
re-A
lgeb
ra
Whe
n yo
u de
al w
ith v
ery
larg
e nu
mbe
rs li
ke 5
,000
,000
or
very
sm
all n
umbe
rs li
ke 0
.000
5, it
is d
iffic
ult
toke
ep t
rack
of
plac
e va
lue.
Num
bers
suc
h as
the
se c
an b
e w
ritte
n in
sci
enti
fic
no
tati
on
.A n
umbe
r is
expr
esse
d in
sci
entif
ic n
otat
ion
whe
n it
is w
ritte
n as
a p
rodu
ct o
f a
fact
or a
nd a
pow
er o
f 10
.The
fact
orm
ust
be g
reat
er t
han
or e
qual
to
1 an
d le
ss t
han
10.
By
defin
ition
, a
num
ber
in s
cien
tific
not
atio
n is
writ
ten
asa
�10
n , w
here
1
a
10 a
nd n
is a
n in
tege
r.
Exp
ress
eac
h n
um
ber
in
sci
enti
fic
not
atio
n.
a.62
,000
,00
To
wri
te i
n s
cien
tifi
c n
otat
ion
,pla
ce t
he
deci
mal
poi
nt
afte
r th
e fi
rst
non
zero
dig
it,t
hen
fi
nd
the
pow
er o
f 10
.
62,0
00,0
00 �
6.2
�10
7T
he d
ecim
al p
oint
mov
es 7
pla
ces.
The
pow
er o
f 10
is 7
.
b.0
.000
25
0.00
025
�2.
5 �
10�
4 P
lace
the
dec
imal
poi
nt a
fter
the
first
non
zero
dig
it.T
he p
ower
of
10 is
�4.
1.4.
12 �
106
4,12
0,00
02.
5.8
�10
258
03.
9.01
�10
�3
0.00
901
4.6.
72 �
10�
70.
0000
0067
25.
8.72
�10
487
,200
6.4.
44 �
10�
50.
0000
444
Exp
ress
eac
h n
um
ber
in
sci
enti
fic
not
atio
n.
7.12
,000
,000
,000
1.2
�10
108.
5000
5.0
�10
39.
0.00
475
4.75
�10
�3
10.
0.00
0074
63
7.46
3 �
10�
511
.23
5,00
02.
35 �
105
12.
0.00
0377
3.77
�10
�4
Ch
oose
th
e gr
eate
r n
um
ber
in
eac
h p
air.
13.
4.9
�10
4 ,9.
9 �
10�
414
.2.
004
�10
3 ,2.
005
�10
–2
15.
3.2
�10
2 ,70
016
.0.
002,
3.6
�10
�4
Stu
dy G
uid
e a
nd I
nte
rven
tion
Sci
enti
fic
No
tati
on
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
__P
ER
IOD
____
_
4-8
4-8
Exp
ress
eac
h n
um
ber
in
sta
nd
ard
for
m.
a.6.
32 �
105
6.32
�10
5�
632,
000
Mov
e th
e de
cim
al p
oint
5 p
lace
s to
the
rig
ht.
b.7
.8 �
10�
6
7.8
�10
�6
�0.
0000
078
Mov
e th
e de
cim
al p
oint
6 p
lace
s to
the
left.
Exam
ple
1Ex
ampl
e 1
Exam
ple
2Ex
ampl
e 2
Exer
cises
Exer
cises
En
rich
men
t
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
__P
ER
IOD
____
_
4-7
4-7
©G
lenc
oe/M
cGra
w-H
ill19
9G
lenc
oe P
re-A
lgeb
ra
Pro
vin
g D
efin
itio
ns
of
Exp
on
ents
Rec
all
the
rule
s fo
r m
ult
iply
ing
and
divi
din
g po
wer
s w
ith
th
e sa
me
base
.Use
th
ese
rule
s,al
ong
wit
h o
ther
pro
pert
ies
you
hav
e le
arn
ed,t
o ju
stif
y ea
ch d
efin
itio
n.A
bbre
viat
ion
s fo
rso
me
prop
erti
es y
ou m
ay w
ish
to
use
are
lis
ted
belo
w.
Ass
ocia
tive
Pro
pert
y of
Mu
ltip
lica
tion
(A
PM
)A
ddit
ive
Iden
tity
Pro
pert
y (A
IP)
Mu
ltip
lica
tive
Ide
nti
ty P
rope
rty
(MIP
)In
vers
e P
rope
rty
of A
ddit
ion
(IP
A)
Inve
rse
Pro
pert
y of
Mu
ltip
lica
tion
(IP
M)
Wri
te t
he
reas
on f
or e
ach
sta
tem
ent.
