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Chapter 10

Copyright © Big Ideas Learning, LLC Big Ideas Math Blue All rights reserved. Worked-Out Solutions

301

Chapter 10 Opener Try It Yourself (p. 409)

1. ( )2 2815 2 3 7 15 2 2 3 74

15 2 4 3 730 4 2134 2113

+ - • = + - •

= • + - •= + -= -=

2. 25 2 10 3 2 1 25 2 10 3 2 150 10 6 15 6 111 110

• ÷ + • - = • ÷ + • -= ÷ + -= + -= -=

3. ( )( ) ( )( )( )

2 2

2

3 1 2 4 3 2 3 1 2 4 5

3 1 2 20

9 1 2 209 1 408 4048

- + + = - +

= - += - + •= - += +=

4. 1.75 0.20.350

So, 1.75 0.2 0.35.• =

5. 1.4 0.60.84

So, 1.4 0.6 0.84.• =

6. 7.03 4.32109

2812 30.229

So, 7.03 4.3 30.229.• =

7. 0.894 0.2

0.1788

So, 0.894 0.2 0.1788.• =

8. 60

0.09 5.40 9 540 540

0

-

So, 5.40 0.09 60.÷ =

9. 13.9

0.3 4.17 3 41.73 11

9 2727

0

-

-

-

So, 4.17 0.3 13.9.÷ =

10. 24

0.15 3.60 15 36030

60 60

0

-

-

So, 3.6 0.15 24.÷ =

11. 1800

0.004 7.200 4 7200 4 3200

32000

-

-

So, 7.2 0.004 1800.÷ =

Section 10.1 10.1 Activity (pp. 410�–411) 1. a.

b. The expression ( )3 n- means 3- raised to an exponent

of n. To find the value of ( )3 ,n- multiply ( )3- as a

factor n times.

Power Repeated Multiplication Form Value

( )13- 3- 3-

( )23- ( ) ( )3 3- • - 9

( )33- ( ) ( ) ( )3 3 3- • - • - 27-

( )43- ( ) ( ) ( ) ( )3 3 3 3- • - • - • - 81

( )53-

( ) ( ) ( )3 3 3- • - • -

( ) ( ) 3 3• - • - 243-

( )63-

( ) ( ) ( ) ( )3 3 3 3- • - • - • -

( ) ( ) 3 3• - • - 729

( )73-

( ) ( ) ( ) ( )3 3 3 3- • - • - • -

( ) ( ) ( ) 3 3 3• - • - • - 2187-

Chapter 10

Big Ideas Math Blue Copyright © Big Ideas Learning, LLC Worked-Out Solutions All rights reserved. 302

2. a. The large cube is made up of 3 3 3• • small cubes. Because each small cube contains $3, the total amount of money in the large cube is 43 3 3 3 3 .• • • =

b. 43 81=

There is $81 in the large cube.

3. a. 2610 100,000,000,000,000,000,000,000,000=

The diameter of the observable universe is 100,000,000,000,000,000,000,000,000 meters.

b. 2110 1,000,000,000,000,000,000,000=

The diameter of the Milky Way Galaxy is 1,000,000,000,000,000,000,000 meters.

This can be written as one sextillion meters.

c. 1610 10,000,000,000,000,000=

The diameter of the solar system is 10,000,000,000,000,000 meters.

This can be written as ten quadrillion meters.

d. 710 10,000,000=

The diameter of Earth is 10,000,000 meters. This can be written as ten million meters.

e. 610 1,000,000=

The length of the Lake Erie shoreline is 1,000,000 meters.

This can be written as one million meters.

f. 510 100,000=

The width of Lake Erie is 100,000 meters. This can be written as one hundred thousand meters.

4. Wives: 17

Sacks: 27 7 7• =

Cats: 37 7 7 7• • =

Kits: 47 7 7 7 7• • • =

5. You can use exponents to write the product of repeated factors.

Sample answer: The formula for the volume of a cube, 3,V s= is an example of how exponents are used in real

life. Exponents are also used in measuring astronomical distances.

