Exponents and Scientific Notation MODULE 2Common Core Standards Lesson 2.1 Lesson 2.2 Lesson 2.3 Lesson 2.4 8.EE.1 Know and apply the properties of integer exponents to generate equivalent

ESSENTIAL QUESTION?

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Exponents and Scientific Notation

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2MODULE

The distance from Earth to other planets, moons, and stars is a very great number of kilometers. To make it easier to write very large and very small numbers, we use scientific notation.

LESSON 2.1

Integer Exponents8.EE.1

LESSON 2.2

Scientific Notation with Positive Powers of 10

8.EE.3

LESSON 2.3

Scientific Notation with Negative Powers of 10

8.EE.3

LESSON 2.4

Operations with Scientific Notation

8.EE.4

COMMON CORE

COMMON CORE

COMMON CORE

COMMON CORE

You can simplify calculations with very large and very small numbers by first writing them in scientific notation.

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29 Module 2

YOUAre Ready?Personal

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Online Assessment and

Interventionmy.hrw.com

Complete these exercises to review skills you will

need for this module.

ExponentsEXAMPLE 10 4 = 10 × 10 × 10 × 10

= 10,000

Write each exponential expression as a decimal.

1. 10 2 2. 10 3 3. 10 5 4. 10 7

Multiply and Divide by Powers of 10EXAMPLE

Find each product or quotient.

5. 45.3 × 10 3 6. 7.08 ÷ 10 2 7. 0.00235 × 10 6 8. 3,600 ÷ 10 4

9. 0.5 × 10 2 10. 67.7 ÷ 10 5 11. 0.0057 × 10 4 12. 195 ÷ 10 6

0.0478 × 10 5 = 0.0478 × 100,000

= 4,780

37.9 ÷ 10 4 = 37.9 ÷ 10,000

= 0.00379

Write the exponential expression as a product.Simplify.

Identify the number of zeros in the power of 10.When multiplying, move the decimal point to the right the same number of places as the number of zeros.

Identify the number of zeros in the power of 10.When dividing, move the decimal point to the left the same number of places as the number of zeros.

100

45,300

50

0.0708

0.000677

2,350

57

0.36

0.000195

1000 100,000 10,000,000

Unit 130

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1

2

3Response toIntervention

Professional

Development

PROFESSIONAL DEVELOPMENT VIDEO

Are You Ready?

Assess ReadinessUse the assessment on this page to determine if students need

intensive or strategic intervention for the module’s prerequisite skills.

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Interactive Whiteboards

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whiteboard-ready lessons and

activities.

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Intervention

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homework, quizzes, tests,

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Interactive Answers and

Solutions

Customize answer keys to print

or display in the classroom.

Choose to include answers only

or full solutions to all lesson

exercises.

Author Juli Dixon models successful

teaching practices as she explores the

concept of scientific notation in an

actual eighth-grade classroom.

Intervention Enrichment

Access Are You Ready? assessment online, and receive

instant scoring, feedback, and customized intervention

or enrichment.

Online and Print Resources

Skills Intervention worksheets

• Skill 12 Exponents

• Skill 37 Multiply and Divide

by Powers of 10

Differentiated Instruction

• Challenge worksheets

PRE-AP

Extend the Math PRE-AP

Lesson Activities in TE

Exponents and Scientific Notation 30

102

10 is: 2 is:

Reading Start-Up VocabularyReview Words

✔ base (base)

✔ exponent (exponente)

integers (enteros)

✔ positive number (número positivo)

standard notation

(notación estándar)

Preview Words

power (potencia)

rational number (número racional)

real numbers (número real)

scientific notation

(notación científica)

whole number (número entero)

Visualize VocabularyUse the ✔ words to complete the Venn diagram. You can put more

than one word in each section of the diagram.

Understand VocabularyComplete the sentences using the preview words.

1. A number produced by raising a base to an exponent

is a .

2. is a method of writing very large or

very small numbers by using powers of 10.

3. A is any number that can be expressed

as a ratio of two integers.

Active ReadingTwo-Panel Flip Chart Create a two-panel flip

chart to help you understand the concepts in this

module. Label one flap “Positive Powers of 10” and

the other flap “Negative Powers of 10.” As you

study each lesson, write important ideas under

the appropriate flap. Include sample problems

that will help you remember the concepts later

when you look back at your notes.

power

Scientific notation

rational number

positive number, base

positive number, exponent

31Module 2

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Reading Start-Up

Have students complete the activities on this page by working alone or

with others.

Visualize VocabularyThe two diagrams help students review the terms base and exponent,

and help prepare students for learning about scientific notation which

is the focus of this module. Students should write more than one

review word in each box.

Understand VocabularyUse the following explanations to help students learn the preview

words.

Scientific notation uses the powers of 10. Power refers to

the exponent, and 10 is the base. In scientific notation, 10 is

always the base. The exponent can be any integer. Scientific

notation is used for very large and very small numbers.

Active ReadingIntegrating Language ArtsStudents can use these reading and note-taking strategies to help

them organize and understand new concepts and vocabulary.

Additional ResourcesDifferentiated Instruction

• Reading Strategies ELL

After

Students will connect that:

• positive numbers written in

scientific notation with negative

exponents represent numbers

between 0 and 1

• measurements written in scientific

notation often use significant

digits to show the precision of a

measurement

Before

Students understand how to:

• write decimals

• use exponents

• add, subtract, multiply, and divide

rational numbers

In this moduleStudents will learn how to:

• apply properties of integer exponents to evaluate

expressions

• convert between large numbers in standard decimal

notation and scientific notation

• convert between small numbers in standard decimal

notation and scientific notation

• add, subtract, multiply, and divide numbers expressed

with scientific notation

COMMONCORE ELA-Literacy.RST.6-8.7 Integrate quantitative or technical information

expressed in words in a text with a version of that information expressed visually (e.g., in a flowchart, diagram, model, graph, or table).

31 Module 2

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Unpacking the Standards

What It Means to YouYou will convert very large numbers to scientific notation.

What It Means to YouYou will use the properties of integer exponents to find equivalent expressions.

Understanding the standards and the vocabulary terms in the standards will help you know exactly what you are expected to learn in this module.

MODULE 2

Evaluate two different ways.

8 3 __ 8 5

8 3 __ 8 5

= 8 ⋅ 8 ⋅ 8 __________ 8 ⋅ 8 ⋅ 8 ⋅ 8 ⋅ 8 = 1 ____ 8 ⋅ 8 = 1 __ 64

8 3 __ 8 5

= 8 (3–5) = 8 -2 = 1 __ 8 2

= 1 ____ 8 ⋅ 8 = 1 __ 64

( 3 2 ) 4 ( 3 2 ) 4 = ( 3 2 )( 3 2 )( 3 2 )( 3 2 ) = 3 2 + 2 + 2 + 2 = 3 8 = 6,561

( 3 2 ) 4 = 3 (2 ⋅ 4) = 3 8 = 6,561

There are about 55,000,000,000 cells in an average-sized adult. Write this number in scientific notation.

Move the decimal point to the left until you have a number that is greater than or equal to 1 and less than 10.

5.5 0 0 0 0 0 0 0 0 0

5.5

You would have to multiply 5.5 by 1010 to get 55,000,000,000.

55,000,000,000 = 5.5 × 1010Visit my.hrw.com to see all the Common Core Standards unpacked.

8.EE.3

Use numbers expressed in the form of a single digit times an integer power of 10 to estimate very large or very small quantities, and to express how many times as much one is than the other.

Key Vocabularyscientific notation (notación

cientifica) A method of writing very large or very small numbers by using powers of 10.

COMMON CORE

8.EE.1

Know and apply the properties of integer exponents to generate equivalent numerical expressions.

Key Vocabularyinteger (entero)

The set of whole numbers and their opposites

exponent (exponente) The number that indicates how many times the base is used as a factor.

COMMON CORE

UNPACKING EXAMPLE 8.EE.1

UNPACKING EXAMPLE 8.EE.3

Move the decimal point 10 places to the left.

Remove the extra zeros.

Unit 132

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Use the examples on the page to help students know exactly what they are expected to learn in this module.

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Unpacking the Standards

Common Core Standards Lesson 2.1

Lesson 2.2

Lesson 2.3

Lesson 2.4

8.EE.1 Know and apply the properties of integer exponents to generate equivalent numerical expressions.

COMMONCORE

8.EE.3 Use numbers expressed in the form of a single digit times a whole-number power of 10 to estimate very large or very small quantities, and to express how many times as much one is than the other.

COMMONCORE

COMMONCORE

8.EE.4 Perform operations with numbers expressed in scientific notation, including problems where both decimal and scientific notation are used. Use scientific notation and choose units of appropriate size for measurements of very large or very small quantities (e.g., use millimeters per year for seafloor spreading). Interpret scientific notation that has been generated by technology.

COMMONCORE

Go online to see a complete unpacking of the Common Core Standards.

Common Core Standards

Content Areas

COMMONCORE Expressions and Equations—8.EE

Work with radicals and integer exponents.


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EngageESSENTIAL QUESTION

How can you develop and use the properties of integer exponents? Sample answer: To develop the properties of integer exponents, look for patterns in a powers table, in the multiplication and division of powers with the same base, and in powers of products. The properties can be used to simplify complicated expressions.

Motivate the LessonAsk: How can patterns help you simplify numbers involving negative exponents? Take a guess. Begin Explore Activity 1 to find out.

ExploreEXPLORE ACTIVITY 1

Focus on Patterns Mathematical PracticesIf students have trouble seeing the patterns, point out that they should look at one row of the table at a time, and examine the two sides of the equation separately. Ask them to identify what changes on the left side and what changes on the right side.

ExplainEXPLORE ACTIVITY 2

Questioning Strategies Mathematical Practices • In Part B, why are there five factors of 4 in the numerator and only three in the denominator? 4 5 means 4 is a factor five times. 4 3 means 4 is a factor three times.

• In Part C, could ( 5 3 ) 2 be thought of as ( 5 3 )( 5 3 )? Justify your answer. Yes, the base is ( 5 3 ), and the exponent 2 means that ( 5 3 ) is used as a factor two times, so ( 5 3 ) 2 = ( 5 3 )( 5 3 ).

Engage with the WhiteboardFor each equation in Part A, have volunteers circle each factor as they count aloud. Then have students write the total number of factors in the box provided.

Integer Exponents2.1L E S S O N

CC

CC

Common Core StandardsThe student is expected to:

Expressions and Equations—8.EE.1


Mathematical Practices

MP.8 Patterns

COMMONCORE

COMMONCORE

33 Lesson 2.1


Exploring Properties of Integer Exponents Complete the following equations.

3 · 3 · 3 · 3 · 3 = 3

(3 · 3 · 3 · 3) · 3 = 3 · 3 = 3

(3 · 3 · 3) · (3 · 3) = 3 · 3 = 3

What pattern do you see when multiplying two powers with the

same base?