1.P
rove
:a0
�1
Sta
tem
ent
Let
mbe
an
in
tege
r,an
d le
t a
be
any
non
zero
nu
mbe
r.
am�
a0�
am�
0
am�
a0�
am
� a1 m��
(am
�a0
) �
� a1 m��
am
�� a1 m��
am��
a0�
� a1 m��
am
1�
a0�
1
a0�
1
2.P
rove
:a–n
�� a1 n�
Sta
tem
ent
Let
nbe
an
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© Glencoe/McGraw-Hill A19 Glencoe Pre-Algebra
Answers (Lessons 4-7 and 4-8)
An
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© Glencoe/McGraw-Hill A20 Glencoe Pre-Algebra
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Lesson 4-8
© Glencoe/McGraw-Hill A21 Glencoe Pre-Algebra
Answers (Lesson 4-8)
An
swer
s
© Glencoe/McGraw-Hill A22 Glencoe Pre-Algebra
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
B:
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11. A
D
A
C
B
A
D
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D
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A
D
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B
A
D
C
B
D
B
A
C
C
A
D
A
B
C
D
C
A
Chapter 4 Assessment Answer Key Form 1 Form 2APage 205 Page 206 Page 207
(continued on the next page)
Tables that seat 8 people
© Glencoe/McGraw-Hill A23 Glencoe Pre-Algebra
12.
13.
14.
15.
16.
17.
18.
19.
20.
B:
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
B:
D
A
A
C
B
C
D
A
D
A
B
C
B
C
B
B
B
D
A
B
B
D
B
A
B
C
A
C
B
Chapter 4 Assessment Answer Key Form 2A (continued) Form 2BPage 208 Page 209 Page 210
An
swer
sVenus, Earth,Neptune, Uranus Neptune, Uranus,
Saturn, Jupiter
© Glencoe/McGraw-Hill A24 Glencoe Pre-Algebra
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
21.
22.
23.
24.
25.
B: 22 � 33 � 11
15 in.
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56
25
yes
no
2, 3, 6
2, 5, 10
Chapter 4 Assessment Answer Key Form 2CPage 211 Page 212
© Glencoe/McGraw-Hill A25 Glencoe Pre-Algebra
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
21.
22.
23.
24.
25.
B: 24 � 32 � 11
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2
Chapter 4 Assessment Answer Key Form 2DPage 213 Page 214
An
swer
s
© Glencoe/McGraw-Hill A26 Glencoe Pre-Algebra
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
21.
22.
23.
24.
25.
26.
27.
28.
29.
30.
31.
32.
33.
B:
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none
2, 3, 5, 6, 10
2, 3, 6
Chapter 4 Assessment Answer Key Form 3Page 215 Page 216
No; it has two termsinvolving addition.
Yes; it is the product ofnumbers and variables.
No; it has two terms involving subtraction.
Yes; it is the product of numbers and variables.
�1 � 3 � 3 � 5 � q � r �r � s � s � s
3.0 � 10�4, 3.03 � 10�4,3.13 � 10�4, 0.00303,0.0313
2, 3, 5, 7, 11, 13, 17, 19,23, 29, 31, 37, 41, 43,and 47
Chapter 4 Assessment Answer KeyPage 217, Open-Ended Assessment
Scoring Rubric
© Glencoe/McGraw-Hill A27 Glencoe Pre-Algebra
An
swer
s
Chapter 4 Assessment Answer KeyPage 217, Open-Ended Assessment
Scoring Rubric
An
swer
s
Score General Description Specific Criteria
• Shows thorough understanding of the concepts factors,fractions, exponent, coefficient, prime, and composite.
• Uses appropriate strategies to compare fractions.• Computations are correct.• Written explanations are exemplary.• Diagram is accurate.• Goes beyond requirements of some or all problems.
• Shows an understanding of the concepts factors, fractions,exponent, coefficient, prime, and composite.
• Uses appropriate strategies to compare fractions.• Computations to compare fractions and identify prime
numbers are mostly correct.• Written explanations are effective.• Diagram is mostly accurate.• Satisfies all requirements of problems.
• Shows an understanding of most of the concepts factors,fractions, exponent, coefficient, prime, and composite.
• May not use appropriate strategies to compare fractions.• Computations to compare fractions and identify prime
numbers are mostly correct.• Written explanations are satisfactory.• Diagram is mostly accurate.• Satisfies most requirements of the problems.
• Final computation is correct.• No written explanations or work is shown to substantiate
the final computation.• Diagrams may be accurate but lack detail or explanation.• Satisfies minimal requirements of some of the problems.
• Shows little or no understanding of most of the conceptsfactors, fractions, exponent, coefficient, prime, andcomposite.