10.1 On Your Own (pp. 412�–413)

1. 1 1 1 1 14 4 4 4 4

• • • •

Because 14

is used as a factor 5 times, its exponent is 5.

So, 51 1 1 1 1 1 .

4 4 4 4 4 4 • • • • =

2. 0.3 0.3 0.3 0.3 x x• • • • •

Because 0.3 is used as a factor 4 times, its exponent is 4. Because x is used as a factor 2 times, its exponent is 2.

So, ( )4 20.3 0.3 0.3 0.3 0.3 .x x x• • • • • =

3. ( )45 5 5 5 5 625- = - • • • = -

4. 31 1 1 1 1

6 6 6 6 216 - = - • - • - = -

5. 33 27 27 27 1 1- ÷ = - ÷ = - =

6. 59 2 0.5 9 32 0.5 9 16 7- • = - • = - = -

7. The diameter is 1.8 meters, so the radius is 0.9 meter.

Inner sphere:

( )

( )

3

3

434 0.934 0.72930.972

V rp

p

p

p

=

=

=

=

Outer sphere: 9 4.52

p p=

The volume of the inflated space is 4.5 0.972 3.528 ,p p p- = or about 11.08 cubic meters.

10.1 Exercises (pp. 414�–415)

Vocabulary and Concept Check

1. 43- is the negative of 43 , so the base is 3, the exponent

is 4, and its value is 81.- ( )43- has a base of 3,- an exponent of 4, and a value of 81.

2. The second one does not belong because it is an incorrect statement about the expression. The power is the entire expression 35 .

Practice and Problem Solving 3. 3 3 3 3• • •

Because 3 is used as a factor 4 times, the exponent is 4. So, 43 3 3 3 3 .• • • =

4. ( ) ( )6 6- • -

Because 6- is used as a factor 2 times, the exponent is 2.

So, ( ) ( ) ( )26 6 6 .- • - = -

Chapter 10


303

5. 1 1 12 2 2

- • - • -

Because 12

- is used as a factor 3 times, the exponent is 3.

So, 31 1 1 1 .

2 2 2 2 - • - • - = -

6. 1 1 13 3 3

• •

Because 13

is used as a factor 3 times, the exponent is 3.

So, 31 1 1 1 .

3 3 3 3 • • =

7. x x x xp p p• • • • • • Because p is used as a factor 3 times, the exponent is 3.

Because x is used as a factor 4 times, the exponent is 4.

So, 3 4.x x x x xp p p p• • • • • • =

8. ( ) ( ) ( )4 4 4 y y- • - • - • •

Because 4- is used as a factor 3 times, the exponent is 3. Because y is used as a factor 2 times, the exponent is 2.

So, ( ) ( ) ( ) ( )3 24 4 4 4 .y y y- • - • - • • = -

9. 6.4 6.4 6.4 6.4 b b b• • • • • •

Because 6.4 is used as a factor 4 times, the exponent is 4. Because b is used as a factor 3 times, the exponent is 3.

So, ( )4 36.4 6.4 6.4 6.4 6.4 .b b b b• • • • • • =

10. ( ) ( ) ( ) ( ) ( )t t t t t- • - • - • - • -

Because t- is used as a factor 5 times, the exponent is 5.

So, ( ) ( ) ( ) ( ) ( ) ( )5.t t t t t t- • - • - • - • - = -

11. 25 5 5 25= • =

12. ( )311 11 11 11 1331- = - • • = -

13. ( ) ( ) ( ) ( ) ( ) ( ) ( )61 1 1 1 1 1 1 1- = - • - • - • - • - • - =

14. 61 1 1 1 1 1 1 1

2 2 2 2 2 2 2 64 = • • • • • =

15. 21 1 1 1

12 12 12 144 - = - • - =

16. 31 1 1 1 1

9 9 9 9 729 - = - • • = -

17. The negative sign is not part of the base; ( )26 6 6 36.- = - • = -

18.

The prime factorization of 675 is 5 5 3 3 3,• • • •

or 2 35 3 .•

19. 1 1 1 14 4 4 4 - • • •

Because 14


So, 41 1 1 1 1 .