Use your pattern to complete this equation: 5 2 · 5 5 = 5 .

Complete the following equation:

4 5 __

4 3 = 4 · 4 · 4 · 4 · 4

__________ 4 · 4 · 4

= 4 · 4 · 4 · 4 · 4 __________

4 · 4 · 4 = 4 · 4 = 4

What pattern do you see when dividing two powers with the same base?

Use your pattern to complete this equation: 6 8 __ 6 3

= 6 .

Complete the following equations:

( 5 3 ) 2 = (5 · 5 · 5) = (5 · 5 · 5) · (5 · 5 · 5) = 5

What pattern do you see when raising a power to a power?

Use your pattern to complete this equation: ( 7 2 ) 4 = 7 .

A

B

111

111

C

EXPLORE ACTIVITY 2

Do the patterns you found in parts A–C apply if the exponents are negative?

If so, give an example of each.

Math TalkMathematical Practices

8.EE.1COMMONCORE

5

4 1 5

5

7

2

5

2 6

8

23

The result has the same base with an exponent equal

to the sum of the exponents in the powers.


to the difference of the exponent in the numerator and

exponent in the denominator.


to the product of the exponents.

Yes;

Part A:

5 -1 · 5 -2 =

1 _ 5

· 1 __ 25

= 1 ___

125 = 1 __

5 3

= 5 -3 = 5 -1 + (-2)

Part B:

2 -5 ___

2 -3 = 2 3

__ 2 5

= 2 · 2 · 2 ___________

2 · 2 · 2 · 2 · 2

= 2 -2 = 2 -5 - (-3)

Part C:

( 3 -2 ) -3

= ( 1 __ 3 2

) -3

= ( 3 2 ) 3 = 3 6

= 3 -2 · (-3)

Unit 134

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EXPLORE ACTIVITY 1

ESSENTIAL QUESTION

Using Patterns of Integer ExponentsThe table below shows powers of 5, 4, and 3.

5 4 = 625 5 3 = 125 5 2 = 25 5 1 = 5 5 0 = 5 -1 = 5 -2 =

4 4 = 256 4 3 = 64 4 2 = 16 4 1 = 4 4 0 = 4 -1 = 4 -2 =

3 4 = 81 3 3 = 27 3 2 = 9 3 1 = 3 3 0 = 3 -1 = 3 -2 =

What pattern do you see in the powers of 5?



Complete the table for the values of 5 0 , 5 -1 , 5 -2 .



Reflect1. Make a Conjecture Write a general rule for the value of a 0 .

2. Make a Conjecture Write a general rule for the value of a -n .

A

B

C

D

E

F

How can you develop and use the properties of integer exponents?

L E S S ON

2.1 Integer Exponents

8.EE.1COMMONCORE

8.EE.1


COMMONCORE

1 1 _ 5

1 _ 4

1 _ 3

1 __ 25

1 __ 16

1 _ 9

1

1

As the exponent decreases by 1, the value of the

power is divided by 5.





See table above.

a 0 = 1

a -n = 1 __ a n

See table above.

See table above.

33Lesson 2.1

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PROFESSIONAL DEVELOPMENT

Math BackgroundUsing patterns, you can intuitively see that any

nonzero number raised to the zero power equals 1.

For example, 2 4 = 16; 2 3 = 8; 2 2 = 4; 2 1 = 2; 2 0 = 1.

The exponent decreases by 1 as the value of the

exponential expression is divided by the value of

the base. Another way to understand this is to use

the fact that any number (other than 0) divided

by itself is 1 and the property of exponents that

tells us that is a x ÷ a y = a x - y , for a ≠ 0. Therefore,

since a x ÷ a x = a x - x or a 0 and a x ÷ a x = 1, we

can conclude that a 0 = 1, for a ≠ 0.

CC Integrate Mathematical Practices MP.8

This lesson provides an opportunity to address

this Mathematical Practices standard. It calls for

students to notice if calculations are repeated.

Students learn to recognize how repeated

division defines the use of negative exponents.

They then use repeated multiplication and

division to discover properties of exponents and

find shortcuts for solving expressions.

Integer Exponents 34

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Interactive Whiteboard Interactive example available online

EXAMPLE 1Questioning Strategies Mathematical Practices • How can you use your knowledge of the order of operations to help simplify expressions

involving exponents? Start by simplifying what you can inside parentheses, and then use

the properties of exponents to simplify terms with exponents.

• How else could you find the product of 3 5 and 3 -8 ? Evaluate each exponent separately

and then multiply the numbers 243 and 1

____ 6561

.

Focus on Critical ThinkingMake sure students understand that for expressions such as (3 + 1) 2 in Part B, they do not

distribute the exponent to each term inside the parentheses but rather simplify inside

the parentheses and then apply the exponent. That is, (3 + 1) 2 ≠ 3 2 + 1 2 but rather

(3 + 1) 2 = 4 2 .

YOUR TURNAvoid Common Errors

To avoid calculation errors in expressions like those in Exercise 6, make sure that students

understand that simplifying terms with common bases usually makes calculating easier.

For example, [ (6 - 1) 2 ] 2 = 5 4 = 625 and (3 + 2) 3 = 5 3 = 125. It is much easier to mentally

simplify 5 4

__ 5 3

= 5 than it is to simplify 625

___ 125

.

Talk About ItCheck for Understanding

Ask: Mike says to simplify the expression in Exercise 7, just simplify to 4 3 - 4 3 • 4 -5

to get 4 -5 or 1

____ 1,024

. Why is his method and answer not correct? Mike did not follow

the order of operations. He should have multiplied before subtracting. He should have

gotten 4 3 - 4 -2 = 64 - 1

__ 16

to get 63 15

__ 16

.

Elaborate

Talk About ItSummarize the Lesson

Ask: How can you simplify an expression with exponents by applying the

properties of exponents? The order of operations must always be followed.

Each expression within brackets or parentheses must be simplified first. Then terms with the

same base that are being multiplied or divided can be simplified using the properties of

exponents. Finally, any terms being added or subtracted are simplified.

GUIDED PRACTICEEngage with the Whiteboard

For Exercises 18 and 19, have students underline each expression inside parentheses

and then write the value of each underlined expression below it while explaining

how to simplify each part of the expression.

Avoid Common ErrorsExercises 1–2, 6–7, 9 Remind students that a negative exponent does not mean the

answer will be a negative number.

Exercise 10 Remind students that when no exponent is given, it is understood that the

number is raised to the power of 1.

ADDITIONAL EXAMPLE 1Simplify each expression.

A 5 - (6 - 4) -3 + (-2) 0 5 7 _ 8

B [(6 - 2)] 4

_______ (8 - 4) 1

4 3 = 64

CC

35 Lesson 2.1

Guided Practice

Find the value of each power. (Explore Activity 1)

1. 8 -1 = 2. 6 -2 = 3. 25 6 0 =

4. 10 2 = 5. 5 4 = 6. 2 -5 =

7. 4 -5 = 8. 8 9 0 = 9. 11 -3 =

Use properties of exponents to write an equivalent expression. (Explore Activity 2)

10. 4 · 4 · 4 = 4 11. (2 · 2) · (2 · 2 · 2) = 2 · 2 = 2

12. 6 7 __ 6 5

= 6 · 6 · 6 · 6 · 6 · 6 · 6 _______________ 6 · 6 · 6 · 6 · 6 = 13. 8 12 ___ 8 9

= 8 -

=

14. 5 10 · 5 · 5 = 5 15. 7 8 · 7 5 =

16. ( 6 2 ) 4 = (6 · 6)

= (6 · 6) · (6 · 6) · (

· ) ·

=

17. ( 3 3 ) 3 = (3 · 3 · 3) 3

= (3 · 3 · 3) · (

·

· )

=

Simplify each expression. (Example 1)

18. (10 - 6) 3 · 4 2 + (10 + 2) 2 19. (12 - 5) 7 ________

[ (3 + 4) 2 ] 2

20. Summarize the rules for multiplying powers with the same base, dividing powers with the same base, and raising a power to a power.

ESSENTIAL QUESTION CHECK-IN???

1 _ 8

1 ____ 1024

1 __ 32

1 ____ 1331

3

26

12

4

6 6

13

8 9

7

6 3

12 9

3 3 3

38

32 5

1 __ 36 1

100 625

(6 · 6)

When multiplying powers with the same base, you add

the exponents. When dividing powers with the same

base, you subtract the exponents. When raising a power

to a power, you multiply the exponents.

(3 · 3 · 3)

1

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Unit 136

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My Notes


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ReflectLet m and n be integers.

3. Make a Conjecture Write a general rule for the value of a m · a n .

4. Make a Conjecture Write a general rule for the value of a m __ a n

, a ≠ 0.

5. Make a Conjecture Write a general rule for the value of ( a m ) n .

Applying Properties of Integer ExponentsYou can use the general rules you found in the Explore Activities to simplify more complicated expressions.

Simplify each expression.

(5 - 2) 5 · 3 -8 + (5 + 2) 0

(3) 5 · 3 -8 + (7) 0

3 5 + (-8) + 1

3 -3 + 1

1 __ 27 + 1 = 1 1 __ 27

[ (3 + 1) 2 ] 3 ________

(7 - 3) 2

( 4 2 ) 3 ____

4 2

4 6 __ 4 2

4 6-2 4 4 = 256

EXAMPLEXAMPLE 1

A

B

Simplify within parentheses.

Use properties of exponents.

Simplify.

Apply the rule for negative exponents and add.

Simplify within parentheses.



Simplify.


YOUR TURN

6. [ (6 - 1) 2 ] 2 _______

(3 + 2) 3

7. ( 2 2 ) 3 - (10 - 6) 3 ∙ 4 -5

8.EE.1COMMONCORE

a m · a n = a m + n a m __ a n = a m - n

( a m ) n = a m · n

5 63 15 __ 16

35Lesson 2.1

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DIFFERENTIATE INSTRUCTION

Multiple RepresentationsWorking in pairs, each student creates two expressions involving exponents that can be simplified. Encourage students to use negative exponents. Have students in each pair exchange problems and simplify each other’s expressions.Sample expressions: (-4) -2 = 1 ____

(-4) 2 = 1 __ 16 ;

6 2 __

6 5 = 6 -3 = 1 __

6 3 = 1 ___ 216

Critical ThinkingHave students discuss whether (-a) 2 for a ≠ 0 will always have a positive value or a negative value. Then have them do the same for - a 2 for a ≠ 0. Students should recognize that (-a) 2 is always positive because the square of any nonzero number is always positive. Students should recognize that the order of operations means - a 2 = -( a 2 ), and its value will always be negative.

Additional ResourcesDifferentiated Instruction includes: • Reading Strategies • Success for English Learners ELL

• Reteach • Challenge PRE-AP



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Lesson Quiz available online

2.1 LESSON QUIZ

Find the value of each power.

