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4A Exponents4-1 Exponents

4-2 Integer Exponents

4-3 Properties of Exponents

4-4 Multiplying and DividingMonomials

4-5 Scientific Notation

LAB Multiply and DivideNumbers in ScientificNotation

4B Roots4-6 Squares and Square

Roots

4-7 Estimating Square Roots

4-8 The Real Numbers

LAB Explore the PythagoreanTheorem

4-9 The Pythagorean Theorem

164 Chapter 4

Exponents and RootsExponents and Roots

KEYWORD: MT8CA Ch4

Stanford LinearAccelerator Center,Menlo Park

Scientific notation is usedin physics research toexpress the numbers andsizes of particles.

Exponents and Roots 165

VocabularyChoose the best term from the list to complete each sentence.

1. According to the __?__, you must multiply or divide before you add or subtract when simplifying a numerical__?__.

2. An algebraic expression is a mathematical phrase that has at least one __?__.

3. In a(n) __?__, an equal sign is used to show that two quantities are the same.

4. You use a(n)__?__ to show that one quantity is greater than another quantity.

Complete these exercises to review skills you will need for this chapter.

Order of Operations Simplify by using the order of operations.

5. 12 � 4(2) 6. 12 � 8 � 4 7. 15(14 � 4)

8. (23 � 5) � 36 � 2 9. 12 � 2 � 10 � 5 10. 40 � 2 � 4

Solve One-Step EquationsSolve.

11. x � 9 � 21 12. 3z � 42 13. �w4� � 16

14. 24 � t � 24 15. p � 7 � 23 16. 12m � 0

Use Repeated MultiplicationFind the product.

17. 7 � 7 � 7 � 7 � 7 18. 12 � 12 � 12 19. 3 � 3 � 3 � 3

20. 11 � 11 � 11 � 11 21. 8�8�8�8�8�8 22. 2 � 2 � 2

23. 100�100�100�100 24. 9 � 9 � 9 � 9 � 9 25. 1 � 1 � 1 � 1

Multiply and Divide by Powers of TenMultiply or divide.

26. 358(10) 27. 358(1000) 28. 358(100,000)

29. �31508

� 30. �1305080� 31. �10

305,0800�

equation

expression

inequality

order of operations

variable

The information below “unpacks” the standards. The Academic Vocabulary ishighlighted and defined to help you understand the language of the standards.Refer to the lessons listed after each standard for help with the math terms andphrases. The Chapter Concept shows how the standard is applied in this chapter.

166 Chapter 4

California Standard

Chapter Concept

Academic Vocabulary

NS1.1 Read, write, and rationalnumbers in scientific (positive andnegative powers of 10), compare rationalnumbers in general.

(Lesson 4-5)

notationcompare compare determine how items are

the same or different

notation a way of showingsomething by a special system ofmarks or characters

You use scientific notation toexpress very large and verysmall numbers.

NS1.4 between rationaland irrational numbers.

(Lesson 4-8)

Differentiate differentiate to tell the differencebetween

You determine the specialgroups that numbers belong to.

NS2.4 Use the relationshipbetween raising to a power and theroot of a perfect square integer; for an integerthat is not square, determine without acalculator the two integers between which itssquare root lies and explain why.

(Lessons 4-6 and 4-7)

extractinginverse inverse opposite

extracting pulling out

You multiply an integer byitself, or square it. You will alsofind what whole number can bemultiplied by itself to get agiven number. When a wholenumber can not be found, youwill find what two wholenumbers the value is between.

AF2.1 Interpret positive whole-numberpowers as repeated multiplication andnegative whole-number powers as repeateddivision or multiplication by the multiplicativeinverse. and expressionsthat include exponents.

(Lessons 4-1, 4-2, 4-3, and 4-4)

evaluateSimplify

simplify make things easier

evaluate find the value of

You write expressions thatcontain exponents in a simplerform. You will also find thevalue of expressions containingvariables and exponents.

MG3.3 Know and understand thePythagorean theorem and its and useit to find the length of the missing side of a righttriangle and the lengths of other line segments and,in some situations, empirically verify thePythagorean theorem by direct measurement.

(Lab 4-9 and Lesson 4-9)

converseconverse the opposite or reverse You learn about the relationship

that exists between the threesides of all right triangles.

Standards NS2.1 and AF2.2 are also covered in this chapter.

Step 2: CuesAfter class, writedown key phrases or questions in theleft column.

Step 3: SummaryUse the cues torestate the mainpoints in your ownwords.

Math Notes–10/18

Interpret the • For whole groups:

quotientDrop the decimal

• For whole groups

needed for a set of

items: Round to

the next whole number

What is the • Exact number: Use

problem asking? the whole quotient

Interpreting quotients is part of problem solving.

Quotients are based on what the problem is asking.

Step 1: NotesDraw a vertical lineabout 2.5 inches fromthe left side of yourpaper. During class,write your notes about the main pointsof the lecture in theright column.

Exponents and Roots 167

1. Describe how you can benefit from using the Cornell system of note taking.

2. In your next class, use the Cornell system of note taking. Comparethese notes to your notes from a previous lecture. Do you think yourold notes or the notes using the Cornell system would better prepareyou for tests and quizzes?

Try This

Study Strategy: Take Effective NotesGood note taking is an important study strategy. The Cornell system ofnote taking is an effective way to organize and review main ideas. Thismethod involves dividing your notebook paper into three main sections.You take notes in the note-taking column during the lecture. You writequestions and key phrases in the cue column as you review your notes.You write a brief summary of the lecture in the summary area.

Vocabularyexponential formexponentbasepower

Fold a piece of 8�12�-by-11-inch paper in half. If you fold it in half again,

the paper is 4 sheets thick. After the third fold in half, the paper is 8 sheets thick. How many sheets thick is the paper after 7 folds?

With each fold the number of sheets doubles.

2 � 2 � 2 � 2 � 2 � 2 � 2 � 128 sheets thick after 7 folds

This multiplication problem can also be written in exponential form.

2 � 2 � 2 � 2 � 2 � 2 � 2 � 27 The number 2 is a factor 7 times.

If a number is in , the represents the number of times

the is to be used as a factor. A number produced by raising a base to an exponent is called a . 128 and 27 are equivalent.

Writing Exponents

Write in exponential form.

5 � 5 � 5 � 5 � 5 � 5 � 5

5 � 5 � 5 � 5 � 5 � 5 � 5 � 57 Identify how many times 5 is a factor.

(�4) � (�4) � (�4)

(�4) � (�4) � (�4) � (�4)3 Identify how many times �4 is a factor.

8 � 8 � 8 � 8 � p � p � p

8 � 8 � 8 � 8 � p � p � p � 84p3 Identify how many times 8 and p areeach used as a factor.

Simplifying Powers

Simplify.

34

34 � 3 � 3 � 3 � 3 Find the product.

� 81

power

baseexponent

exponential form Base Exponent

E X A M P L E 1

E X A M P L E 2

Read (�4)3 as “�4to the 3rd power” or “�4 cubed.”

168 Chapter 4 Exponents and Roots

4-1 Exponents

Why learn this? You can use exponents to determinehow many bacteria exist after one bacterium reproducesmultiple times. (See Exercise 41.)

CaliforniaStandards

AF2.1 Interpret positive whole-number powers as repeatedmultiplication and negative whole-number powers as repeated divisionor multiplication by the multiplicativeinverse. Simplify and evaluateexpressions that includeexponents.Also covered: AF1.2

��14��2

��14��2

� �14� � �

14�

� �116�

Simplify.

(�8)2

(�8)2 � (�8) � (�8) Find the product.

� 64

�23

�23 � �(2 � 2 � 2) Find the product. Then make the

� �8 answer negative.

Using the Order of Operations

Evaluate x � y(z � yz) for x � 20, y � 4, and z � 2.

x � y(z � y z)

20 � 4(2 � 42) Substitute 20 for x, 4 for y, and 2 for z.

20 � 4(2 � 16) Simplify the power.

20 � 4(32) Multiply inside the parentheses.

20 � 128 Multiply from left to right.

�108 Subtract from left to right.

Geometry Application

The number of diagonals of an n-sided figure is �12�(n2 � 3n). Use

the expression to find the number of diagonals in a 6-sided figure.

�12�(n2 � 3n)

�12�(62 � 3 � 6) Substitute the number of sides for n.

�12�(36 � 18) Simplify inside the parentheses.

�12�(18) Subtract inside the parentheses.

9 Multiply.

A 6-sided figure has 9 diagonals. You can verify your answer by sketching the diagonals.

Think and Discuss

1. Explain the difference between (�5)2 and �52.

2. Compare 3 � 2, 32, and 23.

3. Show that (4 � 11)2 is not equal to 42 � 112.

E X A M P L E 3

E X A M P L E 4

The expression (�8)2

is not the same asthe expression �82.Think of �82 as �1 � 82. By the orderof operations, youmust evaluate theexponent beforemultiplying by �1.

4-1 Exponents 169

4-1 ExercisesExercises


1. 12 2. 18 � 18 3. 2 � 2 � b � b � b 4. (�3) � (�3)

Simplify.

5. 26 6. (�7)2 7. ��12��

38. �74 9. 84

Evaluate each expression for the given values of the variables.

10. a5 � 4b for a � 3 and b � 12

11. 2x 9 � (y � z) for x � �1, y � 7, and z � �4

12. s � (t u � 1) for s � 13, t � 5, u � 3

13. 100 � n(p q � 4) for n � 10, p � 3, and q � 8

14. The sum of the first n positive integers is �12��n2 � n�. Check the expression

for the first 5 positive integers. Then use the expression to find the sum ofthe first 14 positive integers.


15. 5 � 5 � 5 � 5 � 5 � 5 16. (�9) � (�9) � (�9) 17. 3 � 3 � d � d

18. �8 19. (�4) � (�4) � c � c � c 20. x � x � y

Simplify.

21. 44 22. (�3)6 23. ��18��

524. �29 25. ��

16��

2

Evaluate each expression for the given values of the variables.

26. b2 for b � �7

27. 2c � 3d(g � 2) for c � 7, d � 5, and g � 1

28. m � np for m � 12, n � 11, and p � 2

29. x � y z for x � 9, y � 3, and z � 2

30. A circle can be divided by n lines into a maximum of �

12��n2 � n� � 1 regions. Use the expression to find

the maximum number of regions for 7 lines.

Simplify.

31. 44 � (5 � 42) 32. (4 � 44) 33. (6 � 71) 34. 84 � [8 � (�2)3]

35. 18 � 73 36. (3 � 42) � 40 37. (9 � 10)5 38. 16 � (2 � 83)

KEYWORD: MT8CA Parent

KEYWORD: MT8CA 4-1

GUIDED PRACTICE

PRACTICE AND PROBLEM SOLVING

INDEPENDENT PRACTICE

Extra PracticeSee page EP8.

See Example 2

See Example 3

See Example 1

See Example 1

See Example 2

See Example 4

See Example 3

See Example 4

12

3

3 lines → 7 regions

4 56 7


CaliforniaStandards Practice

AF1.2, AF2.1

Evaluate each expression for the given values of the variable.

39. m(p � nq) for m � 2, n � 6, p � 3, and q � 3

40. r � (t � sv) for r � 42, s � �4, t � 3, and v � 2

41. Life Science Certain bacteria can divide every 20 minutes, so 1 bacteriumcan divide into 2 bacteria in 20 minutes, 4 in 40 minutes, and so on. If you beginwith 1 bacterium, how many bacteria will there be in 6 hours? Write youranswer by using exponents, and then evaluate.