4 4 4 4 4 - • • • = -

20. The largest doll is 12 inches and the other 3 are 710

the

height of the next larger doll. Use 710

as a factor 3 times.

So, an expression for the height of the smallest doll is 37 7 7 712 12 .

10 10 10 10 • • • = •

37 34312 12 4.116

10 1000 • = • =

The height of the smallest doll is 4.116 inches.

21. 35 3 2 5 3 8 5 24 29+ • = + • = + =

22. ( )22 7 3 2 7 9 2 63 65+ • - = + • = + =

23. ( ) ( )2 213 12 5 169 144 5 25 5 5- ÷ = - ÷ = ÷ =

24. ( ) ( )

( )

( )

3 21 14 6 3 64 6 92 2

1 64 5421 1025

- • = - •

= -

=

=

25. ( ) ( ) ( )31 1 17 5 7 125 132 66 662 2 2

+ = + = = =

26. 3 21 1 1 1 1 16 2 2

2 4 8 16 8 1 - ÷ = - ÷ = - • = - =

675

25

5 5

5 5

9 3

33 3

27

Chapter 10


27.

You should choose getting paid 2 1h - dollars because when you work more than 1 hour, you will get paid more than the other option.

28. a. ( ) ( )4100 0.99988 100 0.99988 99.95tC = =

After 4 years, the amount of carbon-14 remaining is about 99.95 grams.

b. amount remainingpercent remainingoriginal amount

99.95100

99.95%

=

=

=

After 4 years, 99.95% of the carbon-14 remains.

29. a. To travel from A-440 to A, it takes 12 notes.

b. ( ) ( )12440 1.0595 440 1.0595 880nF = =

The frequency of A is about 880 vibrations per second.

c. Sample answer: For a 12-note increase, the frequency approximately doubles.

Fair Game Review 30. The statement 8 8x x• = • represents the

Commutative Property of Multiplication.

31. The statement ( ) ( )2 10 2 10x x• = • represents the

Associative Property of Multiplication.

32. The statement ( )3 1 3x x• = represents the Identity

Property of Multiplication.

33. B; 2418 27

818 9

16

x

x

x

=

=

=


b. To find the product of two powers with the same base, add their exponents.

m n m na a a +• =

c. 2 4 2 4 62 2 2 2+• = =

( ) ( ) ( ) ( )2 4 2 4 63 3 3 3+- • - = - = -

3 2 3 2 57 7 7 7+• = =

1 6 1 6 75.1 5.1 5.1 5.1+• = =

( ) ( ) ( ) ( )2 2 2 2 44 4 4 4+- • - = - = -

3 5 3 5 810 10 10 10+• = =

5 5 5 5 101 1 1 1

2 2 2 2

+ • = =

Using the rule to simplify the products results in the values in the third column of the table in part (a).