• May not use appropriate strategies to compare fractions.• Computations to compare fractions and identify prime
numbers are incorrect.• Written explanations are not satisfactory.• Diagram is not accurate.• Does not satisfy requirements of problems.
0 UnsatisfactoryAn incorrect solutionindicating no mathematicalunderstanding of theconcept or task, or nosolution is given
1 Nearly Unsatisfactory A correct solution with nosupporting evidence orexplanation
2 Nearly SatisfactoryA partially correctinterpretation and/orsolution to the problem
3 SatisfactoryA generally correct solution,but may contain minor flawsin reasoning or computation
4 SuperiorA correct solution that is supported by well-developed, accurateexplanations
© Glencoe/McGraw-Hill A28 Glencoe Pre-Algebra
Chapter 4 Assessment Answer Key Page 217, Open-Ended Assessment
Sample Answers
1. Sample answer: 50 � 3 � 47,74 � 13 � 61, 98 � 37 � 61
2a. Yes. If a number is divisible by 9, thismeans you can separate that manyobjects into nine equal groups, withnone remaining. You can then separateeach of these nine groups into threeequal groups. Thus, the original number is divisible by 3. Following isan example of 18 objects.
2b. No. Consider 12. It is divisible by 3,but not by 9.
3a. No, because 210 is not divisible by 8.Students should give pairs of numbers that are factors of 210,such as 2 by 105, 105 by 2, 3 by 70, 70by 3, 5 by 42, 42 by 5, 6 by 35,35 by 6, 7 by 30, 30 by 7, 10 by 21,21 by 10, 14 by 15, or 15 by 14.
3b. Find a common denominator to compare fractions.
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A greater part of the band turned in their slips.
4a. 3a means 3 � a or a � a � a.a3 means a � a � a.
4b. �3b means �3 � b or �(b � b � b).
b�3 means �b13� or �b �
1b � b�.
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In addition to the scoring rubric found on page A27, the following sample answers may be used as guidance in evaluating open-ended assessment items.
© Glencoe/McGraw-Hill A29 Glencoe Pre-Algebra
1. d
2. i
3. c
4. h
5. g
6. b
7. e
8. f
9. a
10. Sample answer: Anumber expressedin scientific notationis written as theproduct of a factorand a power of 10.The factor must begreater than orequal to 1 and lessthan 10.
11. Sample answer: In apower, the base isthe number that ismultiplied.
12. Sample answer: Analgebraic fraction isa fraction withvariables in thenumerator ordenominator.
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10.
Quiz (Lessons 4-3 and 4-4)
Page 219
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Quiz (Lessons 4-7 and 4-8)
Page 220
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7.
8.
9.
10.
3.60 � 10�5
4.62 � 10�3
3.0 � 109
2.0 � 108
1.602 � 10�7
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�715�
x4
1
10x5
x13
310
D
�230�
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5(6 � y)
2(2a � 7)
7x 4
10
4
2 � 3 � 5 � x � x � y
3 � 7 � b
24 � 7
32 � 7
33
72
192
95
x 3
3, 5
3
2, 3, 5, 6, 10
2
Chapter 4 Assessment Answer Key Vocabulary Test/Review Quiz (Lessons 4-1 and 4-2) Quiz (Lessons 4-5 and 4-6)
Page 218 Page 219 Page 220
An
swer
s
Yes; it is the product of two variables.
No; two variables are added.
electron, proton,neutron
© Glencoe/McGraw-Hill A30 Glencoe Pre-Algebra
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17. 6(3 � 7y)
9r2t
15
2 � 3 � 13
960
1296
�1�2�3�4 0 1 2 43
�1�2�3�4 0 1 2 43
��x9� � 18; �162
IV
2
4
32x
�22m
4x
8
2 � 3 � 11
33 � 22
2, 3, 6
C
B
B
D
C
A
Chapter 4 Assessment Answer Key Mid-Chapter Test Cumulative ReviewPage 221 Page 222
No; 12 is not a factor of 70.
2 � 2 � 2 � 3 � x �x � x � y � y
domain � {2.3, 4.6, 5}range � {3.2, 3.3, 4}
Positive; as the outside temperatureincreases, so does theair conditioning bill.
2 � 3 � 3 � 3 � 3 � 5 �a � b � c � c � c
© Glencoe/McGraw-Hill A31 Glencoe Pre-Algebra
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20. 7.998 � 108
0 0 0
.. ./ /
.
99 9 987654321
87654321
87654321
87654321
1 2 1
0 0 0
.. ./ /
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99 9 987654321
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4 8 0 0
HGFE
DCBA
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Chapter 4 Assessment Answer KeyStandardized Test Practice
Page 223 Page 224
An
swer
s
{�18, �11, �5, 0,2, 7, 32}
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