42. 2b � 2b � 2b 43. 3d � 3d 44. 5a � 5a � 5a 45. 7x � 7x � 7x � 7x

46. Reasoning For any whole number n, 5n�1 is divisible by 4.Verify this statement for n � 4 and n � 6.

47. Estimation A gift shaped like a cube has sides that measure 12.3 cm long.What is the approximate volume of the gift? (Hint: V � s 3)

48. Choose a Strategy Place the numbers 1, 2, 3, 4, and 5 in the boxes to make a true statement: � 3 � 2 � .

49. Write About It Compare 102 and 210. For any two numbers, make a conjecture about which usually gives the greater number: using the greaternumber as the base or as the exponent. Give at least one exception.

50. Challenge Write (42)3 using a single exponent.

Life Science

Most bacteriareproduce by atype of simple celldivision known asbinary fission.Each speciesreproduces bestat a specifictemperature andmoisture level.

4-1 Exponents 171

51. Multiple Choice Which expression has the greatest value?

25 34 43 52

52. Multiple Choice The volume of a cube is calculated by using the formula V � s3, where s is the length of the sides of the cube. What is the volume ofa cube that has sides 8 meters long?

24 m3 512 m3 888 m3 6561 m3

53. Gridded Response What is the value of 54?

54. Multiple Choice Which of the following expressions is equivalent to(�5x)(�5x)(5x)?

(�5x)3 (�5x)(5x)2 (�5x)2(5x) 2(�5x)(5x)

Find each sum. (Lesson 1-4)

55. �18 � (�65) 56. �123 � 95 57. 87 � (�32) 58. �74 � (�27)

Write each fraction as a decimal. (Lesson 2-1)

59. �570� 60. �1

45� 61. �

38� 62. �2

54�

DCBA

DCBA

DCBA

NS1.2, NS1.3, AF1.2, AF2.1

Look at the pattern shown in the table to extend what you knowabout exponents to include negative exponents.

Notice from the chart that as the exponents decrease by 1, the valueof the expression decreases by a factor of 10.

Using a Pattern to Evaluate Negative Exponents

Simplify the powers of 10.

10�4

10�4 ��10 � 101� 10 � 10� Extend the pattern from the table.

� �10,1000� � 0.0001 Multiply. Write as a decimal.

10�5

10�5 � Extend the pattern from Example 1A.

� �1001,000� � 0.00001 Multiply. Write as a decimal.

1��10 � 10 � 10 � 10 � 10

E X A M P L E 1

102 101 100 10�1 10�2 10�3

10 � 10 10 1 �110� �10

1� 10� �10 � 1

10 � 10�

100 10 1 �110� � 0.1 �1

100� � 0.01 �10

100� � 0.001

� 10 � 10 � 10 � 10 � 10

Words Numbers Algebra

Any nonzero number raised to a negative power equals 5�3 � � �

1125� b�n � , if b � 0

1 divided by that number raised to a positive power.

1�bn

1�53

NEGATIVE EXPONENTS



CaliforniaStandards

NS2.1 Understand negativewhole-number exponents. Multiplyand divide expressions involvingexponents with a common base.AF2.1 Interpret positive whole-number powers as repeatedmultiplication and negative whole-number powers as repeateddivision or multiplication by themultiplicative inverse. Simplifyand evaluate expressions thatinclude exponents.

Why learn this? You can use negative exponents torepresent the size of something that is very small, such asthe diameter of a white blood cell. (See Exercise 41.)

Reasoning

Evaluating Negative Exponents

Simplify.

(�2)�3

(�2)�3 �Write the power under 1; change the sign of the exponent.

� Find the product.

� ��18

� Simplify.

6�4

6�4 � Write the power under 1; change the sign of the exponent.

� Find the product.

� �12

196� Simplify.

Notice from the table on the previous page that 100 � 1. This is truefor any nonzero number to the zero power.

Using the Order of Operations

Simplify 2 � (�7)0 � (4 � 2)�2.

2 � (�7)0 � (4 � 2)�2

2 � (�7)0 � 6�2 Add inside the parentheses.

2 � (�7)0 � �612� Rewrite 6�2 as �

612�.

2 � 1 � �316� Simplify the powers.

3 � �316� Add and subtract from left to right.

2�33

56�

1��6 � 6 � 6 � 6

1��(�2) � (�2) � (�2)

Think and Discuss

1. Express �12� using a negative exponent.

2. Tell whether an integer raised to a negative exponent can ever begreater than 1. Justify your answer.

2E X A M P L E

3E X A M P L E


The zero power of any 1000 � 1a0 � 1, if a � 0

nonzero number equals 1. (�5)0 � 1

THE ZERO POWER

4-2 Integer Exponents 173

1�64

1�(�2)3



1. 10�2 2. 10�7 3. 10�6 4. 10�10

Simplify.

5. (2)�6 6. (�3)�4 7. 3�3 8. (�2)�5

9. 4 � 3(4 �90) � 5�3 10. 7 � 8(2)�3 � 13

11. (2 � 2)�2 � (1 � 1)�4 12. 2 � (2�3)


13. 10�1 14. 10�9 15. 10�8 16. 10�12

Simplify.

17. (�4)�1 18. 5�2 19. (�10)�4 20. (�2)�6

21. 128(2 � 6)�3 � (40 � 3) 22. 3 � (�3)�2 � (9 � 7)0

23. 12 � (�5)0 � (3�3 � 9�2) 24. 50 � 49(1 � 6)�2

Simplify.

25. (18 � 16)�5 26. 25 � (6 � 100) 27. (1 � 2�2) 28. (3 � 3)�3

29. 3�2 � 22 � 40 30. 10 � 43 � 2�2

31. 62 � 32 � 1�1 32. 16 � [15 � (�2)�3]

Evaluate each expression for the given value of the variable.

33. 2(x 2 � x) for x � 2.1 34. (4n)�2 � n for n � 3

35. c2 � c for c � �12� 36. m�2 � m0 � m2 for m � 9

Write each expression as repeated multiplication. Then simplify the expression.

37. 11�4 38. 1�10 39. �6�3 40. (�6)�3

41. Life Science The diameter of a white blood cell in human blood is3(500)�2 meter. Write the diameter of the cell without using an exponent.

42. Patterns Describe the following pattern: (�1)1 � ; (�1)�2 � ; (�1)�3 � ; (�1)�4 � . Determine what (�1)�100 would be. Justifyyour thinking.

43. Reasoning Evaluate n1 � n�1 for n � 1, 2, and 3. Then make a conjectureabout what n1 � n�1 is for any value of n. Explain your reasoning.


KEYWORD: MT8CA 4-2

GUIDED PRACTICE




See Example 2

See Example 3

See Example 1

See Example 1

See Example 2

See Example 3



NS2.1, NS2.3, AF1.2, AF2.1

Life Science

44. Some whales have been known to dive todepths greater than 1012 nanometers.Simplify 1012.

45. Blubber makes up 27% of a blue whale’sbody weight. Davis found the averageweight of blue whales and used it tocalculate the average weight of theirblubber. He wrote the amount as 22 � 33 � 5 � 71 pounds. Simplify thisamount.

46. Most baleen whales migrate an averageof 25 � 125 km each way. The gray whalehas the longest known migration of any mammal, a distance of 24 � 3 � 125 km farther each way than the averagebaleen whale migration. How far does the gray whalemigrate each way?

47. A blue whale may eat between 6 and 7 tons of krill eachday. Krill are approximately 2�5 � 3�1 � 5�1 of the length of a blue whale. Simplify this amount.

48. Challenge A cubic centimeter is the same as 1 mL. If a humpback whale has more than 1 kL of blood, how many cubic centimeters of blood does the humpback whale have?

Krill are a food source for differentspecies of baleen whales, such as thehumpback whale, pictured above.

4-2 Integer Exponents 175

49. Multiple Choice Simplify (�5)�2.

�25 ��215� �2

15� 25

50. Extended Response Simplify 83, 82, 81, 80, 8�1, and 8�2. Describe a possiblepattern that is shown in the values. Use the pattern of the values to predict thevalue of 8�3.

Simplify. (Lesson 3-2)

51. 7(x � 3) � x 52. 4(2y � 5) � 6y 53. 8(x � 3y) � x 54. 9(2x � y) � x


55. (�3)4 56. 52 57. (10 � 15)3 58. (�9)3 59. �25

DCBA

NS2.1, AF1.3, AF2.1

The following suggests a rule for multiplying powers with the same base.

24 � 22 � (2 � 2 � 2 � 2) � (2 � 2) � 26

a3 � a2 � (a � a � a) � (a � a) � a5

Notice that the sum of the exponents in each expression equals theexponent in the answer: 4 � 2 � 6 and 3 � 2 � 5.

Multiplying Powers with the Same Base

Simplify each expression. Write your answer in exponential form.

54 � 53 (�a)12 � (�a)�5

54 � 3 Add exponents. (�a)12 � (�5)

57 (�a)7

The following suggests a rule for dividing powers with the same base.

�33

6

2� � � 3 � 3 � 3 � 3 � 34

�xx

5

3� � �x � x

x��

xx

��

xx

� x� � x � x � x2

Notice that the difference between the exponents in each expressionequals the exponent in the answer: 6 � 2 � 4 and 5 � 3 � 2.

3 � 3 � 3 � 3 � 3 � 3��

3 � 3


To multiply powers with the same base, keep the base 35 � 38 � 35 � 8 � 313 b m � bn � b m � n

and add the exponents.

MULTIPLYING POWERS WITH THE SAME BASE


To divide powers with the same base, keep the base �

66

9

4� � 69 � 4 � 65 � b m � n,

and subtract the exponents. if b � 0

bm�bn

DIVIDING POWERS WITH THE SAME BASE

1E X A M P L E



CaliforniaStandards

NS2.1 Understand negativewhole-number exponents.Multiply and divide expressionsinvolving exponents with acommon base.

NS2.3 Multiply, divide, andsimplify rational numbers byusing exponent rules.

Why learn this? You can use properties of exponents tocompare distances between planets. (See Exercise 52.)

1 1

1 11 1 1

1 1 1

Dividing Powers with the Same Base


�11

00

8

5� �33

3

2��

22

5

5��

55

2

4�

108 � 5 33 � 2 � 25 � 5 � 52 � 4

103 31 � 20 � 5�2

3 � 1 � 5�2

3 � 5�2

�532�

To see what happens when you raise a power to a power, use the orderof operations.

(c3)2 � (c � c � c)2 Show the power inside the parentheses.

� (c � c � c) � (c � c � c) Show the power outside the parentheses.

� c6 Simplify.

Notice that the product of the exponents in the power equals theexponent in the answer: 3 � 2 � 6.

Raising a Power to a Power


(75)3 (n9)11

75 � 3 n9 � 11

715 n99

(2�7)�2 (�1210)�6

2�7 � (�2) Multiply exponents. (�12)10 � (�6)

214 (�12)�60

Think and Discuss

1. Explain why the exponents cannot be added in the product 143 � 183.

2. List two ways to express 45 as a product of powers.

2E X A M P L E

(94)5 is read as “nine to the fourthpower, to the fifthpower.”

RAISING A POWER TO A POWER

Words

To raise a power to apower, keep the base andmultiply the exponents.

3E X A M P L E

Multiply exponents.

Subtract exponents.

Numbers

(94)5 � 94 � 5 � 920

Algebra

(bm)n � bm � n

4-3 Properties of Exponents 177



1. 56 � 59 2. 123 � 12�2 3. m � m3 4. (�4)3 � (�4)3

5. �66

5

3� 6. �aa�

8

1� 7. �11

22

5

5� 8. �55

1

4

6

��665

2

��223

3�

9. (34)5 10. (22)0 11. (4�2)3 12. (�y 2)6


13. 1010 � 107 14. (�3)4 � (�3)4 15. r 3 � r �2 16. 18 � 185

17. ��13��2

� ��13��4

18. (�9)�6 � (�9)5 19. y � y�3 � y�1 20. ��15��3

� ��15��3

21. �55

1

6

0� 22. �

33

7

7� 23. �tt�

9

4��rr

4

3� 24.