d. 62 64=

( )63 729- =

57 16,807=

75.1 89,741.1

( )44 256- =

810 100,000,000=

101 1

2 1024 =

h 1 2

2 1h - 12 1 1- = 22 1 3- =

12h- 1 1 02 2 1- = = 2 1 12 2 2- = =

h 3 4

2 1h - 32 1 7- = 42 1 15- =

12h- 3 1 22 2 4- = = 4 1 32 2 8- = =

h 5

2 1h - 52 1 31- =

12h- 5 1 42 2 16- = =

Product Repeated Multiplication Form Power

2 42 2• 2 2 2 2 2 2• • • • • 62

( ) ( )2 43 3- • -

( ) ( ) ( )3 3 3- • - • -

( ) ( ) ( )3 3 3• - • - • - ( )63-

3 27 7• 7 7 7 7 7• • • • 57

1 65.1 5.1• 5.1 5.1 5.1 5.1• • •

5.1 5.1 5.1• • • 75.1

( ) ( )2 24 4- • -

( ) ( )4 4- • -

( ) ( )4 4• - • - ( )44-

3 510 10• 10 10 10 10• • • 10 10 10 10• • • •

810

5 51 12 2 •

1 1 1 1 12 2 2 2 2

• • • •

1 1 1 1 1 2 2 2 2 2

• • • • •

1012

Chapter 10


305

2. a. ( ) ( )( )( )32 63 3 3 3 3 3 3 3= • • • =

b. ( ) ( )( )( )( )42 82 2 2 2 2 2 2 2 2 2= • • • • =

c. ( ) ( )( )23 67 7 7 7 7 7 7 7= • • • • =

d. ( ) ( )( )( )33 9y y y y y y y y y y y= • • • • • • =

e. ( ) ( )( )24 8x x x x x x x x x x= • • • • • • =

General Rule: ( )nm m na a •=

3. a. ( ) ( )( )( )3 3 32 3 2 3 2 3 2 3 2 3• = • • • = •

b. ( ) ( )( )2 2 22 5 2 5 2 5 2 5• = • • = •

c. ( ) ( )( )( )3 3 35 4 5 4 5 4 5 4 5 4• = • • • = •

d. ( ) ( )( )( )( )4 4 46 6 6 6 6 6a a a a a a= = •

e. ( ) ( )( )2 2 23 3 3 3x x x x= = •

General Rule: ( )m m ma b a b• = •

4. a. For ( )3, 5 : 3 52 2 2 2 8 32 256x y• = • = • =

There are 256 pennies in the stack.

b.

5

2 2 32

2 32

2 2

x y

x y

x y

+

+

• =

=

=

The stacks that have 32 pennies are the locations in which the sum x y+ equals 5.

The locations ( )1, 4 , ( )2, 3 , ( )3, 2 , and ( )4,1 have

32 pennies in their stacks.

c. The tallest stack is located at ( )8, 8 .

8 82 2 2 2 256 256 65,536x y• = • = • =

The value of a penny is $0.01. So, there is ( )$0.01 65,536 $655.36= in the tallest stack.

d. From part (c), the tallest stack has 65,536 pennies. So, the height of the tallest stack is 65,536 0.06 3932.16• = inches.

5. If the same pattern holds true for every example you encounter, then you can use inductive reasoning to write a general rule stating that the pattern is always true.

10.2 On Your Own (p. 419)

1. 2 4 2 4 66 6 6 6+• = =

2. 3 6 3 6 91 1 1 1

2 2 2 2

+ - • - = - = -

3. 12 1 12 13z z z z+• = =

4. ( )34 4 3 124 4 4•= =

5. ( )42 2 4 8y y y•= =

6. ( )( ) ( ) ( )23 3 2 64 4 4•- = - = -

7. ( )4 4 4 45 5 625y y y= • =

8. ( )5 5 5ab a b=

9. ( )2 2 2 2 2 20.5 0.5 0.25mn m n m n= • • =

10.

30

30 4

30 4

34

Total number Number of bytes Number of of bytes in a gigabyte gigabytes

2 16

2 2

2

2

+

= •

= •

= •

=

=

The computer has 342 bytes of free storage space.

10.2 Exercises (pp. 420�–421)

Vocabulary and Concept Check 1. Use the Product of Powers Property to multiply powers

with the same base.

2. no; The bases are not the same.