1. 5 -2

2. 3 4

Use properties of exponents to

write an equivalent expression.

3. (5 • 5) • (5 • 5 • 5 • 5)

4. ( 7 4 )

2 ____

7 5


5. (8 - 6) 5 • (2) -4 + (2) 0

6. [(9 - 1)] 5

_______ (6 + 2) 3

Evaluate

GUIDED AND INDEPENDENT PRACTICE

Concepts & Skills Practice

Explore Activity 1

Using Patterns of Integer ExponentsExercises 1–9

Explore Activity 2

Exploring Properties of Integer ExponentsExercises 10–17, 21–27, 29–30

Example 1

Applying Properties of Integer ExponentsExercises 18–19

Additional ResourcesDifferentiated Instruction includes:

• Leveled Practice worksheets

Exercise Depth of Knowledge (D.O.K.) Mathematical Practices

21–22 2 Skills/Concepts MP.7 Using Structure

23 2 Skills/Concepts MP.4 Modeling

24 2 Skills/Concepts MP.3 Logic

25–27 2 Skills/Concepts MP.5 Using Tools

28 3 Strategic Thinking MP.3 Logic

29–30 2 Skills/Concepts MP.4 Modeling


32–34 2 Skills/Concepts MP.7 Using Structure

35–36 3 Strategic Thinking MP.3 Logic

37 3 Strategic Thinking MP.1 Problem Solving

COMMONCORE

8.EE.1COMMONCORE

COMMONCORE 8.EE.1

Answers

1. 1 __

25

2. 81

3 5 6

4. 7 3

5. 3

6. 64

37 Lesson 2.1

Work Area

31. Write equivalent expressions for x 7 · x -2 and x 7 __

x 2 . What do you notice?

Explain how your results relate to the properties of integer exponents.

A toy store is creating a large window display of different colored cubes stacked in a triangle shape. The table shows the number of cubes in each row of the triangle, starting with the top row.

Row 1 2 3 4

Number of cubes in each row 3 3 2 3 3 3 4

32. Look for a Pattern Describe any pattern you see in the table.

33. Using exponents, how many cubes will be in Row 6? How many times as many cubes will be in Row 6 than in Row 3?

34. Justify Reasoning If there are 6 rows in the triangle, what is the total number of cubes in the triangle? Explain how you found your answer.

35. Critique Reasoning A student simplified the expression 6 2 ___

36 2 as 1 _ 3 .

Do you agree with this student? Explain why or why not.

36. Draw Conclusions Evaluate – a n when a = 3 and n = 2, 3, 4, and 5. Now evaluate (–a) n when a = 3 and n = 2, 3, 4, and 5. Based on this sample, does it appear that – a n = (–a) n ? If not, state the relationships, if any, between – a n and ( –a) n .

37. Persevere in Problem Solving A number to the 12th power divided by the same number to the 9th power equals 125. What is the number?

FOCUS ON HIGHER ORDER THINKING

Both expressions equal x 5 , so x 7 · x -2 = x 7 __

x 2 . When multiplying

The number of cubes in each row is 3 raised to the row number.

1,092; I evaluated 3 6 , 3 5 , 3 4 , 3 3 , 3 2 , and 3 1 and added these numbers together.

For - a n , you get -9, -27, -81, -243. For (-a) n , you

get 9, -27, 81, -243. In general, - a n ≠ (-a) n . When n is

even, the two expressions are opposites; when n is odd,

the two expressions are equal.

The number is 5.

No; 6 2 ___ 36 2

= 6 · 6_____36 · 36 = 6 · 6________

6 · 6 · 6 · 6 = 1____6 · 6 = 1__

36

3 6 ; 3 3

powers with the same base, you add exponents; 7 + (-2) = 5.

When dividing powers with the same base, you subtract exponents; 7 - 2 = 5. In cases like this, x n · x -m = x n

__ x m

.

Unit 138

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Name Class Date

Independent Practice2.1

21. Explain why the exponents cannot be added in the product 12 3 · 11 3 .

22. List three ways to express 3 5 as a product of powers.

23. Astronomy The distance from Earth to the moon is about 22 4 miles. The distance from Earth to Neptune is about 22 7 miles. Which distance is the greater distance and about how many times greater is it?

24. Critique Reasoning A student claims that 8 3 · 8 -5 is greater than 1. Explain whether the student is correct or not.

Find the missing exponent.

25. ( b2 ) = b-6 26. x · x6 = x9 27. y 25 ______

y

= y 6

28. Communicate Mathematical Ideas Why do you subtract exponents when dividing powers with the same base?

29. Astronomy The mass of the Sun is about 2 × 10 27 metric tons, or 2 × 10 30 kilograms. How many kilograms are in one metric ton?

30. Represent Real-World Problems In computer technology, a kilobyte is 2 10 bytes in size. A gigabyte is 2 30 bytes in size. The size of a terabyte is the product of the size of a kilobyte and the size of a gigabyte. What is the size of a terabyte?

8.EE.1COMMONCORE

The exponents cannot be added because the bases are not the same.

Earth to Neptune; 22 3 , or 10,648, times greater.

The student is not correct because 8 3 · 8 -5 = 8 3 + (-5) = 8 -2 = 1 __

8 2 = 1 __ 64 , which is less than 1.

Dividing is the same as multiplying by the reciprocal. So when dividing

powers with the same base, you add the opposite of the exponent in

the denominator. This is the same as subtracting the exponents.

10 3 kg, or 1,000 kg

2 40 bytes

3 5 · 3 0 ; 3 4 · 3 1 ; 3 3 · 3 2

-3 3

19

37Lesson 2.1

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8_MFLCESE056718_U1M02L1.indd 37 24/02/16 10:27 AM

Activity available online my.hrw.comEXTEND THE MATH PRE-AP

Activity Describe the following pattern:

(-1) -1 =

(-1) -2 =

(-1) -3 =

(-1) -4 =

Determine what (-1) -100 would be. Justify your thinking.

(-1) -1 = -1; (-1) -2 = 1; (-1) -3 = -1; (-1) -4 = 1; so the pattern is -1, 1, -1, 1, etc. -1 raised to an odd number equals -1, and -1 raised to an even number equals 1. Since -100 is an even number, then (-1) -100 = 1.


DO NOT EDIT--Changes must be made through "File info"CorrectionKey=C

2.2L E S SON Scientific Notation with Positive

Powers of 10


ADDITIONAL EXAMPLE 1The average distance from Earth to

Mars is about 140,000,000 miles. Write

this distance in scientific notation.

1.4 × 108

Engage

ESSENTIAL QUESTIONHow can you use scientific notation to express very large quantities? Sample answer: Write the large quantity as the product of a number greater than or equal to 1 and less than 10 and a power of 10.

Motivate the LessonAsk: If you were given the weight of a whale in pounds as 2 × 105, how many pounds do

you think that is? Begin the Explore Activity to find out.

Explore

EXPLORE ACTIVITYTalk About ItCheck for Understanding

Ask: In 250,000, what place is the 2 in? hundred thousands What digit is in the ten

thousands place? 5

Integrating Language Arts ELL

You may want to pair English learners with a partner for Explore Activity 1 to help them

develop their language skills.

Explain

EXAMPLE 1Questioning Strategies Mathematical Practices • When you write 9 × 107 in standard notation, how many zeros do you write? 7 If you were

to write 9 × 102 in standard notation, how many zeros would you write? 2

• The standard notation for 9.3 × 107 is 93,000,000. Why are there 6 zeros instead of 7 zeros

in the standard notation? One of the 7 factors of 10 moves the decimal to the right of 3.

The remaining 6 factors of 10 moves the decimal 6 places, represented with zeros.

Avoid Common ErrorsA dot used for multiplication can sometimes be confused as a decimal point when using

scientific notation. Encourage students to use X instead.

YOUR TURNTalk About ItCheck for Understanding

Ask: When you begin to write 570,000,000,000 in scientific notation, you write the

first factor as 5.7. Is this factor greater than or less than the original value? less than

To compensate and express an equivalent value, what must the power of 10 factor do,

increase or decrease the value of 5.7? Increase it, so the exponent must be positive.

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The student is expected to:




MP.4 Modeling

COMMONCORE

COMMONCORE

39 Lesson 2.2

Math Trainer


Personal

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Writing a Number in Scientific NotationTo translate between standard notation and scientific notation, you can count

the number of places the decimal point moves.

Writing Large Quantities in Scientific Notation

When the number is greater than or equal to 10, use a positive exponent.

8 4, 0 0 0 = 8.4 × 10 4 The decimal point

moves 4 places. to the left.

The distance from Earth to the Sun is about 93,000,000 miles. Write this

distance in scientific notation.

Move the decimal point in 93,000,000 to the left until you have

a number that is greater than or equal to 1 and less than 10.

9.3 0 0 0 0 0 0.

9.3

Divide the original number by the result from Step 1.

10,000,000

10 7

Write the product of the results from Steps 1 and 2.

93,000,000 = 9.3 × 10 7 miles

EXAMPLE 1

STEP 1

STEP 2

STEP 3

3. 6,400

4. 570,000,000,000

5. A light-year is the distance that light travels in a year and is equivalent to

9,461,000,000,000 km. Write this distance in scientific notation.

Move the decimal point 7 places to the left.

Remove extra zeros.

Divide 93,000,000 by 9.3.

Write your answer as a power of 10.

Write a product to represent 93,000,000 in scientific notation.

Write each number in scientific notation.

YOUR TURN

Is 12 × 10 7 written in scientific notation?

Explain.


8.EE.3COMMONCORE

6.4 × 10 3 5.7 × 10 11

9.461 × 10 12 km

No, because the first factor must be greater than or equal to 1 and less than 10.

Unit 140

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? ESSENTIAL QUESTION

EXPLORE ACTIVITY

How can you use scientific notation to express very large quantities?

L E S S ON

2.2Scientific Notation with Positive Powers of 10

Using Scientific NotationScientific notation is a method of expressing very large and very small

numbers as a product of a number greater than or equal to 1 and

less than 10, and a power of 10.

The weights of various sea creatures are shown in the table.

Write the weight of the blue whale in scientific notation.

Sea Creature Blue whale Gray whale Whale shark

Weight (lb) 250,000 68,000 41,200

Move the decimal point in 250,000 to the left as many places as necessary

to find a number that is greater than or equal to 1 and less than 10.

What number did you find?

Divide 250,000 by your answer to A . Write your answer as a power of 10.

Combine your answers to A and B to represent 250,000.

Repeat steps A through C to write the weight

of the whale shark in scientific notation.

Reflect1. How many places to the left did you move the decimal point to write

41,200 in scientific notation?

2. What is the exponent on 10 when you write 41,200 in scientific notation?

A

B

C

250,000 = × 10

41,200 = × 10

8.EE.3

Use numbers expressed in the form of a single digit times an integer power of 10 to estimate very large or very small quantities, ….