25. 26. �xx0

5� 27. �((�

�

66

))

�

�

2

2� 28.

29. (50)8 30. (64)�1

31. (3�2)2 32. (x 5)2

33. (�13)3 34. (7�6)�1 35. (t 2)�4 36. �(25)3

27 � 95�26 � 92

15 � 4�2�

13 � 4

23 � 36 � 63��24 � 3 � 63

Simplify. Write your answer in exponential form.

37. �44

7

3� 38. 38 � 3�1 39. �aa�

4

3� 40. �1100

1

9

8�

41. x 3 � x7 42. a 6 � b 9 43. (74)3 44. 2 � 24

45. �1502

4� 46. �

11

11

7

6� 47. �y

y

8

8� 48. y 8 � y�8

49. There are 263 ways to make a 3-letter “word” (from aaa to zzz) and 265 ways to make a 5-letter word. How many times more ways are there to make a 5-letter word than a 3-letter word?

50. Astronomy The mass of the sun is about 1027 metric tons, or 1030 kilograms.How many kilograms are in one metric ton? (Hint: Set up a proportion.)

51. Business Using the manufacturing terms atright, tell how many dozen are in a great gross.How many gross are in a great gross?


KEYWORD: MT8CA 4-3

GUIDED PRACTICE




See Example 2

See Example 1

See Example 3

See Example 2

See Example 1

See Example 3


1 dozen � 121 items

1 gross � 122 items

1 great gross � 123 items


NS2.1, NS2.3, Extension of AF2.2

52. Estimation The distance from Earth to the moon is about 224 miles. The distance from Earth to Neptune is about 227 miles. Which distance isgreater? About how many times as great?

Find the missing exponent.

53. b � b 4 � b 8 54. (v2) � v�6 55. �ww

3� � w�3 56. (a 4) � a

57. A googol is the number 1 followed by 100 zeros.

a. What is a googol written as a power of 10?

b. What is a googol times a googol written as a power of 10?

58. What’s the Error? A student said that �39

5

5� is the same as �13�. What mistake

has the student made?

59. Write About It Why do you subtract exponents when dividing powers with the same base?

60. Challenge A number to the 11th power divided by the same number to the 8th power equals 64. What is the number?

Pean

uts

: © U

nit

ed F

eatu

res

Syn

dic

ate,

Inc.

4-3 Properties of Exponents 179

NS1.2, NS2.1, NS2.3, AF2.1

61. Multiple Choice In computer technology, a kilobyte is 210 bytes in size. Agigabyte is 230 bytes in size. The size of a terabyte is the product of the size of akilobyte and the size of a gigabyte. What is the size of a terabyte?

220 bytes 240 bytes 2300 bytes 4300 bytes

62. Short Response A student claims that 103 � 10�5 is greater than 1. Explainwhether the student is correct.

Evaluate each expression for the given value of the variable. (Lesson 2-3)

63. 19.4 � x for x � �5.6 64. 11 � r for r � 13.5 65. p � 65.1 for p � �42.3

66. ��37� � t for t � 1�

57� 67. 3�1

51� � y for y � �2�1

41� 68. ��1

19� � g for g � �

11

89�


69. (�3)�2 70. (�2)�3 71. 1�3 72. �(2)�4

DCBA

A is a number, variable, or a product of numbers andvariables with exponents that are whole numbers.

To multiply two monomials, multiply the coefficients and add theexponents that have the same base.

Multiplying Monomials

Multiply.

(2x3)(5x2)

(2 � 5)(x 3 � x 2) Use the Comm. and Assoc. Properties.

2 � 5 � x3 � 2 Multiply coefficients. Add exponents

10x5 that have the same base.

(8m5n3)(4m2n�8)

(8 � 4)(m5 � m2)(n3 � n�8) Use the Comm. and Assoc. Properties.

8 � 4 � m5 � 2 � n3 � (�8) Multiply coefficients. Add exponentsthat have the same base.32m7n�11

�3

n2m

11

7�

(6a3b)(�4b5)

(6 � (�4))(a3)(b � b5) Use the Comm. and Assoc. Properties.

(6 � (�4))(a3)(b1 � b5) Think: b � b1.

6 � (�4) � a3 � b1� 5 Multiply coefficients. Add exponentsthat have the same base.�24a3b6

To divide a monomial by a monomial, divide the coefficients andsubtract the exponents of the powers in the denominator from theexponents of the powers in the numerator that have the same base.

monomial

E X A M P L E 1


4-4 Multiplying and DividingMonomials

Vocabularymonomial

Why learn this? You can multiply monomials to findthe volume of an aquarium. (See Exercise 47.)

CaliforniaStandards

AF2.2 Multiply and dividemonomials; extend the process oftaking powers and extracting rootsto monomials when the latter resultsin a monomial with an integerexponent.Also covered: AF1.3 Monomials 7x5, �3a2b3, n2, 8, �4

z�

Not monomials m�3, 4z2.5, 5 � y, �w8

3�, 2x

When performingoperations withmonomials, theanswer may not be amonomial. Noticethat the answer toExample 1B has avariable in thedenominator.

Dividing Monomials

Divide. Assume that no denominator equals zero.

�122xx3

7�

�122�x7 � 3

6x4

�8

6

x

x

5

7

y

y

4

4�

�43�x5 � 7y4 � 4 Divide coefficients. Subtract

exponents that have the same base.

�43�x�2y0 � �3

4x� Think: x�2 � �

x

12� and y0 � 1.

To raise a monomial to a power, you must first understand how tofind a power of a product. Notice what happens to the exponentswhen you find a power of a product.

(xy)3 � xy � xy � xy � x � x � x � y � y � y � x3y3

4-4 Multiplying and Dividing Monomials 181

RAISING A PRODUCT TO A POWERWords

To raise a product toa power, raise eachfactor to that power.

Numbers

(2 � 3)2 � 22 � 32

� 4 � 9� 36

Algebra

(ab)n � an � bn

To raise a power to apower, keep the baseand multiply theexponents of thebases.

Think and Discuss

1. Explain why the expression �5b�2 is not a monomial.

2. Tell what happens to the variable m when you simplify �264mm

3

3

nn

2

4�.

E X A M P L E 2

Divide coefficients. Subtract exponents that have the same base.

Raising a Monomial to a Power

Simplify.

(2x)5 (�4mn5)5

25 � x5 (�4)5 � (m)5 � (n5)5

32x5

E X A M P L E 3

Raise each factor to the power.

Raise each factorto the power.

Multiplyexponents.

(�4)5 � m1� 5 � n5 � 5

�1024m5n25

Multiply.

19. (�11z3)(2z9) 20. (5n7)(12n3) 21. (�x5)(2x5)

22. (8p3r4)��12�p4r5� 23. (�3a2b4)(�2b) 24. (x4y11z)(7x2y4z)


25. �375xx7

9� 26. �

250

nn

4

12� 27. �

4

8

p

p

q

q8

5�

28. �18

2a

a4b

5b11

4� 29. �

1

1

5

0

x

x

2

y

y2

7

� 30. ��2

55

aa

3

10b3�

Simplify.

31. (2y)8 32. (7x3)3 33. (4b2)4

34. (5x4y5)3 35. (2a3b10)5 36. (�2p3r)4

Simplify each expression.

37. (12m2)2 38. (9x3)��23�x7� 39. �

336

0a

a

5b2

5�

40. (8p4q5)��34�pq2� 41. �

�

18

2

g

g

3

3

h

h

5

5� 42. (�2xy2)6

43. Geometry The volume of a sphere of radius r can be approximated bythe monomial 4r 3. The surface area of the sphere can be approximated bythe monomial 12r 2. Write and simplify an expression for the volume ofthe sphere divided by the surface area of the sphere.




Multiply.

1. (7y2)(8y5) 2. (8m2)(3m3) 3. (3a2b4)(�2a3b)

4. (�3r 2s7)(2r 9s) 5. (2x)(2y) 6. (c2d3e)(2c5d12e8)


7. �155nn

2

5� 8. �

�

24xx6

7� 9. �

1182aa

2

bb

5

8�

10. �24

1s8

1

s

0

5t12

� 11. �472aa4

4

cc2

7� 12. �

186mm

3

5

pp

�

Simplify.

13. (5a)3 14. (9m3)2 15. (6x2y5)3

16. (�7p5q3)2 17. (2m3n2)5 18. (�3x4yz)4


KEYWORD: MT8CA 4-4

GUIDED PRACTICE

See Example 2

See Example 3

See Example 1

See Example 2

See Example 1



AF1.1, AF1.4, AF2.2

See Example 3


The area of a rectangle is the product of the rectangle’s length and width.Write and simplify an expression for the area of each rectangle.

44. 45. 46.

47. Hobbies To find the volume of a rectangularprism, such as an aquarium, multiply thelength by the width by the height. Find thevolume of the aquarium shown.

48. Geometry To find the volume of a cube,raise the length of a side to the third power.

a. The side of a cube is 4x2y. Write andsimplify an expression for the volume of the cube.

b. Find the cube’s volume when x � 1 and y � 2.

49. Reasoning The table shows several monomials andthe degree of each monomial.

a. Make a conjecture about the definition of thedegree of a monomial.

b. What do you think the degree of x5y3 is?

50. What’s the Error? A student was asked to simplify(2x3y3)3. The student’s answer was 8x6y6. What error did the student make,and what is the correct answer?

51. Write About It Explain why (mn)3 is equivalent to (nm)3.

52. Challenge Simplify the expression (2a2b3)3 � (3ac2)2.

2ac

3abc

2mn

8m3n4

3xy

6x2y

3xy

2x

xy

4-4 Multiplying and Dividing Monomials 183

NS2.3, AF1.3, AF2.2

53. Multiple Choice Which expression is equivalent to �186xx4

8�?

8x2 8x4 2x2 2x4

54. Multiple Choice Which product does NOT equal 8m2n2?

(2m2)(4n2) (4mn)(2mn) (8mn2)(m2n) (mn2)(8m)


55. 3(4y � 6) � 8y 56. 4(x � 8) � 9x 57. 7(3 � 5k) � 14k

Simplify. Write your answer in exponential form. (Lesson 4-3)

58. 54 � 5�2 59. g 3 � g 8 60. �mm

�

6

2� 61. �77

6

5�

DCBA

DCBA

Monomial Degree

y4 4

x3y2 5

xyz 3

x2y2z3 7

The table shows relationships between several powers of 10.

• Each time you divide by 10, the exponent in the power decreases by 1 and the decimal point in the value moves one place to the left.

• Each time you multiply by 10, the exponent in the powerincreases by 1 and the decimal point in the value moves one place to the right.

You can find the product of a number and a power of 10 by movingthe decimal point of the number. You may need to write zeros to theright or left of the number in order to move the decimal point.

Multiplying by Powers of Ten

Multiply.

13.7 � 103

13.7 0 0 Since the exponent is a positive 3, move the

13.700 decimal point 3 places to the right.

2.3 � 10�4

0 0 0 0 2.3 Since the exponent is a negative 4, move the

0.00023 decimal point 4 places to the left.

Powers of 10 are used when writing numbers in scientific notation.is a way to express numbers that are very large or

very small. Numbers written in scientific notation are expressed as 2 factors. One factor is a number greater than or equal to 1. The otherfactor is a power of 10.