Practice and Problem Solving

3. 2 2 2 2 43 3 3 3+• = =

4. 10 4 10 4 148 8 8 8+• = =

5. ( ) ( ) ( ) ( )5 7 5 7 124 4 4 4+- • - = - = -

6. 3 3 3 3 6a a a a+• = =

7. 6 6 1 7h h h h+• = =

8. 2 6 2 6 82 2 2 2

3 3 3 3

+ • = =

x y x y+

1 4 5

2 3 5

3 2 5

4 1 5

Chapter 10


9. 8 9 8 9 175 5 5 5

7 7 7 7

+ - • - = - = -

10. ( ) ( ) ( ) ( )7 1 7 82.9 2.9 2.9 2.9+- • - = - = -

11. ( )34 4 3 125 5 5•= =

12. ( )312 12 3 36b b b•= =

13. ( )43 3 4 123.8 3.8 3.8•= =

14. 25 5 2 103 3 3

4 4 4

• - = - = -

15. The bases should not be multiplied.

2 9 2 9 115 5 5 5+• = =

16. You should multiply the exponents instead of adding.

( )46 6 4 24r r r•= =

17. ( )3 3 3 36 6 216g g g= • =

18. ( ) ( )5 5 5 53 3 243v v v- = - • = -

19. 2 2

2 21 1 15 5 25

k k k = • =

20. ( )4 4 4 41.2 1.2 2.0736m m m= • =

21. ( )12 12 12rt r t=

22. 3 3

3 33 3 274 4 64

p p p - = - • = -

23. No. They are not equal.

2 3 53 3 3 243,• = = but 2 33 3 9 27 36.+ = + =

24. a. ( )33 3V s w= =

An expression for the volume of the case is ( )33 .w

b. ( )3 3 3 33 3 27w w w= • =

25. ( )24 5 2 4 5 2 2

9 4

2 2 2 2 2

2 2512 16496

+ •• - = -

= -= -=

26. 4 4

4 4 41 1 116 16 162 2 16

x x x x = • • = • • =

27. ( ) ( )( )

2 3 2 2 3 2

2 5

2 5

7

5 5 5 5 5

5 5

5

578,125

+

+

• =

=

=

==

28. 8

1 8

9

highest altitude 3 lowest altitude

3 3

3

3

+

= •

= •

=

=

The highest altitude of an altocumulus cloud is about 93 feet.

29. a.

( )

( )

434 2 2 334 1231650.27

V abcp

p

p

p

=

= • •

=

=

The volume of the egg is 16 50.27 cubic inches.p

b. 2 2

2 2

2 2

2

2

3

a

b

c

=

=

=

2 2 2

2 2 2

4

434 2 2 334 2 334 2 934 16 93192603.19

V abcp

p

p

p

p

p

+

=

= • • •

= • •

= • •

= • •

=

The volume of the egg is 192 603.19 cubic inches.p Because 16 12 192 ,p p• = the volume is 12 times greater than the volume in part (a).

Chapter 10


307

30. Because the side lengths of the base increase by 50%, or 0.5, the lengths increase by a factor of 150% of 1, or 1.5.

( ) ( )

2

2

2 2

2

2 2

131 1.531 1.531 2.253

30.75 , or 4

V b h

b h

b h

b h

b h b h

=

=

= • • •

= • • •

=

31.

( ) ( )8 2 8 4 2

8 8 4 2 2

8 8 4 2 2

16 4 4

Total pieces Amount delivered Number ofof mail each second seconds

2 5 2 3 5

2 2 3 5 52 3 52 3 5

+ +

= •

= • • • •

= • • • •= • •= • •

The United States Postal Service delivers 16 4 42 3 5• • pieces of mail in 6 days.

32. a. 5

5 8

5 8

2 2 2562 2 2

2 2

x

x

x+

• =• =

=

Because 5 3 8,+ = 3.x =

b. 2

2 6

2 6

1 1 13 3 729

1 1 13 3 3

1 13 3

x

x

x+

• =

• =

=

Because 2 4 6,+ = 4.x =

Fair Game Review

33. 4 4•1

41

4 41

= = 34. 5 5 5• •1

51

25 251

= =

35. 21

32•

1

3 31

= = 36. 81

6•1

66

•

18•1

6 61

= =

37. B; ( ) ( )2 180 8 2 1808

6 1808

10808

135

nn

- • ° - • °=

• °=

°=

= °

Each angle measures 135 .°


Quotient Repeated Multiplication Form Power

4

222

21

2•1

2 22

• •

1 2•

1

22

( )( )

5

24

4

--

4-( )1

4• -( ) ( )( ) ( )

1

4

4 44

• -

• - • --( )