COMMONCORE

8.EE.3COMMONCORE

2.5

100,000; 10 5

52.5

44.12

The decimal point moves 4 places to the left.

The exponent on 10 is also 4.

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Lesson 2.2


Math BackgroundScientific notation is used to represent very large

numbers. One very large named number is a

googol–the number 1 followed by 100 zeros. An

even larger named number is the googolplex–

the number 1 followed by a googol (10100) of

zeros. This massive number can be represented

as 10googol. This idea has been extended even

further to name a googolplexian, which is the

number 1 followed by a googolplex of zeros. The

googol (also spelled google) was named by the

nine-year-old nephew of the American

mathematician Edward Kasner.



this Mathematical Practices standard. It calls for

students to solve problems arising in everyday

life, society, and the workplace. Students use

scientific notation to write very large numbers to

express facts about the natural world, and they

see how this notation is used by scientists in

reporting scientific information.

Scientific Notation with Positive Powers of 10 40


ADDITIONAL EXAMPLE 2Write 7.8 × 109 in standard notation.

7,800,000,000

EXAMPLE 2Questioning Strategies Mathematical Practices • Is the value of 3.5 × 106 greater than or less than than 3.5? greater Which direction does a

decimal point move when a number is increasing? to the right

• What is the value of 100? Explain. 100 = 1. Any number to the zero power is equal to 1.

Engage with the WhiteboardHave a student use curved arrows, as in Example 1, to show moving the decimal

point the correct number of places. Ask students whether there can be additional

zeros after the decimal point. Remind them that it is the location of the decimal, not the

number of zeros, that determines the value.

YOUR TURNFocus on Critical Thinking

Discuss with students how to compare the values 7.034 × 109 and 2.36 × 105 when they are

expressed in scientific notation. Be sure they understand that they can compare the

exponents of 10. Because 109 has a greater exponent than 105, 7.034 × 109 is greater than

2.36 × 105.

Elaborate


Ask: How can you write a very large number such as 51,200,000 in scientific

notation? First, move the decimal point 7 places to the left to determine the first

factor, which must be greater than or equal to 1 and less than 10. Then write the power of

10 factor with the exponent equal to 7, which is the number of places you moved the

decimal to determine the first factor.


For Exercises 1–12, have students draw and count curved arrows to show moving

the decimal point the correct number of places.

Avoid Common ErrorsExercises 1–6 Remind students to draw arrows to avoid counting errors as they move the

decimal point.

Exercises 7, 11–12 To avoid decimal placement errors, suggest that students place the

decimal point to the right of the whole number factor before converting to standard

notation.

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41 Lesson 2.2

Write each number in scientific notation. (Explore Activity and Example 1)

1. 58,927

Hint: Move the decimal left 4 places.

2. 1,304,000,000

Hint: Move the decimal left 9 places.

3. 6,730,000

4. 13,300

5. An ordinary quarter contains about

97,700,000,000,000,000,000,000 atoms.

6. The distance from Earth to the Moon is

about 384,000 kilometers.

Write each number in standard notation. (Example 2)

7. 4 × 10 5

Hint: Move the decimal right 5 places.

8. 1.8499 × 10 9

Hint: Move the decimal right 9 places.

9. 6.41 × 10 3

10. 8.456 × 10 7

11. 8 × 10 5 12. 9 × 10 10

13. Diana calculated that she spent about 5.4 × 10 4 seconds doing her math

homework during October. Write this time in standard notation. (Example 2)

14. The town recycled 7.6 × 10 6 cans this year. Write the number of cans in

standard notation. (Example 2)

15. Describe how to write 3,482,000,000 in scientific notation.

ESSENTIAL QUESTION CHECK-IN?

Guided PracticeGuided Practice

5.8927 × 10 4

6.73 × 10 6

400,000

6,410

800,000

54,000 s

7,600,000 cans

First move the decimal point 9 places to the left to find 3.482,

a number that is greater than or equal to 1 and less than 10.

Then multiply 3.482 by 109, using an exponent on 10 that

equals the number of places you moved the decimal.

9.77 × 10 22

1.304 × 10 9

1.33 × 10 4

1,849,900,000

84,560,000

90,000,000,000

3.84 × 10 5

Unit 142

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My Notes

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Writing a Number in Standard NotationTo translate between scientific notation and standard notation, move the

decimal point the number of places indicated by the exponent in the power

of 10. When the exponent is positive, move the decimal point to the right and

add placeholder zeros as needed.

Write 3.5 × 10 6 in standard notation.

Use the exponent of the power of 10

to see how many places to move the

decimal point.

6 places

Place the decimal point. Since you are

going to write a number greater than 3.5,

move the decimal point to the right. Add

placeholder zeros if necessary.

3 5 0 0 0 0 0.

The number 3.5 × 10 6 written in standard notation is 3,500,000.

Reflect6. Explain why the exponent in 3.5 × 10 6 is 6, while there are only 5 zeros

in 3,500,000.

7. What is the exponent on 10 when you write 5.3 in scientific notation?

EXAMPLEXAMPLE 2

STEP 1

STEP 2

Write each number in standard notation.

YOUR TURN

8. 7.034 × 10 9 9. 2.36 × 10 5

10. The mass of one roosting colony of Monarch butterflies in Mexico was

estimated at 5 × 10 6 grams. Write this mass in standard notation.

8.EE.3COMMONCORE

The decimal point moves 6 places to the right, but

one of those places is the 5 tenths in 3.5, so only 5

placeholder zeros are needed.

The exponent on 10 is 0. 5.3 = 5.3 × 10 0 .

7,034,000,000

5,000,000 g

236,000

41Lesson 2.2

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Cognitive StrategiesSuggest that students visualize how the final

answer will look before writing anything down.

If the question says “Write in scientific notation,”

realize that the answer will be the product of a

number greater than or equal to 1 and less than

10, and a power of 10.

If the question says “Write in standard notation,”

be aware that an answer may have zeros as

placeholders to show the number’s size.

TechnologyHave students use a graphing calculator and

enter 3,500,000. Then have them multiply that

number by 1,000,000. Ask what the answer in

the display, 3.5E12, means. 3.5 × 1012 Ask them

to predict the answer display on a calculator for

10 million times 3 million. Then have them check

their prediction using the calculator. 3E13 Have

them investigate when their calculator changes

to scientific notation. Sample answer: when

there are more than 11 digits in the combined

factors


• Reading Strategies

• Success for English Learners ELL

• Reteach

• Challenge PRE-AP



Math Trainer


Personal

Online homework


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2.2 LESSON QUIZ

1. The approximate mass of Mars is 6.42 × 1023 kilograms. Write this mass in standard notation.

2. An adult blue whale can eat about 40,000,000 krill a day. Write this number in scientific notation.

3. Write 9 in scientific notation.

4. Write 1.0 × 105 in standard notation and in words.

5. Is 10.2 × 105 written in scientific notation? Explain.

Evaluate



Explore Activity

Using Scientific NotationExercises 1–6

Example 1

Writing a Number in Scientific NotationExercises 1–6, 16–22, 24–27

Example 2

Writing a Number in Standard NotationExercises 7–14, 23–24, 25



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26 1 Recall of Information MP.7 Using Structure



29–30 3 Strategic Thinking MP.4 Modeling

31 3 Strategic Thinking MP.7 Using Structure

COMMONCORE

8.EE.3COMMONCORE

COMMONCORE 8.EE.3

Answers1. 642,000,000,000,000,000,000,000 kg

2. 4 × 107 krill

3. 9 × 100

4. 100,000; one hundred thousand

5. No; the first number is not more than 1 and less than 10.

43 Lesson 2.2

Work Area27. Explain the Error Polly’s parents’ car weighs about 3500 pounds. Samantha,

Esther, and Polly each wrote the weight of the car in scientific notation. Polly

wrote 35.0 × 10 2 , Samantha wrote 0.35 × 10 4 , and Esther wrote 3.5 × 10 4 .

a. Which of these girls, if any, is correct?

b. Explain the mistakes of those who got the question wrong.

28. Justify Reasoning If you were a biologist counting very large numbers of

cells as part of your research, give several reasons why you might prefer to

record your cell counts in scientific notation instead of standard notation.

29. Draw Conclusions Which measurement would be least likely to be

written in scientific notation: number of stars in a galaxy, number of

grains of sand on a beach, speed of a car, or population of a country?

Explain your reasoning.

30. Analyze Relationships Compare the two numbers to find which is

greater. Explain how you can compare them without writing them in

standard notation first.

4.5 × 10 6 2.1 × 10 8

31. Communicate Mathematical Ideas To determine whether a number is

written in scientific notation, what test can you apply to the first factor,

and what test can you apply to the second factor?


None of the girls has the correct answer.

Neither Polly nor Samantha moved the decimal point

to the correct place. Esther moved the decimal point to

the correct place, but wrote the wrong power of 10.

Sample answer: Scientific notation is shorter to write,

easier to read (you see how many zeros), used by scientists

everywhere, and easier to compare sizes of large numbers.

The speed of a car because it is likely to be less than 100.

2.1 × 10 8 is greater because the exponent 8 is greater

than the exponent 6.

Is the first factor greater than or equal to 1 and less than

10? Is the second factor a power of 10?

Unit 144

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Name Class Date

Paleontology Use the table for problems

16–21. Write the estimated weight of each

dinosaur in scientific notation.

Estimated Weight of Dinosaurs

Name Pounds

Argentinosaurus 220,000

Brachiosaurus 100,000

Apatosaurus 66,000

Diplodocus 50,000

Camarasaurus 40,000

Cetiosauriscus 19,850

16. Apatosaurus

17. Argentinosaurus

18. Brachiosaurus

19. Camarasaurus

20. Cetiosauriscus

21. Diplodocus

22. A single little brown bat can eat up to

1,000 mosquitoes in a single hour.

Express in scientific notation how many

mosquitoes a little brown bat might eat in

10.5 hours.

23. Multistep Samuel can type nearly

40 words per minute. Use this information

to find the number of hours it would take

him to type 2.6 × 10 5 words.

24. Entomology A tropical species of mite

named Archegozetes longisetosus is the

record holder for the strongest insect in

the world. It can lift up to 1.182 × 10 3 times

its own weight.

a. If you were as strong as this insect,

explain how you could find how many

pounds you could lift.

b. Complete the calculation to find how

much you could lift, in pounds, if you

were as strong as an Archegozetes

longisetosus mite. Express your answer

in both scientific notation and standard

notation.

25. During a discussion in science class, Sharon

learns that at birth an elephant weighs

around 230 pounds. In four herds of

elephants tracked by conservationists, about

20 calves were born during the summer. In

scientific notation, express approximately

how much the calves weighed all together.

26. Classifying Numbers Which of the

following numbers are written in scientific

notation?