Scientific notation

E X A M P L E 1



Vocabularyscientific notation

Why learn this? You can use scientific notation torepresent the speed of light. (See Exercise 53.)

CaliforniaStandards

NS1.1 Read, write, and comparerational numbers in scientificnotation (positive and negativepowers of 10), compare rationalnumbers in general.

Power 103 102 101 100 10�1 10�2 10�3

Value 1000 100 10 1 0.1 0.01 0.001

�10 �10 �10 �10 �10 �10

�10 �10 �10 �10 �10 �10

The average diameter of an atom is about 0.00000003 cm. A quartercontains about 97,700,000,000,000,000,000,000 atoms. These numberscan be written in scientific notation.

Move the decimal point 8 places to the right.

0.00000003 � 3.0 � 10�8

For numbers less than 1, use a negative exponent.

Move the decimal point 22 places to the left.

97,700,000,000,000,000,000,000 � 9.77 � 1022

For numbers greater than or equal to 10, use a positive exponent.

Writing Numbers in Scientific Notation

Write each number in scientific notation.

0.0000005

0.0000005 Think: The decimal point needs to move 7 placesto get a number between 1 and 10.

5 � 10�7 Think: The number is less than 1, so the exponentwill be negative.

So 0.0000005 written in scientific notation is 5 � 10�7.

2,780,000,000

2,780,000,000 Think: The decimal point needs to move 9 placesto get a number between 1 and 10.

2.78 � 109 Think: The number is greater than 1, so theexponent will be positive.

So 2,780,000,000 written in scientific notation is 2.78 � 109.

Reading Numbers in Scientific Notation

Write each number in standard form.

3.12 � 109

3.12 � 109

3.120000000 Think: Move the decimal point right 9 places.

3,120,000,000

1.35 � 10�4

1.35 � 10�4

00001.35 Think: Move the decimal point left 4 places.

0.000135

2E X A M P L E

4-5 Scientific Notation 185

3E X A M P L E

To compare numbers written in scientific notation, first compare theexponents. If the exponents are equal, then compare the decimals.

Comparing Numbers in Scientific Notation

Mercury is 9.17 � 107 kilometers from Earth. Jupiter is 6.287 � 108 kilometers from Earth. Which planet is closer to Earth?

Mercury: 9.17 � 107Compare the exponents.

Jupiter: 6.287 � 108

Notice that 7 � 8. So 9.17 � 107 � 6.287 � 108.

Mercury is closer to Earth than Jupiter.

Think and Discuss

1. Explain the benefit of writing numbers in scientific form.

2. Describe how to write 2.977 � 106 in standard form.

3. Determine which measurement would be least likely to be written in scientific notation: size of bacteria, speed of a car, ornumber of stars in a galaxy.


Multiply.

1. 15 � 103 2. 12 � 104 3. 208 � 103 4. 113 � 107


5. 0.000057 6. 4,000 7. 6,980,000 8. 0.000000025


9. 4.17 � 103 10. 1.33 � 10�5 11. 6.2 � 107 12. 3.9 � 10�7

13. The maximum length of a particle that can fit through a surgical mask is 1 � 10�4 millimeters. The average length of a dust mite is approximately1.25 � 10�1 millimeters. Can a dust mite fit through a surgical mask?


KEYWORD: MT8CA 4-5

GUIDED PRACTICE

See Example 2

See Example 3

See Example 1

E X A M P L E 4

See Example 4



NS1.1

Multiply.

14. 21 � 102 15. 8 � 104 16. 25 � 105 17. 40 � 104

18. 268 � 103 19. 550 � 107 20. 2,115 � 105 21. 70,030 � 101


22. 0.00007 23. 6,500,000 24. 100,000,000 25. 0.00000587


26. 9.2 � 106 27. 6.7 � 10�4 28. 3.6 � 10�2 29. 5.24 � 108

30. Astronomy The orbits of Neptune and Pluto cross each other. Neptune’saverage distance from the Sun is approximately 4.5 � 109 kilometers. Pluto’saverage distance from the Sun is approximately 5.87 � 109 kilometers. Whichhas the greater average distance from the Sun, Neptune or Pluto?


31. 1.4 � 105 32. 3.24 � 10�2 33. 7.8 � 101 34. 2.1 � 10�6

35. 5.3 � 10�8 36. 8.456 � 10�4 37. 5.59 � 105 38. 7.1 � 103

39. 7.113 � 106 40. 4.5 � 10�1 41. 2.9 � 10�4 42. 5.6 � 102

43. Life Science Duckweed plants live on the surface of calm ponds and arethe smallest flowering plants in the world. They weigh about 0.00015 g.

a. Write this number in scientific notation.

b. If left unchecked, one duckweed plant, which reproduces every 30–36 hours, could produce 1 � 1030 (a nonillion) plants in four months. How much would one nonillion duckweed plants weigh?

44. Life Science The diameter of a human red blood cell ranges fromapproximately 6 � 10�6 to 8 � 10�6 meters. Write this range in standardform.

45. History Ancient Egyptians hammered gold into sheets so thin that ittook 3.67 � 105 sheets to make a pile 2.5 centimeters high. Write thenumber of sheets in standard form.

46. Astronomy Mars is 7.83 � 107 kilometers from Earth. Venus is 4.14 � 107

kilometers from Earth. Which planet is closer to Earth?

Find the missing number or numbers.

47. 24,500 � 2.45 � 10 48. 16,800 � � 104 49. � 3.40 � 102

50. 280,000 � 2.8 � 10 51. 5.4 � 108 � 52. 60,000,000 � � 10

53. Science In a vacuum, light travels at a speed of about nine hundred andeighty million feet per second. Write this speed in scientific notation.




See Example 2

See Example 3

See Example 1

See Example 4

Life Science

This frog iscovered withduckweed plants.Duckweed plantscan grow both insunlight and inshade andproduce tinywhite flowers.

4-5 Scientific Notation 187

54. Social Studies The diagram shows information about the country of Taiwan.

a. Express the population and area of Taiwan in scientific notation.

b. Divide the number of square miles by the population to find the number of square miles per person in Taiwan. Express your answer in scientific notation.


55. 0.00858 56. 0.0000063 57. 5,900,000

58. 7,045,000,000 59. 0.0076 60. 400

61. 4200 62. 0.0000000082 63. 0.0000000006

64. 0.000005 65. 7,000,000 66. 0.0095678

67. Order the list of numbers below from least to greatest.1.5 � 10�2, 1.2 � 106, 5.85 � 10�3, 2.3 � 10�2, 5.5 � 106

68. Write a Problem An electron has a mass of about 9.11 � 10�31 kg. Use this information to write a problem.

69. Write About It Two numbers are written in scientific notation. How can you tell which number is greater?

70. Challenge Where on a number line does the value of a number in scientific notation with a negative exponent lie?

Population:

Area:

Capital:

Number oftelevisions:

Languages:

22,113,250

14,032 mi2

Taipei

10,800,000

Taiwanese (Min),Mandarin,Hakka dialects

Taiwan


NS1.1, NS2.3, AF1.1, AF2.2

71. Multiple Choice The distance light can travel in one year is 9.46 � 1012 kilometers. What is this distance in standard form?

94,600,000,000,000,000 km 9,460,000,000,000 km

946,000,000,000 km 0.000000000946 km

72. Short Response Explain how you can determine the sign of theexponent when 29,600,000,000,000 is written in scientific notation.

Write an inequality for each situation. (Lesson 3-5)

73. At least 35 students registered for the photography course.

74. The amount remaining in the savings account was no more than $550.

Simplify each expression. Write your answer in exponential form. (Lesson 4-3)

75. �77

4

2� 76. 53 � 58 77. �tt

8

5� 78. 109 � 10�3

DB

CA

Multiply and Divide Numbersin Scientific Notation

You can use a graphing calculator to perform operations with numbers

written in scientific notation. Use the key combination

to enter numbers in scientific notation. On a graphing calculator,

9.5 � 1016 is displayed as 9.5E16.

Use a calculator to find (4.8 � 1012)(9.4 � 109).

Press 4.8 12 9.4 9 .

The calculator displays the answer 4.512E22, which is the same as 4.512 � 1022.

1. When you use the Associative and Commutative Properties tomultiply 4.8 � 1012 and 9.4 � 109, you get (4.8 � 9.4)(1012 � 109) �45.12 � 1021. Explain why this answer is different from the answeryou obtained in the activity.

Use a calculator to simplify each expression.

1. (5.76 � 1013)(6.23 � 10�20) 2. �92..79

�

�

1100

1

7

0� 3. (1.6 � 105)(9.65 � 109)

4. �56..2154

�

�

1100

1

8

3� 5. (1.1 � 109)(2.2 � 103) 6. �8

2..53

64

�

�

11

00

9

8

7

0�

7. (2.74 � 1011)(3.2 � 10�5) 8. �58.8.9

26�

�

1100

�

1

1

1

1� 9. (4.5 � 1012)(3.7 � 108)

10. The star Betelgeuse, in the constellation of Orion, is approximately3.36 � 1015 miles from Earth. This is approximately 1.24 � 106 timesas far as Pluto’s minimum distance from Earth. What is Pluto’sapproximate minimum distance from Earth? Write your answer inscientific notation.

11. If 446 billion telephone calls were placed by 135 million UnitedStates telephone subscribers, what was the average number of callsplaced per subscriber?

ENTEREE

,2nd�EE

,2nd

EE

,2nd

Activity

Think and Discuss

Try This

Use with Lesson 4-5

4-5

KEYWORD: MT8CA Lab4

4-5 Technology Lab 189

CaliforniaStandards

Extension of NS1.1 Read, write, and compare rationalnumbers in scientific notation (positive and negativepowers of 10), compare rational numbers in general.

Quiz for Lessons 4-1 Through 4-5

4-1 Exponents

Simplify.

1. 101 2. 86 3. �34 4. (�5)3

5. Write 5 � 5 � 5 � 5 in exponential form. 6. Evaluate a7 � 4b for a � 3 and b � �1.


Simplify.

7. 10�6 8. (�3)�4 9. �6�2 10. 40

11. 8 � 100(�6) 12. 5�1 � 3(5)�2 13. �4�3 � 20 14. 3�2 � (60 � 6�2)



15. 93 � 95 16. �55

1

1

0

0� 17. q9 � q6 18. 33� 3�2

19. (33)�2 20. (42)0 21. (�x2)4 22. (4�2)5

23. The mass of the known universe is about 1023 solar masses, which is1050 metric tons. How many metric tons are in one solar mass?

4-4 Multiplying and Dividing Monomials


24. (15p2)2 25. (15m)��13�m4� 26. �

�4

6

8

x

x5

�

27. ��27�a8b3�(�56a2b) 28. �

5

6

5

5

p

p

4

4

q

q

4

4� 29. (�3m3n)3



30. 0.00000015 31. 99,980,000 32. 0.434 33. 100


34. 1.38 � 105 35. 4 � 106 36. 1.2 � 10�3 37. 9.37 � 10�5

38. Jupiter is 6.287 � 108 kilometers from Earth. Mars is 7.83 � 107 kilometers from Earth. Which planet is farther from Earth?


Mary is making a string of beads. If eachbead is 7.0 � 10�1 cm wide, how manybeads does she need to make a stringthat is 35 cm long?

The total area of the United States is 9.63 � 106 square kilometers. The totalarea of Canada is 9.98 � 106 squarekilometers. What is the total area of boththe United States and Canada?

Suppose �13� of the fish in a lake are

considered game fish. Of these, �25� meet

the legal minimum size requirement.What fraction of the fish in the lake aregame fish that meet the legal minimumsize requirement?