1

4• -( )1

( )34-

7

377

71

7•1

7•1

7 7 7 7

7• • • •

1 7•

1 7•

1

47

9

68.58.5

8.51

8.5•1

8.5•1

8.5•1

8.5•1

8.5•1

8.5 8.5 8.58.5

• • •

1 8.5•

1 8.5•

1 8.5•

1

8.5•1

8.5•1

38.5

8

51010

101

10•1

10•1

10•1

10•1

10 10 1010

• • •

1 10•

1 10•

1 10•

1 10•

1

310

12

433

31

3•1

3•1

3•1

3 33 3 3 3 3 33

• •• • • • • •

1 3•

1 3•

1 3•

1

83

( )( )

7

55

5

--

( )5- ( )1

5• - ( )1

5• - ( )1

5• - ( )1

5• -

( ) ( )( )

1

5 55

• - • -- ( )

1

5• - ( )1

5• - ( )1

5• - ( )1

5• -1

( )25-

4

11111

111

11 11 1111

• • •

1

311

Chapter 10


b. To find the quotient of two powers with the same base, subtract their exponents.

m

m nn

aa

a-=

c. 4

4 2 22

2 2 22

-= =

( )( )

( ) ( )5

5 2 32

44 4

4--

= - = --

7

7 3 43

7 7 77

-= =

9

9 6 36

8.5 8.5 8.58.5

-= =

8

8 5 35

10 10 1010

-= =

12

12 4 84

3 3 33

-= =

( )( )

( ) ( )7

7 5 25

55 5

5--

= - = --

4

4 1 31

11 11 1111

-= =

Using the rule to simplify the quotients results in the values in the third column of the table in part (a).

2.

The volume of the smaller cube equals the number of smaller cubes that will fit inside the larger cube.

3. To divide two powers that have the same base, subtract their exponents.

Sample answer: 5

5 3 23

99 1 8

1

2 2 227 7 77

-

-

= =

= =

10.3 On Your Own (pp. 424�–425)

1. 7

7 4 34

9 9 99

-= =

2. 6

6 5 15

4.2 4.2 4.2 4.24.2

-= = =

3. ( )( )

( ) ( )8

8 4 44

88 8

8--

= - = --

4. 8

8 3 53

xx x

x-= =

5. 15 15 15

15 8 73 5 3 5 82 2 2 2 2

2 2 2 2-

+= = = =•

6. 5 9

5 1 9 8 4 1 4 1 58

d dd d d d d d

d d- - +• = • = • = =

7. 9 5

9 4 5 2 5 3 84 2

5 5 5 5 5 5 55 5

- -• = • = • =

8.

21

17

21

17

21 17

4

People per Population in 2030square kilometer Land area

2.25 22

22.252

2.25 2

2.25 236

-

=

•=

= •

= •

= •=

There will be about 36 people per square kilometer in Alabama in 2030.

10.3 Exercises (pp. 426�–427)

Vocabulary and Concept Check 1. To divide powers means to divide out the common

factors of the numerator and denominator. To divide powers with the same base, subtract their exponents.

2. The third quotient does not belong because it is a quotient of powers with different bases, whereas the other three are quotients of powers with the same bases.


3. 10

10 4 64

6 6 66

-= = 4. 9

9 7 27

8 8 88

-= =

5. ( )( )

( ) ( )4

4 1 31

33 3

3--

= - = --

6. 5

5 3 23

4.5 4.5 4.54.5

-= =

7. 9

9 3 63

5 5 55

-= =

Volume

of Smaller Cube

Volume of Larger

Cube

LargerVolumeSmallerVolume

Answer

a. 34 ( )32 64 4= 6

344

34

b. 33 ( )32 63 3= 6

333

33

c. 36 ( )32 66 6= 6

366

36

d. 310 ( )32 610 10= 6

31010

310

Chapter 10


309

8. 4

4 3 13

64 64 64 6464

-= = =

9. ( )( )

( ) ( )5

5 2 32

1717 17

17--

= - = --

10. ( )( )

( ) ( )10

10 4 64

7.97.9 7.9

7.9--

= - = --

11. ( )( )