0.641 × 10 3 9.999 × 10 4

2 × 10 1 4.38 × 5 10

2.2 Independent Practice8.EE.3COMMON

CORE

6.6 × 10 4 lb

2.2 × 10 5 lb

1 × 10 5 lb

4 × 10 4 lb

1.985 × 10 4 lb

5 × 10 4 lb

Write 1.182 × 10 3 in standard

notation, 1,182, and then

multiply by your weight.

Answers will vary. Sample

answer: 94,560 lb; 9.456 × 10 4

4.6 × 10 3 lb1.05 × 10 4 mosquitoes

108 1 _ 3

hours or 108 hours and

20 minutes 9.999 × 10 4 and 2 × 10 1

43Lesson 2.2

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Activity Ask students to consider the three numbers and compare each pair.

3.2 × 103 8.7 × 103 2.4 × 104

When comparing two numbers written in scientific notation, should students

compare the coefficient or the power of 10 first? the power of 10. Which pair of

numbers have the same power of 10? 3.2 × 103 and 8.7 × 103 When the power of 10 is

the same, how do you compare the numbers? by comparing the coefficient When

the power of 10 is different, do you need to compare the coefficient? no

Have students write their own set of rules to describe how to compare two numbers

written in scientific notation.


2.3L E S SON Scientific Notation with Negative

Powers of 10Engage

ESSENTIAL QUESTIONHow can you use scientific notation to express very small quantities? Write the small quantity as the product of a number greater than or equal to 1 and less than 10 times a power of 10 with a negative exponent.

Motivate the LessonAsk: Have you ever seen a large number of insects in flight? Monarch butterflies migrating

through Texas travel in huge numbers; as many as 22,410,000 butterflies in an area about

the size of two and a half football fields. You have written very large numbers like this in

scientific notation. What is a shorter way to write very small numbers, such as the weight of

a butterfly egg? A single Monarch egg may weigh as little as 0.00046 grams.

Explore

EXPLORE ACTIVITYFocus on PatternsDiscuss with students the relationship between moving a decimal point to the left and

dividing by 10. Ask them the following questions. What does moving a decimal point one

place to the left do to a number? divides the number by 10 What does dividing a number

by 10 do to an exponent with a base of 10? reduces the exponent by 1

Explain

EXAMPLE 1Questioning Strategies Mathematical Practices • How do you know whether to move a decimal point left or right? Move it left if the

number is greater than or equal to 10, right if the number is less than 1.

• Where does moving the decimal point 8 places right in 0.00000003 put the decimal point?

after the 3

Avoid Common ErrorsTo prevent writing the wrong exponent, students should first predict how the answer will

look. Think: 0.00000003 is a very small number. When you multiply 3.0 times a power of 10,

the exponent must be negative to equal this very small value.

YOUR TURNTalk About ItCheck for Understanding

Ask: With 0.0000829, after you write the first part as 8.29, have you made the

original number larger or smaller? larger So, if you multiply by a power of 10, what

should be the sign of the exponent? negative


ADDITIONAL EXAMPLE 1The weight of one of the smaller

species of butterflies was measured at

0.0007 ounces. Write the weight of this

butterfly in scientific notation.

7.0 × 10-4 ounces

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Animated MathPowers of 10

Students will explore and visualize

powers of ten through virtual

manipulatives.

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MP.2 Reasoning

COMMONCORE

COMMONCORE

45 Lesson 2.3


and Intervention

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Writing a Number in Scientific NotationTo write a number less than 1 in scientific notation, move the decimal point right and use a negative exponent.

Writing Small Quantities in Scientific Notation

When the number is between 0 and 1, use a negative exponent.

0.0 7 8 3 = 7.83 × 10 -2 The decimal point

moves 2 places to the right.

The average size of an atom is about 0.00000003 centimeter across. Write the average size of an atom in scientific notation.

Move the decimal point as many places as necessary to find a number that is greater than or equal to 1 and less than 10.

Place the decimal point. 3.0

Count the number of places you moved the decimal point. 8

Multiply 3.0 times a power of 10. 3.0 × 10

The average size of an atom in scientific notation is 3.0 × 10 -8 .

Reflect3. Critical Thinking When you write a number that is less than 1 in

scientific notation, how does the power of 10 differ from when you write a number greater than 1 in scientific notation?

EXAMPLE 1

STEP 1

STEP 2

STEP 3

Since 0.00000003 is less than 1, you moved the decimal point to the right and the exponent on 10 is negative.

4. 0.0000829 5. 0.000000302

6. A typical red blood cell in human blood has a diameter of approximately 0.000007 meter. Write this diameter

in scientific notation.

Write each number in scientific notation.

YOUR TURN

8.EE.3COMMONCORE

-8

The exponent is negative instead of positive.

8.29 × 10 -5 3.02 × 10 -7

7 × 10 -6 m

46 Unit 1

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EXPLORE ACTIVITY

How can you use scientific notation to express very small quantities?

ESSENTIAL QUESTION

L E S S O N

2.3Scientific Notation with Negative Powers of 10

Negative Powers of 10You can use what you know about writing very large numbers in scientific notation to write very small numbers in scientific notation.

A typical human hair has a diameter of 0.000025 meter. Write this number in scientific notation.

Notice how the decimal point moves in the list below. Complete the list.

2.345 × 10 0 = 2.3 4 5 2.345 × 10 0 = 2.3 4 5

2.345 × 10 1 = 2 3.4 5 2.345 × 10 -1 = 0.2 3 4 5

2.345 × 10 2 = 2 3 4.5 2.345 × 10 -2 = 0.0 2 3 4 5

2.345 × 10 = 2 3 4 5. 2.345 × 10 = 0.0 0 2 3 4 5

Move the decimal point in 0.000025 to the right as many places as necessary to find a number that is greater than or equal to 1 and

less than 10. What number did you find?

Divide 0.000025 by your answer to B .

Write your answer as a power of 10.

Combine your answers to B and C to represent 0.000025 in

scientific notation.

Reflect1. When you move the decimal point, how can you know whether you are

increasing or decreasing the number?

2. Explain how the two steps of moving the decimal and multiplying by a power of 10 leave the value of the original number unchanged.

A

B

C

D

8.EE.3

Use numbers expressed in the form of a single digit times an integer power of 10 to estimate very large or very small quantities, ….

COMMONCORE

8.EE.3COMMONCORE

It moves one place to the right with each increasing power of 10.

It moves one place to the left with each decreasing power of 10.

-3

0.00001

2.5

0.000025 = 2.5 × 10 -5

10 -5

Moving left is a decrease; moving right is an increase.

One increases and the other decreases the number.

3

45Lesson 2.3

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Math BackgroundScientific notation is used to represent very small numbers in an efficient way which is easy to write and easy to read. Later, students will use this form as they compare and combine very large and very small numbers.

There may be confusion about whether “a very small” number might indicate a number that is negative with a large absolute value, such as -999,999 (a debt, for instance). In this context “very small” indicates a number that is very close to zero (has a very small absolute value).


This lesson provides an opportunity to address this Mathematical Practices standard. It calls for students to represent a situation symbolically. Students write very small numbers in two forms of symbolic representation: standard form and scientific notation. Students also use mathemati-cal language to express the processes they use to convert from one representation to the other.

Scientific Notation with Negative Powers of 10 46


EXAMPLE 2Questioning Strategies Mathematical Practices • Is the value of 2.33 × 10-6 greater than or less than 2.33? less Which direction does the

decimal point move when a number is decreasing? to the left

• How could you check your answer? Convert 0.00000233 back to scientific notation by

moving the decimal right 6 places and writing the number times 10-6.

Engage with the WhiteboardIn Exercise 7, rewrite 0.9 × 10-5 in standard notation, then convert it to scientific

notation. Draw curved arrows as you move the decimal point.

YOUR TURNFocus on Communication Mathematical Practices • Compare the number of places that a decimal point moves with the number of zeros in

the resulting standard notation form. Discuss with students, in Exercise 9 for example, why

the decimal moves 6 places, but there are only 5 zeros in the decimal part of the answer.

Help them to communicate that one of the 6 factors of 10-1 moves the decimal point to

the left of the digit 1. The remaining 5 factors of 10-1 move the decimal point leftward

5 spaces, and those spaces are filled with zeros.

Elaborate


Ask: How can you write a number that is very close to zero, such as 0.0000078 in

scientific notation? Count the number of places that the decimal point would have

to move to the right to make a number greater than or equal to 1 and less than 10 for the

first factor. The second factor is a power of 10 with the exponent equal to -6 to represent

the number of places leftward you would need to move the decimal back to its original

position.


For Exercises 1–14, have students draw and count curved arrows to show moving

the decimal point the correct number of places.

Integrating Language Arts ELL

Encourage a broad class discussion on the Math Talk. English learners will benefit from

hearing and participating in classroom discussions.

Avoid Common Errors Exercises 1–6 Remind students to avoid counting errors as they move the decimal point

by drawing arrows and counting a second time to check.


ADDITIONAL EXAMPLE 2A single atom of oxygen has a mass of

2.66 × 10-23 grams. Write 2.66 × 10-23

in standard notation.

0.0000000000000000000000266

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47 Lesson 2.3

Guided Practice

Write each number in scientific notation. (Explore Activity and Example 1)

1. 0.000487 Hint: Move the decimal right 4 places.

2. 0.000028 Hint: Move the decimal right 5 places.

3. 0.000059 4. 0.0417

5. Picoplankton can be as small as 0.00002 centimeter.

6. The average mass of a grain of sand on a beach is about 0.000015 gram.

Write each number in standard notation. (Example 2)

7. 2 × 10 -5 Hint: Move the decimal left 5 places.

8. 3.582 × 10 -6 Hint: Move the decimal left 6 places.

9. 8.3 × 10 -4 10. 2.97 × 10 -2

11. 9.06 × 10 -5 12. 4 × 10 -5

13. The average length of a dust mite is approximately 0.0001 meter. Write this number in scientific notation. (Example 1)

14. The mass of a proton is about 1.7 × 10 -24 gram. Write this number in standard notation. (Example 2)

15. Describe how to write 0.0000672 in scientific notation.


4.87 × 10 -4

5.9 × 10 -5

2 × 10 -5

1 × 10 -4

2.8 × 10 -5

1.5 × 10 -5

4.17 × 10 -2

0.00002

0.00083

0.0000906 0.00004

0.0297

0.000003582

0.0000000000000000000000017

Move the decimal point 5 places right to find 6.72, a number greater than

or equal to 1 and less than 10. Then multiply 6.72 by 10-5, using a negative

exponent on 10 that equals the number of places you moved the decimal.

48 Unit 1

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Writing a Number in Standard NotationTo translate between scientific notation and standard notation with very small numbers, you can move the decimal point the number of places indicated by the exponent on the power of 10. When the exponent is negative, move the decimal point to the left.

Platelets are one component of human blood. A typical platelet has a diameter of approximately 2.33 × 10-6 meter. Write 2.33 × 10-6 in standard notation.