Part of a checkbook register is shownbelow. Find the amount in the accountafter the transactions shown.

Determine the action for each problem. Write the problem using theactions. Then show what operation you used to get the answer.

Solve• Choose an operation

To decide whether to add, subtract, multiply, or divide to solve a problem, you need to identify any relationships and determinethe action taking place in the problem.

Focus on Problem Solving 191

2

4

1 3

Withdrawal

Withdrawal

11/16 autodebit for phone bill $43.16 $43.16$287.34

Check 1256 11/18 groceries

new clothes

ATM withdrawal

$27.56 $27.56

Check 1257 11/23

11/27

$74.23 $74.23

$41.25$40.00 $1.25

TRANSACTION

RECORD ALL CHARGES OR CREDITS THAT AFFECT YOUR ACCOUNT

DATE DESCRIPTION AMOUNT FEE DEPOSITS BALANCE

CaliforniaStandards

MR1.1 Analyze problems byidentifying relationships,distinguishing relevant from irrelavantinformation, identifying missinginformation, sequencing and prioritizinginformation, and observing patterns.Also covered: NS1.1

Action Operation

Combining numbers or putting numbers together Addition

Taking away or finding out how far apart two numbers are Subtraction

Combining equal groups Multiplication

Splitting things into equal groups or finding how many equal groups you can make

Division

Because the area of a square can be expressed using anexponent of 2, a number with an exponent of 2 is said to besquared. You read 32 as “three squared.”

The of a number is one of the two equal factorsof that number. Squaring a nonnegative number and findingthe square root of that number are inverse operations.

Positive real numbers have two square roots, one positive and onenegative. The positive square root, or , isrepresented by ��. The negative square root is represented by ��.

A is a number whose square roots are integers. Someexamples of perfect squares are shown in the table.

perfect square

principle square root

square root

Finding the Positive and Negative Square Roots of a Number

Find the two square roots of each number.

81

�81� � 9 9 is a square root, since 9 � 9 � 81.

��81� � �9 �9 is also a square root, since �9 � (�9) � 81.

The square roots of 81 are 9 and �9, or �9.

1

�1� � 1 1 is a square root, since 1 � 1 = 1.

��1� � �1 �1 is also a square root, since �1 � (�1) � 1.

The square roots of 1 are 1 and �1, or �1.

E X A M P L E 1

4-6 Squares and Square Roots


Vocabularysquare rootprincipal square rootperfect square

Why learn this? When you know the area of a square,you can use square roots to find the length of the square’ssides. (See Example 2.)

CaliforniaStandards

NS2.4 Use the inverserelationship between raising to apower and extracting the root ofa perfect square integer; for aninteger that is not square, determinewithout a calculator the two integersbetween which its square root lies andexplain why.Also covered: AF2.2

0 1 4 9 16 25 36 49 64 81 100

02 12 22 32 42 52 62 72 82 92 102

Positive squareroot of 16

Negative squareroot of 16

You can write thesquare roots of 81 as � 9, which is read as“plus or minus nine.”

3

3Area = 32

Computer Application

The square computer icon contains 625pixels. How many pixels tall is the icon?

Find the square root of 625 to find thelength of the side. Use the principal squareroot; a negative length has no meaning.

252 � 625

So �625� � 25.

The icon is 25 pixels tall.

The symbol �� is also used with algebraic expressions. Rememberthat �� represents the nonnegative square root. When you simplify,use absolute value to be sure your answer is not negative. Forexample, to simplify �x2�, write �x2� �⏐x⏐, since you do not knowwhether x is positive or negative.

Finding the Square Root of a Monomial


�9x2��9x2� � �(3x)2� Write the monomial as a square.

�3⏐x⏐ Use the absolute-value symbol.

�m10��m10� � �(m5)2� Write the monomial as a square: m10 = (m5)2.

�⏐m5⏐ Use the absolute-value symbol.

�36y4��36y4� � �(6y2)2� Write the monomial as a square.

�6y2 6y2 is nonnegative for all values of y. Theabsolute-value symbol is not needed.

The square computer icon contains 625 colored dots that make up the picture. These dots are called pixels.

Think and Discuss

1. Describe what is meant by a perfect square. Give an example.

2. Explain how many square roots a positive number can have. Howare these square roots different?

3. Decide how many square roots 0 has. Explain whether you canfind the square roots of negative numbers.

E X A M P L E 3

2E X A M P L E

4-6 Squares and Square Roots 193

A variable raised toan even power isnonnegative for allvalues of thevariable.



1. 4 2. 64 3. 121 4. 484

5. A square court for playing the game four square has an area of 256 ft2.How long is one side of the court?


6. �4x2� 7. �y6� 8. �25n4� 9. �49a8�


10. 25 11. 169 12. 400 13. 361

14. Elisa found a square digital image of a famous painting on a Web site. Theimage contained 256 pixels. How many pixels high is the image?


15. �s6� 16. �100m�2� 17. �36x8� 18. �81y12�


19. 36 20. 196 21. 225 22. 625

23. 441 24. 289 25. 576 26. 324

27. Estimation Mr. Barada bought a square rug. The area of the rug wasabout 68.06 ft2. He estimated that the length of a side was about 7 ft. Is Mr. Barada’s estimate reasonable? Explain.

28. Art An artist is making two square stained-glass windows. The firstwindow has a perimeter of 48 inches. The second window has an area of121 inches. Which window is bigger? Explain.

29. Sports A karate match is held on a square mat that has an area of 676 ft2. What is the perimeter of the mat?


30. �9a10� 31. �64x16� 32. �121n2�0� 33. �16y14�

34. �m2n6� 35. �a4b6� 36. �225x1�8y 22� 37. �169p3�0q24�


38. �1121� 39. �

81

16� 40. �

38214

� 41. �16

67

96�


KEYWORD: MT8CA 4-6

GUIDED PRACTICE




See Example 2

See Example 3

See Example 1

See Example 2

See Example 3

See Example 1



NS2.4, AF2.2

CaliforniaGames

The Santa MonicaInternational ChessPark is a hotspot foramateur andprofessional chessplayers. There are 14large and 10 smallchess tables in thepark, as well as alife-size chessboard.

KEYWORD: MT8CA Chess

42. Architecture Mr. and Mrs. Garner want to build a house with a squarefoundation. They want the house to cover 225 square yards. How longshould each side of the house be?

43.. Games A chessboard contains 32 black and 32 white squares. How manysquares are along each side of the game board?

44. Hobbies A quilter wants to use as many of his 65 small fabric squares aspossible to make one large square quilt.

a. How many small squares can the quilter use? How many small squareswould he have left?

b. How many more small squares would the quilter need to make thenext largest possible square quilt?

45. Reasoning Simplify the expressions �x2�, �x4�, �x6�, �x8�, �x10�, and�x12�. Look for a pattern. Make a conjecture about when you do not needto use an absolute value in your answer.

46. What’s the Error? A student said that since the square roots of a certainnumber are 1.5 and �1.5, the number must be their product, �2.25. Whaterror did the student make?

47. Write About It Explain the difference between finding the square of anumber and finding the square root of a number. Use models andnumbers in your explanation.

48. Challenge The square root of a number is four less than three timesseven. What is the number?

4-6 Squares and Square Roots 195

NS1.1, NS1.5, NS2.4

49. Multiple Choice Which number does NOT have a square root that is an integer?

81 196 288 400

50. Short Response Deanna knows that the floor in her kitchen is a squarewith an area of 169 square feet. The perimeter of her kitchen floor isfound by adding the lengths of all its sides. What is the perimeter of herkitchen floor? Explain your answer.

Write each decimal as a fraction in simplest form. (Lesson 2-1)

51. 0.35 52. 2.6 53. �7.18 54. 0.125

Write each number in scientific notation. (Lesson 4-5)

55. 1,970,000,000 56. 2,500,000 57. 31,400,000,000

58. 0.0000543 59. 0.006 60. 0.000000088

DCBA

Recall that a perfect square is anumber whose square roots areintegers. For example, 25 and100 are perfect squares.

You can use the square roots of perfect squares to estimate the square roots of other numbers.

Estimating Square Roots of Numbers

The �30� is between two integers. Name the integers. Explainyour answer.

�30�16, 25, 36, 49 List perfect squares near 30.

25 � 30 � 36 Find the perfect squares nearest 30.

�25� � �30� � �36� Find the square roots of the perfect squares.

5 � �30� � 6

�30� is between 5 and 6 because 30 is between 25 and 36.

Recreation Application

While searching for a lost hiker, a helicopter covers a square areaof 150 mi2. What is the approximate length of each side of thesquare area? Round your answer to the nearest mile.

121, 144, 169, 196 List perfect squares near 150.


�144�� 150�� 169�� Find the square roots of the perfect squares.

12 � �150�� 13

�150�� 12 150 is closer to 144 than 169, so �150� iscloser to 12 than 13.

Each side of the area is about 12 miles long.

E X A M P L E 1



Why learn this? By estimatingsquare roots, you can determineapproximately how much trim isneeded for a square window.(See Exercise 39.)

CaliforniaStandards

NS2.4 Use the inverse relationshipbetween raising to a power andextracting the root of a perfect squareinteger; for an integer that is notsquare, determine without acalculator the two integersbetween which its square rootlies and explain why.

E X A M P L E 2

Think and Discuss

1. Discuss whether 9.5 is a good first guess for �75�.

2. Determine which square root or roots would have 7.5 as a goodfirst guess.

E X A M P L E 3

4-7 Estimating Square Roots 197

You can use the square roots of perfect squares to approximate thesquare root of a value that is not a perfect square.

Approximating Square Roots to the Nearest Hundredth

Approximate �200� to the nearest hundredth.

Step 1: Find the value of the whole number.


�196�� 200�� 225�� Find the square roots of the perfect squares.

14 � �200�� 15 The number will be between 14 and 15.

The whole number part of the answer is 14.

Step 2: Find the value of the decimal.

200 � 196 � 4

225 � 196 � 29

�249�

4 � 29 � 0.137

Step 3: Find the approximate value.

14 � 0.137 � 14.137 Combine the whole number and decimal.

14.137 � 14.14 Round to the nearest hundredth.

The approximate value of �200� to the nearest hundredth is 14.14.

You can also use a calculator to approximate the square root of avalue that is not a perfect square.

Using a Calculator to Estimate the Value of a Square Root

Use a calculator to find �700�. Round to the nearest tenth.�700� � 26.45751311 Use a calculator.

�700� � 26.5 Round to the nearest tenth.

�700� rounded to the nearest tenth is 26.5.

Find the difference between the givennumber, 200, and the lower perfect square.Find the difference between the greaterperfect square and the lower perfect square.Write the difference as a ratio.

Divide to find the approximate decimal value.

E X A M P L E 4


Each square root is between two integers. Name the integers. Explain youranswer.

1. �40� 2. �90� 3. �156� 4. �306� 5. �250�

6. A Coast Guard ship patrols an area of 125 square miles. The area the shippatrols is a square. About how long is each side of the square?

Approximate each square root to the nearest hundredth.

7. �42� 8. �73� 9. �156� 10. �236� 11. �275�

Use a calculator to find each value. Round to the nearest tenth.

12. �74� 13. �34.1� 14. �3600� 15. �190� 16. �5120�

Each square root is between two integers. Name the integers. Explain your answer.

17. �52� 18. �3� 19. �600� 20. �2000� 21. �410�

22. The area of a square field is 200 ft2. What is the approximate length ofeach side of the field? Round your anwer to the nearest foot.

Approximate each square root to the nearest hundredth.

23. �19� 24. �84� 25. �123� 26. �251� 27. �290�

Use a calculator to find each value. Round to the nearest tenth.