( ) ( )8

8 6 26

6.46.4 6.4

6.4--

= - = --

12. 11

11 7 47

p p pp

-= = 13. 24

24 11 1311

bb b

b-= =

14. 18

18 7 117

nn n

n-= =

15. When dividing powers, you subtract exponents instead of dividing them.

1515 5 10

56 6 66

-= =

16. 5 3 5 3 8

8 2 62 2 2

7 7 7 7 7 77 7 7

+-• = = = =

17. 19 5 19 5 24

24 15 912 3 12 3 15

2 2 2 2 2 22 2 2 2

+-

+• = = = =•

18. ( )( )

( )( )

( ) ( )

( ) ( )( )( )

8 48 7 4 3

7 3

1 1

1 1

2

8.3 8.38.3 8.3

8.3 8.3

8.3 8.3

8.3

8.3

- -

+

- -• = - • -

- -

= - • -

= -

= -

19. 30 30 30

30 22 818 4 18 4 22p p p p p

p p p p-

+= = = =•

20. 22 22 22

22 17 58 9 8 9 17c c c

c cc c c c

-+= = = =

•

21. 13 17

13 5 17 11 8 6 8 6 145 11

k kk k k k k k

k k- - +• = • = • = =

22. 14

14 6 86

10 10 1010

-= =

The sound of a jet at takeoff is 810 times more intense than the sound of a normal conversation.

23. 8 8

8 5 35 54 4 4 4 64

4 4x

x x x x-• = • = • = • =

24. 3 3

3 2 12 2

6 6 6 6 66 6

ww w w w-• = • = • = =

25. 3 4 4 4 4

32 2

3 4 2 4 1

3 2 3

3 2

5 55 5

5

5

125

a b ba

b ba b

a b

a b

- -

• • = • ••

= • •

= • •

=

26. 12 10 2 12 10

29 9 9 9

12 9 10 9 2

3 1 2

2

5 55 5

5

5

125

c d cd

c cc d

c d

cd

- -

• • = • ••

= • •

= • •

=

27. 15 9 15 9

15 8 9 3 7 68 3 8 3

x y x yx y x y

x y x y- -= • = • =

28. 10 7 10 7

10 1 7 6 9 1 91 6 1 6

m n m nm n m n m n

m n m n- -= • = • = =

29. a. 4

4 2 22

2 2 2 42

-= = =

MP3 Player D has 22 4= times more memory than MP3 Player B.

b.

no; The graph is not a line, so memory and price do not show a linear relationship.

30. a. Sample answer: To satisfy the equation, the difference m n- must equal 2. When 4m = and 2,n = the difference 4 2m n- = - is 2.

b. infinitely many solutions; Any two numbers that satisfy the equation 2m n- = are solutions.

y

x40

30

0

60

90

120

150

180

210

240

270

8 12 16 20 24 28 32 36

Memory (GB)

Pric

e (d

olla

rs)

Chapter 10


31.

24

10

24

1 10

24

11

24 11

13

Number of Number of stars in the Universegalaxies Number of stars in the Milky Way Galaxy

1010 1010

10101010

10

+

-

=

=•

=

=

=

=

There are about 1310 galaxies in the Universe.

32.

( )

39

2 1

3 2 1 9

3 2 1 9

1 9

8 88

8 8

8 8

8 8

x

x

x x

x x

x

+

- +

- -

-

=

=

=

=

For the equation to be true, the value of 1x - must equal 9.

1 9 1 1

10

x

x

- =+ +

=

So, 10.x =

Fair Game Review

33. ( )4 5 4 5 9- - = - + - = -

34. ( )23 15 23 15 8- - - = - + = -

35. ( )33 28 33 28 61- - = + =

36. ( )18 22 18 22 4- = + - = -

37. B; 2 903 90

30

x x

x

x

° + ° = °==


b. 3

35 125 1;5 125

= =2

26 36 1;6 36

= = ( )( )

4

4

3 81 1;813

-= =

-

( )( )

5

5

4 1024 1;10244

- -= =--

They are all equal to 1.

c. Because all the expressions in part (b) simplify to 1, you can equate the powers in the last column of the table in part (a) to 1. So, based on these results, you can conclude that 0 1,a = where 0.a

2. a.