Use the exponent of the power of 10 to see 6 placeshow many places to move the decimal point.

Place the decimal point. Since you are going to 0.0 0 0 0 0 2 3 3write a number less than 2.33, move the decimal point to the left. Add placeholder zeros if necessary.

The number 2.33 × 10-6 in standard notation is 0.00000233.

Reflect7. Justify Reasoning Explain whether 0.9 × 10 -5 is written in scientific

notation. If not, write the number correctly in scientific notation.

8. Which number is larger, 2 × 1 0 -3 or 3 × 1 0 -2 ? Explain.

EXAMPLEXAMPLE 2

STEP 1

STEP 2

9. 1.045 × 10 -6 10. 9.9 × 10 -5

11. Jeremy measured the length of an ant as 1 × 10-2 meter. Write this length in standard notation.

Write each number in standard notation.

YOUR TURN

Describe the two factors that multiply together to form a number written in



8.EE.3COMMONCORE

No, because the first factor is less than 1. The decimal

point should move to the right and the exponent

should decrease by 1: 9 × 1 0 -6

3 × 1 0 -2 because its exponent is greater.

0.000001045

0.01 m

0.000099

The first factor is a number greater than or equal to 1 and less than 10, and the second factor is a power of 10.

47Lesson 2.3

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Cognitive StrategiesSuggest that students, before writing anything, think about where the number would be on a number line.

When a number that is very close to zero is written in scientific notation, the exponent will be a negative number.

A number that is very close to zero when written in standard notation will have zeros as placeholders after the decimal point.

Number SenseHave students use a graphing calculator and type 0.000000000099 and press ENTER. Ask what the answer in the display, 9.9E–11, means. 9.9 × 10-11 This number has 10 zeros after the decimal point. Have students compare what happens when they enter the number 0.0099 with what happens when they enter 0.00099. Ask them to draw some conclusions, based on their investigations, about the rules in the programming software for their calculator.

Additional ResourcesDifferentiated Instruction includes: • Reading Strategies • Success for English Learners ELL

• Reteach • Challenge PRE-AP



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2.3 LESSON QUIZ

1. The weight of an ant is about 1.7 × 10-5 pounds. Write 1.7 × 10-5

in standard notation.

2. A bee sting delivers about 0.00005 grams of venom. Write this number in scientific notation.

3. Write 0.77 in scientific notation.

4. Write 1.0 × 10-3 in standardnotation and in words.

5. Is 0.1 × 10-4 written in scientificnotation? Explain.

Evaluate



Explore Activity

Negative Powers of 10Exercises 1–6

Example 1

Writing a Number in Scientific NotationExercises 1–6, 13, 16–21, 23, 26, 28, 30, 32

Example 2

Writing a Number in Standard NotationExercises 7–12, 14, 29, 31, 33



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23 2 Skills/Concepts MP.2 Reasoning




35 3 Strategic Thinking MP.5 Using Tools

36–37 3 Strategic Thinking MP.3 Logic

COMMONCORE

8.EE.3COMMONCORE

COMMONCORE 8.EE.3

Answers1. 0.000017

2. 5 × 10-5

3. 7.7 × 10-1

4. 0.001; one thousandth

5. No; the first number is not more than 1 and less than 10.

49 Lesson 2.3

Work Area

Physical Science The table shows the length of the radii of several very

small or very large items. Complete the table.

28.

29.

30.

31.

32.

33.

34. List the items in the table in order from the smallest to the largest.

35. Analyze Relationships Write the following diameters from least to greatest.

1.5 × 10 -2 m 1.2 × 10 2 m 5.85 × 10 -3 m 2.3 × 10 -2 m 9.6 × 10 -1 m

36. Critique Reasoning Jerod’s friend Al had the following

homework problem:

Express 5.6 × 10 -7 in standard form.

Al wrote 56,000,000. How can Jerod explain Al’s error and how to correct it?

37. Make a Conjecture Two numbers are written in scientific notation.

The number with a positive exponent is divided by the number with a

negative exponent. Describe the result. Explain your answer.


Item Radius in Meters(Standard Notation)

Radius in Meters(Scientific Notation)

The Moon 1,740,000

Atom of silver 1.25 × 1 0 -10

Atlantic wolfish egg 0.0028

Jupiter 7.149 × 1 0 7

Atom of aluminum 0.000000000182

Mars 3.397 × 1 0 6

5.85 × 10 -3 m, 1.5 × 10 -2 m, 2.3 × 10 -2 m,

9.6 × 10 -1 m, 1.2 × 10 2 m

0.000000000125

1.74 ×1 0 6

2.8 ×1 0 -3

1.82 ×1 0 -10

71,490,000

3,397,000

The result will be greater than the number with the

positive exponent because the divisor is less than 1.

Atom of silver, atom of aluminum, Atlantic wolfish egg,

the Moon, Mars, Jupiter

Al treated the exponent as if it were positive instead

of negative and moved the decimal in the wrong

direction. The answer should be 0.00000056.

50 Unit 1

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Name Class Date


Use the table for problems 16–21. Write the

diameter of the fibers in scientific notation.

Average Diameter of Natural Fibers

Animal Fiber Diameter (cm)

Vicuña 0.0008

Angora rabbit 0.0013

Alpaca 0.00277

Angora goat 0.0045

Llama 0.0035

Orb web spider 0.015

16. Alpaca

17. Angora rabbit

18. Llama

19. Angora goat

20. Orb web spider

21. Vicuña

22. Make a Conjecture Which measurement

would be least likely to be written in scientific

notation: the thickness of a dog hair, the

radius of a period on this page, the ounces in

a cup of milk? Explain your reasoning.

23. Multiple Representations Convert the

length 7 centimeters to meters. Compare

the numerical values when both numbers

are written in scientific notation.

24. Draw Conclusions A graphing calculator

displays 1.89 × 10 12 as 1.89E12. How do you

think it would display 1.89 × 10 -12 ? What

does the E stand for?

25. Communicate Mathematical Ideas When

a number is written in scientific notation,

how can you tell right away whether or not

it is greater than or equal to 1?

26. The volume of a drop of a certain liquid is

0.000047 liter. Write the volume of the drop

of liquid in scientific notation.

27. Justify Reasoning If you were asked to

express the weight in ounces of a ladybug

in scientific notation, would the exponent

of the 10 be positive or negative? Justify

your response.

8.EE.3COMMONCORE

1.3 × 10 -3 cm

3.5 × 10 -3 cm

4.7 × 10 -5 L

4.5 × 10 -3 cm

1.5 × 10 -2 cm

8 × 10 -4 cm

2.77 × 10 -3 cm

The ounces in a cup of milk; it is

more than 1 but less than 10.

1.89E–12; the exponent on 10

7 cm = 0.07 m , 7 cm = 7 × 10 0 cm;

0.07 m = 7 × 10 -2 m. The first

factors are the same; the

exponents differ by 2.

If the exponent on 10 is

nonnegative, the number is

greater than or equal to 1.

Negative, because a ladybug

would weigh less than 1 ounce.

49Lesson 2.3

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Activity Have students write 0.008 and 800 in scientific notation. Notice that both of

these numbers have two zeros that are essential (since 0.008 can be written as .008).

Do both exponents have the same absolute value when the numbers are written in

scientific notation? Explain.

Sample answer: No; 0.008 = 8.0 × 10-3 and 800 = 8.0 × 10 2 . When you move the

decimal to the left, the 8 counts as one place. When you move to the right, the 8 is

not included in the count of the number of moves.


Engage

ESSENTIAL QUESTIONHow do you add, subtract, multiply, and divide using scientific notation? Sample answer: To add or subtract, rewrite the numbers to the same power of 10, add or subtract the multipliers, and rewrite in scientific notation. To multiply or divide, multiply or divide the multipliers, multiply or divide the powers of 10, and rewrite the answer in scientific notation.

Motivate the LessonAsk: How do you think you could add or subtract numbers in scientific notation without

rewriting them in standard notation? What about if you wanted to multiply or divide?

ExploreExplain to students that operations with scientific notation can be handled either by

converting the numbers into standard notation or by keeping them in scientific notation

and rewriting the powers of 10 using the properties of exponents. Discuss with students the

advantages and disadvantages of each method.

Explain

EXAMPLE 1Questioning Strategies Mathematical Practices • In which method would you expect more errors to be made? Justify your answer. When

the exponents are very large or small, Method 2 could introduce more errors because it

would be easy to write an incorrect number of 0s when converting between notations.

Focus on Critical ThinkingMake sure students understand why the table figure for Canada has been written with a

different exponent in Method 1, Step 1.

YOUR TURNAvoid Common ErrorsStudents may compare the multipliers and think Canada has a greater population than

Mexico. Remind students that the power of 10 must be the same in order to compare and

subtract these numbers.

EXAMPLE 2Questioning Strategies Mathematical Practices • How is dividing two values with scientific notation a matter of solving two division

problems? The multipliers are divided and the powers of 10 are divided.

Avoid Common ErrorsMake sure students understand that the powers of 10 do not need to be the same when

multiplying or dividing.

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ADDITIONAL EXAMPLE 1Use the population table in Example 1

on the student page. How many more

people live in the United States than in

Canada? 2.762 × 10 8 people

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ADDITIONAL EXAMPLE 2When Neptune makes an orbit around

the Sun, it travels about 2.82 × 10 10 km.

Neptune travels at a rate of about

470,000 km/day. How many days does

one orbit of Neptune take? Write your

answer in scientific notation.

about 6.0 × 10 4 days

2.4L E S SON

Operations with Scientific Notation

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Perform operations with numbers expressed in scientific notation, including problems where both decimal and scientific notation are used. Use scientific notation and choose units of appropriate size for measurements of very large or very small quantities … . Interpret scientific notation that has been generated by technology.


MP.1 Problem Solving

COMMONCORE

COMMONCORE

51 Lesson 2.4


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1. Using the population table above, how many more people live in Mexico

than in Canada? Write your answer in scientific notation.

YOUR TURN

Multiplying and Dividing with Scientific NotationNumbers in scientific notation can be multiplied and divided directly by using

properties of exponents.

When the Sun makes an orbit around the center of the Milky Way, it travels

2.025 × 10 14 kilometers. The orbit takes 225 million years. At what rate

does the Sun travel? Write your answer in scientific notation.

Analyze Information

The answer is the number of kilometers per year that the Sun travels

around the Milky Way.

Formulate a Plan

Set up a division problem using Rate = Distance _______

Time to represent the situation.

Justify and EvaluateSolve

Substitute the values from the problem into the Rate formula.

Write the expression for rate with years in scientific notation.

Find the quotient by dividing the decimals and using the laws

of exponents.

2.025 ÷ 2.25 = 0.9

Combine the answers to write the rate in scientific notation.

Rate = 0.9 × 10 6 = 9.0 × 10 5 km per year

Justify and Evaluate

Check your answer using multiplication.