28. �58� 29. �91.5� 30. �550� 31. �150� 32. �330�

Write the letter that identifies the position of each square root.

33. ��3� 34. �5� 35. �7�

36. ��8� 37. �14� 38. �0.75�

39. A couple wants to install a square stained-glass window that has an areaof 500 square inches. To the nearest tenth of an inch, what length of woodtrim is needed to go around the window?

40. Each square on Laura’s chessboard is 13 square centimeters. A chessboard has 8 squares on each side. To the nearest hundredth, what is the width ofLaura’s chessboard?


KEYWORD: MT8CA 4-7

GUIDED PRACTICE




See Example 1

See Example 2

See Example 3

See Example 1

�2 �1�3 0

A B C D E F

1 2 3 4

See Example 2

See Example 3



NS2.4

See Example 4

See Example 4

4-7 Estimating Square Roots 199

NS2.4, AF1.2

50. Multiple Choice Which expression has a value between 14 and 15?

�188� �200� �227� �324�

51. Multiple Choice The square of a whole number is between 1000 and 1100. Thenumber must be between which of the following numbers?

25 and 30 30 and 35 35 and 40 40 and 45

Evaluate each expression for the given values of the variables. (Lesson 1-1)

52. 4x � 5y for x � 3 and y � 9 53. 7m � 2n for m � 5 and n � 7

54. 8h � 9j for h � 11 and j � 2 55. 6s � 2t for s � 7 and t � 12

Find the two square roots of each number. (Lesson 4-5)

56. 100 57. 64 58. 484 59. 1296

DCBA

DCBA

41. Multi-Step On a baseball field, the infield area created by the baselinesis a square. In a youth baseball league for 9- to 12-year-olds, this area is3600 ft2. The distance between each base in a league for 4-year-olds is 20 ftless than it is for 9- to 12-year-olds. What is the distance between each basefor 4-year-olds?

Order the numbers from least to greatest.

42. �49�, �137�, 6.5, 8, �

245� 43. 5�

23�, �25�, 32, 7.15, �

249�

44. Multi-Step Find the perimeter of a square whose area is 49 square inches.

45. Science The formula D � 3.56 � �A� gives the distance D in kilometers tothe horizon from an airplane flying at an altitude A meters. If a pilot isflying at an altitude of 1800 m, about how far away is the horizon? Roundyour answer to the nearest kilometer.

46. Multi-Step For his new room, Darien’s grandmother gave him ahandmade quilt. The quilt is made of 16 squares set in 4 rows of 4. Thearea of each square is 324 in2. How long is each side of the quilt in inches?

47. Reasoning Show how two squares can be formed by drawing only 6 lines.

48. Write About It Explain how you know whether �29� is closer to 5 or 6without using a calculator.

49. Challenge The speed of a tsunami in miles per hour can be found usingr � �14.88d�, where d is the water depth in feet. Suppose the water depthis 25,000 ft.

a. How fast is the tsunami moving in miles per hour?

b. How long would it take a tsunami to travel 3000 miles if the waterdepth were a consistent 10,000 ft?


Vocabularyreal numberirrational numberDensity Property

Who uses this? Just as biologistsclassify living things as plants or animals,mathematicians classify all real numbersas rational or irrational.

CaliforniaStandards

NS1.4 Differentiatebetween rational and irrationalnumbers.


Biologists classify animals based on sharedcharacteristics. For example, the hornedlizard is an animal, a reptile, a lizard, and a gecko.

You already know that some numbers can be classified as wholenumbers, integers, or rational numbers. The number 2 is a wholenumber, an integer, and a rational number. It is also a real number.

The set of is all numbersthat can be written on a number line. Itconsists of the set of rational numbersand the set of irrational numbers.

Recall that rational numbers can bewritten as the quotient of two integers(a fraction) or as either terminating orrepeating decimals.

3�45� � 3.8 �

23� � 0.6� �1.44� � 1.2

can be written only as decimals that do notterminate or repeat. They cannot be written as the quotient of twointegers. If a whole number is not a perfect square, then its squareroot is an irrational number. For example, 2 is not a perfect square, so �2� is irrational.

Classifying Real Numbers

Write all classifications that apply to each number.

�3� 3 is a whole number that is not a perfect square.

irrational, real

�52.28 �52.28 is a terminating decimal.

rational, real

��25� ��25� � �5

integer, rational, real

Irrational numbers

real numbers

E X A M P L E 1

Real Numbers

Rationalnumbers

Irrationalnumbers

Integers

Wholenumbers

Animals

Reptile

Lizard

Gecko

A repeating decimalmay not appear to repeat on acalculator becausecalculators show afinite number ofdigits.

A fraction with a denominator of 0 is undefined because you cannotdivide by zero. So it is not a number at all.

Determining the Classification of All Numbers

State if each number is rational, irrational, or not a real number.

�15� 15 is a whole number that is not a perfect square.

irrational

�30�

undefined, so not a real number

��19��

13��

13��

19�

rational

The of real numbers states that between any tworeal numbers is another real number. This property is not true whenyou limit yourself to whole numbers or integers. For instance, there isno integer between �2 and �3.

Applying the Density Property of Real Numbers

Find a real number between 1�13� and 1�

23�.

There are many solutions. One solution is halfway between the twonumbers. To find it, add the numbers and divide by 2.

�1�13� � 1�

23�� 2

� �2�33�� 2

� 3 � 2 � 1�12�

A real number between 1�13� and 1�

23� is 1�

12�.

Check: Use a graph.

1 212

113

1 23

1

Density Property

Think and Discuss

1. Explain how rational numbers are related to integers.

2. Tell if a number can be irrational and whole. Explain.

3. Use the Density Property to explain why there are infinitely manyreal numbers between 0 and 1.

2E X A M P L E

E X A M P L E 3

4-8 The Real Numbers 201

4-8 ExercisesExercisesKEYWORD: MT8CA Parent

KEYWORD: MT8CA 4-8


1. �10� 2. �49� 3. 0.25 4. ��16�


5. �9� 6. �107� 7. �72� 8. ��3�

9. ��25� 10. ��23�� 11. ��

23

56�� 12. �

10�

Find a real number between each pair of numbers.

13. 3�18� and 3�

28� 14. 4.14 and �

279� 15. �

18� and �

14�


16. �35� 17. �58� 18. 3 19. �

��

831�

�


20. ��

146

� 21. �04� 22. �81� 23. ��3�

24. ��

825�� 25. �14� 26. ��

14�� 27. �

40�

Find a real number between each pair of numbers.

28. 3�25� and 3�

35� 29. ��1

10� and 0 30. 4 and �9�


31. 6 32. ��36� 33. �10� 34. �13�

35. �2.56� 36. �36� � 6 37. 0.2�1� 38.

39. �4.3134 40. �4.5� 41. �312 42. �07�

43. Explain the difference between �30� and �

03�.

Give an example of each type of number.

44. an irrational number that is less than �3

45. a positive rational number that is less than 0.3

46. a real number between �59� and �

69�

47. a real number between �3�27� and �3�

37�

�100��20

GUIDED PRACTICE




See Example 2

See Example 3

See Example 1

See Example 2

See Example 3

See Example 1



NS1.4

48. Find a rational number between ��19�� and �1�.

49. Find a real number between �4� and �5�.



52. Find a real number between ��25� and ��16�.

53. a. Find a real number between 1 and �3�.

b. Find a real number between 1 and your answer to part a.

c. Find a real number between 1 and your answer to part b.

Determine if each expression is rational or irrational when x � 2.

54. �2x� 55. 3 � �x� 56. �x � 2�

57. �3x � 6� 58. �5x � 2� 59. �1 � �5x

��60. What’s the Error? A student said that all integers are whole numbers.

What mistake did the student make? Explain.

61. Write About It Can you ever use a calculator to determine if a numberis rational or irrational? Explain.

62. Challenge The circumference of a circle divided byits diameter is an irrational number, represented bythe Greek letter π (pi). Could a circle with a diameterof 2 have a circumference of 6? Why or why not?

?

2

4-8 The Real Numbers 203

NS1.1, NS1.4, AF1.2, AF1.3, AF2.1

63. Multiple Choice Which value is between 8 and 10?

7.12 �61� 3 � π �11213

�

64. Multiple Choice Which value is NOT a rational number?

0.7� �11

13� �19� �225�

65. Multiple Choice For which values of x is �x � 19� a real number?

x � �19 x � �19 x � 19 x � 19

Tell which property is represented. (Lesson 3-1)

66. x � 5 � 5 � x 67. 2(y � 3) � 2y � 2 � 3 68. 3��13� � x� � �3 � �

13��x


69. 85 70. (�3)3 71. (�5)4 72. 92

DCBA

DCBA

DCBA


4-9

Explore the PythagoreanTheorem

Use with Lesson 4-9

The Pythagorean Theorem states thatin a right triangle, if the lengths of thelegs are a and b and the length of thehypotenuse is c, then a2 � b2 � c2. Youcan use centimeter graph paper and aruler to verify the PythagoreanTheorem.

On centimeter graph paper, draw a right triangle whose legs are3 cm long and 4 cm long.

Use a centimeter ruler to measure the length of the hypotenuse. Copy the table below,and record the length of the hypotenuse as c in the first row of the table.

Complete the first row of the table by calculating a2, b2, a2 � b2, and c2.

Repeat the process by drawing right triangles whose legs have lengths a and b as given in the table.

1. What do you notice about the values of a2 � b2 and c2 in your table?

2. When you have a right triangle and you know the length of the hypotenuse and one leg,how can you find the length of the other leg?

In the right triangle shown, both legs are 2 cm long.

1. Use a centimeter ruler to measure the length of the hypotenuse to thenearest tenth of a centimeter.

2. Use the Pythagorean Theorem to calculate the length of the hypotenuseto the nearest tenth of a centimeter.

3. How do the results of 1 and 2 compare?

CaliforniaStandards

MG3.3 Know andunderstand the Pythagorean theoremand its converse and use it to find thelength of the missing side of a righttriangle and the lengths of other linesegments and, in some situations,empirically verify the Pythagoreantheorem by direct measurement.

KEYWORD: MT8CA Lab4

Activity

1

2

3

4

Think and Discuss

Try This

b

cHypotenuse

a

Legs

2 cm

2 cm

a b c a2 b2 a2 � b2 c2

3 4

5 12

6 8

4-9 The Pythagorean Theorem 205

4-9

CaliforniaStandards

MG3.3 Know andunderstand the Pythagoreantheorem and its converse and useit to find the length of themissing side of a right triangleand the lengths of other linesegments and, in some situations,empirically verify the Pythagoreantheorem by direct measurement.

VocabularyPythagorean Theoremleghypotenuse

Since length can onlybe positive, use onlythe principal squareroot.

Why learn this? You can use thePythagorean Theorem to determine howhigh a ladder leaning against a wall cansafely reach. (See Exercise 23.)

The Pythagorean Theorem shows that a special relationship existsbetween the sides of a right triangle. You can use the theorem to findthe length of any side of a right triangle.

Calculating the Length of a Side of a Right Triangle

Use the Pythagorean Theorem to find each missing measure.

a 2 � b 2 � c 2 Use the Pythagorean Theorem.

52 � 122 � c 2 Substitute for a and b.

25 � 144 � c 2 Simplify the powers.

169 � c 2 Add.

�169� � �c 2� Take the square root of both sides.13 � c

The length of the hypotenuse is 13 cm.

a 2 � b 2 � c 2 Use the Pythagorean Theorem.

a 2 � 122 � 152 Substitute for b and c.

a2 � 144 � 225 Simplify the powers.

� 144 � 144 Subtract 144 from both sides.

a2 � 81

�a2� � �81� Take the square root ofboth sides.a � 9

The length of the leg is 9 m.

E X A M P L E 1

5 cm

12 cm

c

15 m12 m

a

LegsThe two sides that form the right angle in a right triangle

HypotenuseThe side opposite the right angle

The Pythagorean Theorem

PYTHAGOREAN THEOREMWords

In a right triangle, thesum of the squares ofthe lengths of the legs is equal to the squareof the length of thehypotenuse.

Numbers Algebra

a 2 � b 2 � c 2

ca

b

106

8

62 � 82 � 102

36 � 64 � 100


You can use the Converse of the Pythagorean Theorem to tell whether atriangle is a right triangle. The Converse of the Pythagorean Theoremstates that if a triangle has sides of length a, b, and c, and a2 � b2 � c2,then the triangle is a right triangle.

PROBLEM SOLVING APPLICATIONA regulation baseball diamond is a square with sides that measure 90 feet. About how far is it from home plate to second base? Round your answer to the nearest tenth.

1. Understand the Problem

Rewrite the question as a statement.

• Find the distance from home plate to second base.

List the important information:

• Drawing a segment between home plate and second base dividesthe diamond into two right triangles.

• The angle at first base is the right angle, so the segment betweenhome plate and second base is the hypotenuse.

• The base lines are legs, and they are each 90 feet long.

2. Make a Plan

You can use the Pythagorean Theorem to write an equation.

3. Solve

a 2 � b2 � c 2 Use the Pythagorean Theorem.

902 � 902 � c 2 Substitute for the known variables.

8,100 � 8,100 � c 2 Simplify the powers.

16,200 � c 2 Add.

�16,200� � �c 2� Take the square root of both sides.

127.279 � c

127.3 � c Round.

The distance from home plate to second base is about 127.3 ft.

4. Look Back

The hypotenuse is the longest side of a right triangle. Since thedistance from home plate to second base is greater than the distancebetween the bases, the answer is reasonable.

1

2

3

4

2E X A M P L E


Think and Discuss

1. Explain whether it is ever possible to use the PythagoreanTheorem to find an unknown side length of a scalene triangle.

2. Explain if 2, 3, and 4 cm could be side lengths of a right triangle.

Identifying a Right Triangle

Tell whether the given side lengths form a right triangle.

7, 24, 25

a2 � b2 �?

c2 Compare a2 � b2 to c2.

72 � 242 �?

252 Substitute the longest side length for c.

49 � 576 �?

625 Simplify the powers.

625 � 625 ✔ Add.

The side lengths form a right triangle.

5, 8, 12

a2 � b2 �?

c2 Compare a2 � b2 to c2.

52 � 82 �?

122 Substitute the longest side length for c.

25 � 64 �?

144 Simplify the powers.

89 � 144 ✘ Add.

The side lengths do not form a right triangle.

E X A M P L E 3

4-9 ExercisesExercisesKEYWORD: MT8CA Parent

KEYWORD: MT8CA 4-9


MG3.3

The hypotenuse isalways the longestside of a righttriangle. Be sure to substitute it onlyfor c.


1. 2. 3.

4. A 10 ft ladder is leaning against a wall. If the ladder is 5 ftfrom the base of the wall, how far above the ground doesthe ladder touch the wall? Round your answer to thenearest tenth.

GUIDED PRACTICE

See Example 2

See Example 1

12 m

16 m

c 17 ft

15 ft

b

5 ft

10 ft

26 cm

10 cma

Find the missing side length for each right triangle to the nearest tenth.

15. a � 4, b � 3, c � 16. a � , b � 40, c � 50

17. a � 5, b � 12, c � 18. a � 10, b � , c � 26

19. a � , b � 40, c � 41 20. a � 65, b � , c � 97

21. Reasoning The numbers 3, 4, and 5 form a Pythagorean triple because32 � 42 � 52. When you double each of these values, does the resulting setof numbers also form a Pythagorean triple? Explain.

22. History To determine the boundaries of their fields, ancient Egyptians used a loop ofrope that was knotted at 12 equal intervals andstretched around 3 stakes. Explain why thetriangle formed by the knotted rope must be aright triangle.

23. For safety reasons, the base of a 24-foot ladder must be placed at least8 feet from the wall. To the nearest tenth of a foot, how high can a 24-footladder safely reach?

24. Write a Problem Use a street map to write and solve a problem thatrequires the use of the Pythagorean Theorem.

PRACTICE AND PROBLEM SOLVINGExtra Practice

See page EP9.



5. 3, 4, 5 6. 8, 10, 14 7. 0.5, 1.2, 1.3

See Example 3


8. 9. 10.

11. James rides his bike 15 miles west. Then he turnsnorth and rides another 15 miles before he stopsto rest. How far is James from his starting pointwhen he stops to rest? Round your answer to thenearest tenth.


See Example 2

See Example 1

18 yd

24 yd

c

25 ft

20 ft

b

30 in.

34 in.a

15 mi

15 mi


12. 8, 15, 17 13. 5, 6, 9 14. 2.4, 2.5, 3.6

See Example 3


NS2.4, AF4.1, MG3.3

31. Multiple Choice A flagpole is 24 feet tall. A rope is tied to the top of the flagpole and secured to the ground 9 feet from the base of the flagpole. What is the length of the rope to the nearest foot?

5 feet 15 feet 26 feet 300 feet

32. Gridded Response Brad leans his 13-foot ladder against his house. The baseof the ladder is placed 5 feet from the base of the house. How many feet up thehouse does the ladder reach?

Solve each inequality. (Lesson 3-8)

33. 3 � 7x � 24 34. 1 � x � 11 35. ��

32x� � �

16� � �

13� 36. �5

x� � 0.5 � �0.5

Each square root is between two integers. Name the integers. (Lesson 4-7)

37. �30� 38. �42� 39. �55� 40. �67�

DCBA

25. Ancient Egyptians built pyramids to serve as tombs for their kings. Onepyramid, called Menkaure, has a square base with an area of about 12,100 m2.

a. What is the length of each side of the base?

b. What is the length of a diagonal of the base? Round your answer to the nearest tenth.

26. The photograph shows the Pyramid of Khafre in Egypt. Each side of its square base is about 214 meters long. Each triangular side is an isosceles triangle with a height of about 179 meters. What is the area of one side of the pyramid?

27. Use the Pythagorean Theorem to find the distance from one corner of the Pyramid of Khafre to its peak. Round your answer to the nearest tenth.

28. Multi-Step The pyramids were constructed using a unit of measurement called a cubit. There are about 21 inches in 1 cubit. If the height of a pyramid is 471 feet, what is its height in cubits?

29. Write About It Given a right triangle, explain how you know which values to substitute into the equation a 2 � b2 � c 2.

30. Challenge The pyramid at right has a square base. Find the height of the pyramid to the nearest tenth.

History

KEYWORD: MT8CA Egypt

Quiz for Lessons 4-6 Through 4-9

4-6 Squares and Square Roots


1. 16 2. 121 3. 225 4. 529

5. The Merryweathers want a new square rug for their living room. Ifthe living room is 20 ft � 16 ft, will a square rug with an area of289 square feet fit? Explain your answer.


6. �64n2� 7. �81x10� 8. �9y16� 9. �b4�



10. �72� 11. �200� 12. �340� 13. �610�

14. The area of a chess board is 110 square inches. Find the length of oneside of the board to the nearest hundredth.



15. �12� 16. 0.15 17. �169� 18. �

19. Give an example of an irrational number that is less than �5.

20. Find a real number between 5 and �36�.

4-9 The Pythagorean Theorem

Find the missing side length for each right triangle. Round your answer to the nearest tenth.

21. a � 3, b � 4, c � 22. a � , b � 24, c � 25

23. a � 20, b � , c � 52 24. a � , b � 32, c � 40

25. A construction company is pouring a concrete foundation. The measures oftwo sides that meet in a corner are 20 ft and 48 ft. For the corner to be a rightangle, what would the length of the diagonal have to be?


26. 20, 21, 29 27. 3.5, 12, 12.5 28. 3, 7, 9

�144��4


Divide and ConquerA biologist is growing coloniesof two bacteria. As shown inthe table, the cells ofbacterium A divide in twoevery hour. The cells ofbacterium B divide in twoevery two hours.

1. After 8 hours, how many more cells are there of bacterium A than ofbacterium B?

2. How many hours does it take until there are more than1000 cells of bacterium A?

3. After 24 hours, how many times as many cells are there of bacterium A as bacterium B?

4. At the end of 24 hours, there are about 1.68 � 107 cells of bacterium A. The biologist divides this colony into 3roughly equal portions. About how many cells are in eachportion?

5. As a rule of thumb, if an experiment yields n colonies ofbacteria, future experiments are likely to yield between n � �n� and n � �n� colonies. Suppose an experimentproduces 170 colonies of bacterium A. Explain how youcan estimate the range of the number of colonies thatfuture experiments will produce.

Number of Cells

Elapsed Bacterium BacteriumTime A B

Start 1 1

1 hour 21 —

2 hours 22 21

3 hours 23 —

4 hours 24 22

Concept Connection 211

Each bingo card has numbers on it. Thecaller has a collection of equations. Thecaller reads an equation, and then theplayers solve the equation for the variable. If players have the solution on their cards,they place a chip on it. The winner is thefirst player with a row of chips either down,across, or diagonally.A complete copy of the rules and game boards areavailable online.

Magic SquaresA magic square is a square with numbers arranged so that the sums of the numbers in each row, column, and diagonal are the same.

Complete each magic square below.

Use the numbers �4, �3, �2, �1, 0, 1, 2, 3, and 4 to make a magicsquare with row, column, and diagonal sums of 0.

2

1

Equation BingoEquation Bingo

8 + 1 +

6 = 15

3 + 5 +

7 = 15

4 + 9 +

2 = 15

6 + 7 + 2 = 15

1 + 5 + 9 = 15

8 + 3 + 4 = 1523 31 22

80 51 32

61 71 21

6 + 5

+ 4 = 15

8 + 5 + 2 = 15

�36� 22

80 �9�

32 � 2

�(�4� � 4) �(90)

�(�16�) 03

�(�9�) 20 � 1

According to an ancient Chinese legend, a tortoisefrom the Lo river had the pattern of this magic squareon its shell.

KEYWORD: MT8CA Games


A

B

Materials• strip of white paper

(18 in. by 7 in.)• piece of decorative

paper (6 in. by 6 in.)• tape• scraps of

decorative paper• markers• glue

It’s a WrapPROJECT

Design your own energy-bar wrapper to hold your notes on exponents and roots.

DirectionsMake accordion folds on the strip of white paper so that there are six panels, each about 3 in. wide.Figure A

Fold up the accordion strip.

Wrap the decorative paper around the accordionstrip. The accordion strip will stick out on eitherside. Tape the ends of the decorative paper togetherto make a wrapper. Figure B

Write the number and title of the chapter on scrapsof decorative paper, and glue these to the wrapper.

Taking Note of the MathUse the panels of theaccordion strip to takenotes on the key conceptsin this chapter. Includeexamples that will help youremember facts aboutexponents, roots, and thePythagorean Theorem. Foldup the strip and slide itback into the wrapper.

4

3

2

1

213


Complete the sentences below with vocabulary words from the list above.

1. A(n) ___?___ is a number that cannot be written as a fraction.

2. ___?___ is a short-hand way of writing extremely large or extremely small numbers.

3. The ___?___ states that the sum of the squares of the ___?___ of a right triangle is equal to the square of the ___?___.