900,000 × 225,000,000 = 202,500,000,000,000, or 2.025 × 10 14 .

The answer is correct.

EXAMPLE 2 ProblemSolving

STEP 1

Rate = 2.025 × 10 14 kilometers _________________

225,000,000 years

STEP 2

Rate = 2.025 × 10 14 kilometers _________________

2.25 × 10 8 years

STEP 3

10 14 ____

10 8 = 10 14 - 8 = 10 6

STEP 4

Could you write

2.025 × 10 14 in standard notation to do the division?

Would this be a good way to solve the problem?


8.EE.4COMMONCORE

Divide the multipliers.

Divide the powers of 10.

225 million = 2.25 × 10 8

7.62 × 10 7 more people

Yes; no; 2.025 × 10 14 in standard notation is 202,500,000,000,000. Dividing this by 225,000,000 is more complicated than doing the division using scientific notation.

Unit 152

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ESSENTIAL QUESTION

Adding and Subtracting with Scientific NotationNumbers in scientific notation can be added and subtracted, either directly or

by rewriting them in standard form.

The table below shows the population of the three largest countries in

North America in 2011. Find the total population of these countries.

Country United States Canada Mexico

Population 3.1 × 10 8 3.38 × 10 7 1.1 × 10 8

Method 1:

First, write each population with the same power of 10.

United States: 3.1 × 10 8

Canada: 0.338 × 10 8

Mexico: 1.1 × 10 8

Add the multipliers for each population.

3.1 + 0.338 + 1.1 = 4.538

Write the final answer in scientific notation: 4.538 × 10 8 .

Method 2:

First, write each number in standard notation.

United States: 310,000,000

Canada: 33,800,000

Mexico: 110,000,000

Find the sum of the numbers in standard notation.

310,000,000 + 33,800,000 + 110,000,000 = 453,800,000

Write the answer in scientific notation: 4.538 × 10 8 .

EXAMPLEXAMPLE 1

STEP 1

STEP 2

STEP 3

STEP 1

STEP 2

STEP 3

How do you add, subtract, multiply, and divide usingscientific notation?

L E S S ON

2.4Operations with Scientific Notation

8.EE.4COMMONCORE

8.EE.4

Perform operations … in scientific notation. … choose units of appropriate size for measurements … . Interpret scientific notation … generated by technology.

COMMONCORE

51Lesson 2.4

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Math BackgroundThe rules of exponents are used when perform-

ing operations with numbers in scientific

notation:

1. Product Rule: a m · a n = a m + n , for a ≠ 0.

Example: 4 3 · 4 –1 = 4 (3 + –1) = 4 2

2. Quotient Rule: a m ÷ a n = a m - n , for a ≠ 0.

Example: 4 3 ÷ 4 -1 = 4 3-(-1) = 4 4

3. Power Rule: ( a m ) n = a m × n , for a ≠ 0.

Example: ( 4 3 ) -1

= 4 3 · (-1) = 4 –3



this Mathematical Practice standard. It calls for

students to make sense of problems and

persevere in solving them. Example 2 uses a

four-step problem-solving process to determine

the speed of the sun as it moves in the Milky Way.

Students analyze the information, formulate a

plan, solve the problem, and justify and evaluate

the solution.

Operations with Scientific Notation 52

YOUR TURNEngage with the Whiteboard

For Exercises 2 and 3, have volunteers demonstrate the solution method while they

explain how to multiply or divide the multipliers and the powers of 10.

EXAMPLE 3Questioning Strategies Mathematical Practices • How does the situation inform you as to which values you will be operating on and which

operation(s) you will use? The phrase “total area of these three continents” tells you to add

the areas of Asia, Africa, and Europe.

• How can you access the EE feature on a calculator when it is printed next to a key but not

on a key? You must first enter the “2nd” or “shift” key, then the key next to EE.

Focus on Technology Mathematical PracticesScientific calculators handle scientific notation in a variety of ways. Before beginning

Example 3, review with students how their calculators display very large and very small

numbers and how best to input numbers in scientific notation. Note that e x and e are not

used for entering scientific notation.

YOUR TURNFocus on Modeling Mathematical PracticesIn Exercises 4–6, expressions that model scientific notation are to be rewritten in calculator

notation and vice versa for Exercises 7–9. Ensure that students can locate the multiplier, the

base, and the exponent in each.

Elaborate


Ask: When must the powers of 10 be the same when operating on numbers

written in scientific notation? If the expressions are to be added or subtracted, the

powers of 10 must be the same.


For Exercises 1–2, have volunteers explain the process of arriving at the correct value

as they complete the write-in boxes for each exercise.

Avoid Common ErrorsExercises 5–8 Remind students that the multipliers are operated on separately from the

powers of 10.

Exercises 9–11 Remind students that the exponent is not written as a superscript to E but

rather the same size as and to the right of E.


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ADDITIONAL EXAMPLE 3Use the table in Example 3 on the

student page. What is the difference

between Asia's area and Europe’s area?

Use your calculator to find the answer.

3.36 × 10 13 square meters

CC

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53 Lesson 2.4

Add or subtract. Write your answer in scientific notation. (Example 1)

1. 4.2 × 10 6 + 2.25 × 10 5 + 2.8 × 10 6

4.2 × 10 6 + × 10 + 2.8 × 10 6

4.2 + +

× 10

2. 8.5 × 10 3 - 5.3 × 10 3 - 1.0 × 10 2

8.5 × 10 3 - 5.3 × 10 3 - × 10

- -

× 10

3. 1.25 × 10 2 + 0.50 × 10 2 + 3.25 × 10 2

4. 6.2 × 10 5 - 2.6 × 10 4 - 1.9 × 10 2

Multiply or divide. Write your answer in scientific notation. (Example 2)

5. ( 1.8 × 10 9 ) ( 6.7 × 10 12 ) 6. 3.46 × 10 17 _________

2 × 10 9

7. ( 5 × 10 12 ) ( 3.38 × 10 6 ) 8. 8.4 × 10 21 ________

4.2 × 10 14

Write each number using calculator notation. (Example 3)

9. 3.6 × 10 11 10. 7.25 × 10 -5 11. 8 × 10 -1

12. 7.6E-4 13. 1.2E16 14. 9E1

Write each number using scientific notation. (Example 3)

15. How do you add, subtract, multiply, and divide numbers written in

scientific notation?


Guided Practice

0.225 0.10

0.225 0.10

7.2256

2.8

6 3

33.1

5.38.5

5 × 10 2

1.206 × 10 22 1.73 × 10 8

2 × 10 7 1.69 × 10 19

5.9381 × 10 5

8E-17.25E-53.6E11

7.6 × 10 -4 1.2 × 10 16 9 × 10 1

To add or subtract, rewrite the numbers to the same power of 10,

add or subtract the multipliers, and rewrite the answer in scientific

notation. To multiply or divide, multiply or divide the multipliers, use

the rules of exponents to multiply or divide the powers of 10, and

rewrite the answer in scientific notation.

Unit 154

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2. Light travels at a speed of 1.86 × 10 5 miles per second. It takes light

from the Sun about 4.8 × 10 3 seconds to reach Saturn. Find the

approximate distance from the Sun to Saturn. Write your answer


3. Light travels at the speed of 1.17 × 10 7 miles per minute. Pluto’s

average distance from the Sun is 3,670,000,000 miles. On average, how

long does it take sunlight to reach Pluto? Write your answer in scientific

notation.

YOUR TURN

Scientific Notation on a CalculatorOn many scientific calculators, you can enter numbers in scientific notation

by using a function labeled “ee” or “EE”. Usually, the letter “E” takes the place of

“×10”. So, the number 4.1 × 10 9 would appear as 4.1E9 on the calculator.

The table shows the approximate areas for three continents given

in square meters. What is the total area of these three continents?

Write the answer in scientific notation using more appropriate units.

Continent Asia Africa Europe

Area ( m 2 ) 4.4 × 10 13 3.02 × 10 13 1.04 × 10 13

Find 4.4 × 10 13 + 3.02 × 10 13 + 1.04 × 10 13 .

Enter 4.4E13 + 3.02E13 + 1.04E13 on your calculator.

Write the results from your calculator: 8.46E13.

Write this number in scientific notation: 8.46 × 10 13 m 2 .

Square kilometers is more appropriate: 8.46 × 10 7 km 2 .

EXAMPLEXAMPLE 3

Write each number using calculator notation.

YOUR TURN

4. 7.5 × 10 5 5. 3 × 10 -7 6. 2.7 × 10 13

7. 4.5E-1 8. 5.6E12 9. 6.98E-8

Write each number using scientific notation.

8.EE.4COMMONCORE

Because 1 km = 1,000 m,1 km2 = 1,0002 m2, or 106 m2. Divide by 10 6.

3.14 × 10 2 minutes

8.928 × 10 8 miles

7.5E5 3E-7 2.7E13

6.98 × 10 -8 5.6 × 10 12 4.5 × 10 -1

53Lesson 2.4

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Cooperative LearningHave pairs of students solve a list of four or five

addition and subtraction problems similar to

34,000,000 × 18,000,000 and 0.0000000015 ÷ 0.0006. Have one student solve a problem using

standard notation and the other student solve

the same problem using scientific notation.

Have the students compare answers and discuss

which method is faster for that particular

problem. Then switch roles to solve the next

problem.

Visual CuesHave students solve multiplication and division

of scientific notation by splitting the factors into

separate problems. Students can circle the

multipliers in red and the powers of 10 in blue.

4.2 × 1 0 7 ________

1.4 × 1 0 5

3.0 1 0 2

3.0 × 1 0 2


• Reading Strategies

• Success for English Learners ELL

• Reteach

• Challenge PRE-AP



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Personal

Online homework


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Evaluate



Example 1

Adding and Subtracting with Scientific NotationExercises 1–4, 18–20, 22

Example 2

Multiplying and Dividing with Scientific Notation

Exercises 5–8, 16–17, 21, 23–27

Example 3

Scientific Notation on a CalculatorExercises 9–14


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2.4 LESSON QUIZ

Add or subtract. Write your answer


1. 3.2 × 10 -2 + 3.2 × 10 -4 + 2.8 × 1 0 -2

2. 5.2 × 10 4 - 1.5 × 10 2 - 4.6 × 10 4

Multiply or divide. Write your

answer in scientific notation.