4. The set of ___?___ is the set of all numbers that can be written on a number line.

Vocabulary

Exponents (pp. 166–169)4-1

E X A M P L E EXERCISES

■ Write 4 � 4 � 4 in exponential form.

43

■ Simplify (�2)3.

(�2) � (�2) � (�2)�8


5. (�3) � (�3) 6. k � k � k � k

7. �9 8. 6 � x � x

Simplify.

9. 54 10. (�2)5 11. (�1)9

12. 28 13. (�3)1 14. 43

. . . . . . . . . . . . . . . . . . 166

. . . . . . 199

. . . . . . . . . . . . . . 166

. . . . . . 166

. . . . . . . . . . . . 203

. . . . . . 198

. . . . . . . . . . . . . . . . . . . . 203

. . . . . . . . . . . . . 178

. . . . . . . . . 190

. . . . . . . . . . . . . . . . . 166

. . . 190

. . 203

. . . . . . . . . . . 198

. . . . . . 182

. . . . . . . . . . . . 190square root

scientific notation

real number

Pythagorean Theorem

principal square root

power

perfect square

monomial

leg

irrational number

hypotenuse

exponential form

exponent

Density Property

base

Identify how many times 4 isused as a factor.

Find the product.

AF1.2, AF2.1

Integer Exponents (pp. 170–173)4-2


Simplify.

■ (�3)�2■ 20

�(�

13)2�

1

�19�

Simplify.

15. 5�3 16. (�4)�3 17. 11�1

18. 10�4 19. 1000 20. �6�2

21. (9 � 7)�3 22. (7 � 10)0

23. 4�1 � (5 � 7)�2 24. 10 � 9(3�2 � 60)

Write thereciprocal;change thesign of theexponent.

NS2.1, AF2.1

Properties of Exponents (pp. 174–177)4-3


Simplify each expression. Write youranswer in exponential form.

■ 25 � 23■ �

11

00

9

2�

25�3 109�2

28 107

Simplify each expression. Write youranswer in exponential form.

25. 42 � 45 26. 92 � 94 27. p � p 3

28. 15 � 152 29. 62 � 62 30. x4 � x6

31. �88

5

2� 32. �99

3� 33. �

mm7

2�

34. �3

3�

5

2� 35. �4

4

�

�

5

5� 36. �y

y�

6

3�

37. (43)4 38. (x�2)5 39. (�100)8

Study Guide: Review 215

Multiplying and Dividing Monomials (pp. 178–181)4-4


■ Multiply.

(�9x2y)(4x3y7) Commutative Property(�9 � 4)(x2 � x3)(y1 � y7) Multiply coefficients.�9 � 4 � x2 � 3y1 � 7 Add exponents that�36x5y8 have the same base.

■ Divide. Assume that no denominatorequals zero.

��1

24mm

2

7n�

�7m7 � 2n�7m5n

Multiply.

40. (�m9n)(5mn9) 41. ��13�x2z6�(21xz5)

42. (�a3b2)(�a4) 43. (2rs5)(10r5s)

Divide. Assume that no denominatorequals zero.

44. �8

9

1

p

p8

9

� 45. ��

244ss

6

6tt

5

3�

46. ��2

1

2

1

x

x

5

2

y2

� 47. �1142mm

1

2

1

nn

4

7�

Simplify.

48. (4t3)2 49. (�7p5q6)2

50. (�6x4y)3 51. (10m2n5)4

AF1.3, AF2.2

NS2.1, NS2.3

Scientific Notation (pp. 182–186)4-5



■ 0.000007 ■ 62,500

0.0 0 0 0 0 7 6.2500

7 � 10�6 6.25�104


■ 3.58 � 104■ 3.58 � 10�4

3.5800 00003.58

35,800 0.000358


52. 0.000000008 53. 73,000,000

54. 0.0000096 55. 56,400,000,000


56. 1.62 � 103 57. 1.62 � 10�3

58. 9.1 � 105 59. 9.1 � 10�5

60. Order 2.3 � 10�5, �3.1 � 105, 4.9 � 1045,and �1.7 � 10�4 from least to greatest.

NS1.1

Add exponents.

Subtract exponents.

Divide coefficients. Subtractexponents that have the samebase.

Estimating Square Roots (pp. 194–197)4-7


■ �53� is between two integers. Name theintegers.

�49� � �53� � �64�7 � �53� � 8

�53� is between 7 and 8.

Each square root is between two integers.Name the integers. Explain your answer.

67. �82� 68. �13� 69. �125�

70. �37� 71. �259� 72. �174�

The Pythagorean Theorem (pp. 203–207)4-9


The Real Numbers (pp. 198–201)4-8


State if the number is rational, irrational,or not a real number.

■ ��2� The decimal equivalentirrational does not repeat or end.

■ �20� The number is undefined, not real so it is not real.

State if the number is rational, irrational,or not a real number.

73. �81� 74. �122� 75. ��16�

76. ��5� 77. ��04� 78. �

70�

79. Find a real number between �9� and�16�.

Squares and Square Roots (pp. 190–193)4-6



NS2.4, AF2.2

NS2.4

NS1.4

MG3.3

■ Find the two square roots of 400.

�400� � 20 ��400� � �20The square roots of 400 are �20.

■ Simplify the expression �16r2�.

�16r2� � �(4r)2� Write the monomial as a square.

� 4⏐r⏐ Write the nonnegative square root.


61. 16 62. 900 63. 676


64. �49m4� 65. �4a6� 66. �x12�

■ Find the length of side b in the righttriangle where a � 8 and c � 17.

a 2 � b 2 � c 2

82 � b 2 � 172

64 � b 2 � 289b 2 � 225

�b 2� � �225�b � 15

Solve for the unknown side in each righttriangle.

80. If a � 6 and b � 8, find c.

81. If b � 24 and c � 26, find a.

Tell whether the given side lengths form aright triangle.

82. 1, 1, 2 83. 12, 16, 20 84. 7, 12, 19

Simplify.

1. 109 2. 11�3 3. 27 4. 3�4


5. �33

3

6� 6. 79 � 72 7. (510)6 8. �1111

�

7

7�

9. 273 � 27�18 10. (52�7)�3 11. 130 � 139 12. �88

1

7

2�


13. (�4a)3 14. (7y4)(�9y5) 15. �7�

28mm

nn

2�

16. (12pq7)��23�q2� 17. �

129aa

8

5b3

� 18. (�5x3y)2


19. 4.257 � 105 20. 4.8 � 108 21. 6.09 � 10�3 22. 3.5 � 10�4


23. 19,000,000,000 24. 0.0000039 25. 1,980,000,000 26. 0.00045

27. Order 3.7 � 105, 3.7 � 106, 3.7 � 10�5, and 3.7 � 10�6 from least togreatest.


28. 196 29. 1 30. 225 31. 625

32. The area of a square rug is 144 square feet. What is the length of the rug?


33. �y4� 34. �121s8� 35. �9y6� 36. �25m2�


37. �230� 38. �125� 39. ��60� 40. ��3�


41. ��121� 42. �1.7� 43. �70� 44.

Find the missing length for each right triangle.

45. a � 10, b � 24, c � 46. a � , b � 15, c � 17 47. a � 12, b � , c � 20


48. 8, 10, 14 49. 30, 40, 50 50. 2.5, 6, 6.5

�225��3

Chapter 4 Test 217

KEYWORD: MT8CA Practice

1. Which expression is NOT equivalent to 3 � 3 � 3 � 3 � 3 � 3?

36 18

93 729

2. A number to the 8th power divided bythe same number to the 4th power is16. What is the number?

2 6

4 8

3. Which expression is equivalent to 81?

29 ��13��4

3�4 ��13��4

4. The airports in the United States servemore than 635,000,000 people eachyear. Which of the following is thesame number written in scientificnotation?

635 � 106 6.35 � 108

6.35 � 10�8 6.35 � 109

5. Simplify the expression shown below.(�3x2z4)(9xz3)

6x2z12 �27x2z12

6x3z7 �27x3z7

6. The population of India is close to 1.08 � 109. Which of the followingrepresents this population written instandard form?

1,080,000,000 1,080,000

180,000,000 108,000

7. Jenny finds that a baby lizard growsabout 0.5 inch every week. Whichequation best represents the numberof weeks it will take for the lizard togrow to 1 foot long if it was 4 incheslong when it hatched?

0.5w � 4 � 1 �w

1�2

4� � 0.5

0.5w � 4 � 12 �0.5w� 4� � 1

8. A number k is decreased by 8, and theresult is multiplied by 8. This product isthen divided by 2. What is the finalresult?

8k � 4 4k � 32

4k � 8 8k � 64

9. Which is an irrational number?

�9� ��14��

7.125 �32�

10. A quilt is made with 10 square piecesof fabric. If the area of each squarepiece is 169 square inches, what is thelength of each square piece?

12 inches 14 inches

13 inches 15 inches

11. Which number is NOT between 1.5 and 1.75?

1�14� 1.62

1.73 1�1235�

12. The �18� is between which pair ofnumbers?

8 and 9 4 and 5

7 and 8 3 and 4DB

CA

DB

CA

DB

CA

DB

CA

DB

CA

DB

CA

DB

CA

DB

CA

DB

CA

DB

CA

DB

CA

DB

CA

Cumulative Assessment, Chapters 1–4Multiple Choice


Cumulative Assessment, Chapters 1–4 219

13. Mrs. Graham ordered five pizzas forher top-performing class. The studentsate �

78� of the pepperoni pizza, �

34� of the

cheese pizza, �45� of the veggie pizza, �

23� of

the Hawaiian pizza, and �12� of the

barbecue chicken pizza. How muchtotal pizza was left over?

3�17210� 1�1

4290�

2�18� 1�1

75�

14. Which inequality is represented by thisgraph?

��x7� � 14 �7x � �14

�7x � 14 ��x7� � �14

15. Which expression is equivalent to�64k2�?

8k 8k2

8⏐k⏐ 64⏐k⏐

Gridded Response

16. What exponent makes the statement3? � 272 true?

17. Chrissy is 25 years older than her dog.The sum of their ages is 37. How old isChrissy’s dog?

18. Evaluate the expression �45� �⏐�

12� � x⏐

for x � �15�.

19. From her house, Lea rode her bike 8miles north and then 15 miles west to afriend’s house. How far in miles was shefrom her house along a straight path?

Short Response

20. A bag of pinto beans weighs210 pounds.

a. How much does 10,000 bags ofpinto beans weigh? Write youranswer in standard form.

b. Write the numbers 210 and 10,000in scientific notation.

c. Explain how to use rules ofexponents to write the weight of10,000 bags of pinto beans inscientific notation.

21. Jack works part time with his dadinstalling carpet. They need to installcarpet in a square room that has anarea of about 876 square feet. Carpetcan only be ordered in whole squareyards.

a. About how many feet long is theroom?

b. About how many square yards ofcarpet do Jack and his dad need inorder to cover the floor of theroom? Explain your reasoning.

Extended Response

22. Marissa’s cat is stuck in a tree. The cat ison a branch 23 feet from the ground.Marissa is 5.5 feet tall, and she owns a16-foot ladder.

a. Create a table that shows how highup on the tree the top of the ladderwill reach if Marissa places the baseof the ladder 1 foot, 2 feet, 3 feet, 4 feet, and 5 feet from the tree.

b. How high will Marissa be if sheplaces the base of the ladder thedistances from the tree in part aand stands on the rung 2.5 feetfrom the top of the ladder?

c. Do you think Marissa can use thisladder to reach her cat? Explainyour reasoning.

DB

CA

DB

CA

DB

CA

Pay attention to the units given in atest question, especially if there aremixed units, such as inches and feet.

�3 �2 �1 0 1 2 3 4

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