3. (1.25 × 10 4 )(4 × 10 4 )

4. 4.5 × 10 28

_ 3 × 10 15

5. Write 7.18 × 10 -2 using calculator notation.

6. Write 2.898E15 using scientific notation.



CC

COMMONCORE 8.EE.4

8.EE.4COMMONCORE



28 3 Strategic Thinking MP.6 Precision


30 3 Strategic Thinking MP.6 Precision

COMMONCORE

Exercises 18–21 combine concepts from the Common Core cluster

“Work with radicals and integer exponents.”Answers1. 6.032 × 10 -2

2. 5.85 × 10 3

3. 5 × 10 8

4. 1.5 × 10 13

5. 7.18E-2

6. 2.898 × 10 15

55 Lesson 2.4

Work Area

26. Courtney takes 2.4 × 10 4 steps during her a long-distance run. Each step covers an average of 810 mm. What total distance (in mm) did Courtney cover during her run? Write your answer in scientific notation. Then convert the distance to the more appropriate unit kilometers. Write that answer in standard form.

27. Social Studies The U.S. public debt as of October 2010 was $9.06 × 10 12 . What was the average U.S. public debt per American if the population in 2010 was 3.08 × 10 8 people?

28. Communicate Mathematical Ideas How is multiplying and dividing numbers in scientific notation different from adding and subtracting numbers in scientific notation?

29. Explain the Error A student found the product of 8 × 10 6 and 5 × 10 9 to be 4 × 10 15 . What is the error? What is the correct product?

30. Communicate Mathematical Ideas Describe a procedure that can be

used to simplify ( 4.87 × 10 12 ) - ( 7 × 10 10 )

___________________ ( 3 × 10 7 ) + ( 6.1 × 10 8 )

. Write the expression in scientific

notation in simplified form.


1.944 × 10 7 mm; 19.44 km

$2.94 × 10 4 , or $29,400 per person

You can add or subtract numbers written in scientific

notation only if their powers of 10 are the same. You can

multiply and divide numbers written in scientific notation

that have different powers. The laws of exponents are

used to combine the powers.

First, simplify the numerator by rewriting both numbers to the

same power of 10 ( 10 10 ) and subtracting to get 480 × 10 10 or

4.8 × 10 12 . Then, simplify the denominator by rewriting both

numbers to the same power of 10 ( 10 7 ) and adding to get

64 × 10 7 or 6.4 × 10 8 . Finally, divide the multipliers (4.8 ÷ 6.4) to

get 0.75, use the division rule for exponents ( 10 12 ____ 10 8

) to get 10 4 ,

and rewrite 0.75 × 10 4 in scientific notation as 7.5 × 10 3 .

The student is off by a power of ten. The correct product

is 40 × 10 15 or 4.0 × 10 16 .

Unit 156

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Name Class Date


16. An adult blue whale can eat 4.0 × 10 7 krill in a day. At that rate, how many krill can an adult blue whale eat in 3.65 × 10 2 days?

17. A newborn baby has about 26,000,000,000 cells. An adult has about 4.94 × 10 13 cells. How many times as many cells does an adult have than a newborn? Write your answer in scientific notation.

Represent Real-World Problems The table shows the number of tons of waste generated and recovered (recycled) in 2010.

Paper Glass Plastics

Tons generated 7.131 × 10 7 1.153 × 10 7 3.104 × 10 7

Tons recovered 4.457 × 10 7 0.313 × 10 7 0.255 × 10 7

18. What is the total amount of paper, glass, and plastic waste generated?

19. What is the total amount of paper, glass, and plastic waste recovered?

20. What is the total amount of paper, glass, and plastic waste not recovered?

21. Which type of waste has the lowest recovery ratio?

Social Studies The table shows the approximate populations of three countries.

Country China France Australia

Population 1.3 × 10 9 6.48 × 10 7 2.15 × 10 7

22. How many more people live in France than in Australia?

23. The area of Australia is 2.95 × 10 6 square miles. What is the approximate average number of people per square mile in Australia?

24. How many times greater is the population of China than the population of France? Write your answer in standard notation.

25. Mia is 7.01568 × 10 6 minutes old. Convert her age to more appropriate units using years, months, and days. Assume each month to have 30.5 days.

8.EE.4COMMONCORE

1.46 × 10 10 krill

about 1.9 × 10 3 as many

11.388 × 10 7 , or in scientific

notation 1.1388 × 10 8 tons

5.025 × 10 7 tons

6.363 × 10 7 tons

Plastics

4.33 × 10 7 more people

about 7 people per square mile

13 years, 3 months, 22.5 days

about 20.1 times as great

55Lesson 2.4

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Activity Assume 1 ≤ a < 10. When the quotient of (6 × 10 6 ) ÷ (a × 10 2 ) is written in scientific notation, the power of 10 is 4. What are the possible values of a? Justify your answer.

1 ≤ a ≤ 6; Since (6 × 10 6 ) ÷ (a × 10 2 ) = (6 ÷ a) × 10 4 , if a is any number between 6 and 10, the value of 6 ÷ a would be less than 1. You could not write this expression in scientific notation with an exponent of 4.



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ReadyMODULE QUIZ

2.1 Integer ExponentsFind the value of each power.

1. 3−4 2. 35 0 3. 44

Use the properties of exponents to write an equivalent expression.

4. 83 · 87 5. 126

___ 122 6. (103)5

2.2 Scientific Notation with Positive Powers of 10Convert each number to scientific notation or standard notation.

7. 2,000 8. 91,007,500

9. 1.0395 × 109 10. 4 × 102

2.3 Scientific Notation with Negative Powers of 10Convert each number to scientific notation or standard notation.

11. 0.02 12. 0.000701

13. 8.9 × 10-5 14. 4.41 × 10-2

2.4 Operations with Scientific NotationPerform the operation. Write your answer in scientific notation.

15. 7 × 106 - 5.3 × 106 16. 3.4 × 104 + 7.1 × 105

17. (2 × 104 )(5.4 × 106) 18. 7.86 × 109

________ 3 × 104

19. Neptune’s average distance from the Sun is 4.503 × 109 km. Mercury’s average

distance from the Sun is 5.791 × 107 km. About how many times farther from

the Sun is Neptune than Mercury? Write your answer in scientific notation.

20. How is scientific notation used in the real world?

ESSENTIAL QUESTION

1 __ 81

1,039,500,000

9.10075 × 107

2 × 10−2

2 × 103

0.000089

1.7 × 106

1.08 × 1011 2.62 × 105

7.44 × 105

400

7.01 × 10−4

0.0441

810 124

1 256

1015

about 7.776 × 101 times farther

Sample answer: Very large numbers, such as distances in

space, and very small numbers, such as the sizes of atomic

particles, can be written in scientific notation.

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Module 2

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Intervention Enrichment

Access Ready to Go On? assessment online, and receive

instant scoring, feedback, and customized intervention

or enrichment.

Online and Print Resources


• Reteach worksheets

• Reading Strategies ELL

• Success for English

Learners ELL


• Challenge worksheets

PRE-AP

Extend the Math PRE-AP

Lesson Activities in TE

Additional ResourcesAssessment Resources includes

• Leveled Module Quizzes

Ready to Go On?

Assess MasteryUse the assessment on this page to determine if students have

mastered the concepts and standards covered in this module.


Lessons Exercises Common Core Standards

2.1 1–6 8.EE.1

2.2 7–10 8.EE.3

2.3 11–14 8.EE.3

2.4 15–19 8.EE.4

57 Unit 1 Module 2




MODULE 2 MIXED REVIEW

Selected Response

1. Which of the following is equivalent to 6–3?

A 216 C - 1 ___

216

B 1 ___

216 D -216

2. About 786,700,000 passengers traveled by

plane in the United States in 2010. What is

this number written in scientific notation?

A 7,867 × 105 passengers

B 7.867 × 102 passengers

C 7.867 × 108 passengers

D 7.867 × 109 passengers

3. In 2011, the population of Mali was about

1.584 × 107 people. What is this number

written in standard notation?

A 1.584 people

B 1,584 people

C 15,840,000 people

D 158,400,000 people

4. The square root of a number is between

7 and 8. Which could be the number?

A 72 C 51

B 83 D 66

5. Each entry-level account executive in a

large company makes an annual salary of

$3.48 × 104. If there are 5.2 × 102 account

executives in the company, how much do

they make in all?

A $6.69 × 101

B $3.428 × 104

C $3.532 × 104

D $1.8096 × 107

6. Place the numbers in order from least to

greatest.

0.24, 4 × 10-2, 0.042, 2 × 10-4, 0.004

A 2 × 10-4, 4 × 10-2, 0.004, 0.042, 0.24

B 0.004, 2 × 10-4, 0.042, 4 × 10-2, 0.24

C 0.004, 2 × 10-4, 4 × 10-2, 0.042, 0.24

D 2 × 10-4, 0.004, 4 × 10-2, 0.042, 0.24

7. Guillermo is 5 5 _ 6 feet tall. What is this

number of feet written as a decimal?

A 5.7 feet C 5.83 feet

B 5. _

7 feet D 5.8 _

3 feet

8. A human hair has a width of about

6.5 × 10-5 meter. What is this width written

in standard notation?

A 0.00000065 meter

B 0.0000065 meter

C 0.000065 meter

D 0.00065 meter

Mini-Task

9. Consider the following numbers: 7000, 700,

70, 0.7, 0.07, 0.007

a. Write the numbers in scientific notation.

b. Look for a pattern in the given list and

the list in scientific notation. Which

numbers are missing from the lists?

c. Make a conjecture about the missing

numbers.

Assessment Readiness

7 × 103, 7 × 102, 7 × 101,

7 × 10−1, 7 × 10−2, 7 × 10−3

7 and 7 × 100

7 × 100 is 7 written in


BB

CC

CC

CC

DD

DD

DD

CC

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Unit 1




Assessment Readiness Tip Students can often eliminate some

answer choices of multiple-choice questions using logic.

Item 3 1.584 doesn’t make sense as a population figure, since it’s

not possible to have part of a person, and 1,584 people is a very

small figure for the population of a country. Therefore answers A

and B can be eliminated.

Item 5 The salary of several account executives must be much

larger than the salary of a single executive. Answer choices A and B

are smaller, while answer choice C is only slightly larger. Therefore,

only answer choice D makes sense in the context.

Avoid Common ErrorsItem 2 Caution students not to simply count the zeros in the

number and select answer choice A. Remind them that they must

count the number of places that the decimal point is moved to

determine the correct exponent.

Item 8 To avoid losing their place when counting many zeros,

encourage students to use the point of a pen or pencil to keep

track of where they are when counting, and also to repeat

counting to confirm that it is correct.

Additional Resources

Assessment Readiness


Items Grade 8 Standards Mathematical Practices

1 8.EE.1

2 8.EE.3 MP.4

3 8.EE.3 MP.4

4* 8.NS.2 MP.6

5 8.EE.4 MP.4

6 8.EE.3

7* 8.NS.1 MP.4

8 8.EE.3 MP.4

9 8.EE.3 MP.7, MP.8

* Item integrates mixed review concepts from previous modules or a previous course.


Documents

Exponents and Scientific Notation MODULE 2Common Core Standards Lesson 2.1 Lesson 2.2 Lesson 2.3 Lesson 2.4 8.EE.1 Know and apply the properties of integer exponents to generate equivalent