b. yes; Each product is equal to the value of the number with the nonzero exponent. This implies that the numbers with the zero exponents are equal to 1. So, based on these results, you can conclude that 0 1,a = where 0.a

Quotient Quotient of Powers Property Power

3

355

3 35 - 05

2

266

2 26 - 06

( )( )

4

4

3

3

--

( )4 43 -- ( )03-

( )( )

5

5

4

4

--

( )5 54 -- ( )04-

Product Product of Powers Property Power

0 43 3• 0 43 +

43

2 08 8• 2 08 +

28

( ) ( )3 02 2- • - ( )3 02 +-

( )32-

0 51 13 3

- • -

0 51

3

+ -

51

3 -

Chapter 10


311

3. a.

b. They are all equal to 1. c. From the Multiplicative Inverse Property, you know

that the product of a number and its reciprocal is 1. d. Because each product is equal 1, you can conclude

that the numbers in the product are reciprocals. So, na- is equal to the reciprocal of 1, or .n

naa

4. a. The exponents decrease by 1.

ones: 010

tenths: 110-

hundredths: 210-

thousandths: 310- b.

3 2 1 0

1 2 3

3452.867

3 10 4 10 5 10 2 10

8 10 6 10 7 103 1000 4 100 5 10 2 1

1 1 1 8 6 710 100 1000

3000 400 50 2 0.8 0.06 0.007

- - -

= • + • + • + •

+ • + • + •= • + • + • + •

+ • + • + •

= + + + + + +

5. Any nonzero number with an exponent of 0 is equal to 1. Any nonzero number with a negative integer exponent can be written as the reciprocal of the nonzero number with its positive integer exponent.

10.4 On Your Own (pp. 430�–431)

1. 22

1 144 16

- = = 2. ( )( )

55

1 12322

-- = = --

3. 8 8 8 8 06 6 6 6 1- - +• = = =

4. ( )( )

( ) ( )( )

55 6 1

6 13 1 13 3

33 3- --

= - = - = = -- -

5. ( )7 4 7 4 37 41 1 1 1 1 1 15 5 5 5 5 1255- - + -

•• = = = =•

6. ( )5 35 3 2

2 2 02 2 2

4 4 4 4 4 4 14 4 4

+ ---• = = = = =

7. 22

88xx

- =

8. 0 1010 101 11b b

b b-• = • =

9. 6

6 9 39 3

1 1 115 15 15 15

zz z

z z- -= • = • =

10. 55

13600 5 36005

136003125

36003125191125

1.152

-• = •

= •

=

=

=

So, 1.152 liters of water leak from the faucet in 1 hour.

10.4 Exercises (pp. 432�–433)

Vocabulary and Concept Check

1. no; For any nonzero number a, the value of 0 1.a =

2. Rewrite 310- as 31 ,

10or 1 .

1000

3. The numbers, in order from least to greatest, are 55 ,- 05 ,

and 45 .

4. The last statement does not belong.

( ) ( ) ( ) ( )33 3 3 3- • - • - = -

3

33

1 1 133 3 3 3 3

- = = = • •


5. 7

7 7 07

8 8 8 18

-= = =

6. 0 3 0 3 35 5 5 5 125+• = = =

7. ( ) ( ) ( ) ( )8 8 8 8 02 2 2 2 1- - +- • - = - = - =

8. ( )4 44 4 09 9 9 9 1+ --• = = =

9. 22

1 166 36

- = = 10. 0158 1=

Product Product of Powers Property Power

3 35 5- • 3 35- + 05

2 26 6-• ( )2 26 + - 06

( ) ( )4 43 3 -- • - ( ) ( )4 43 + -- ( )03-

( ) ( )5 54 4-- • - ( ) 5 54 - +- ( )04-

Chapter 10


11. 3

3 5 25 2

4 1 14 44 4 16

- -= = = =

12. ( )

( ) ( )( )

1 2 12 1

3 1 13 333 3

- -- = - = - = = -- -

13. 44

14 2 5 4 5214 5

161 54154

-• + = • +

= • +

= +

=

14. 3 2 3 2 55

1 13 3 3 33 243

- - - - -• = = = =

15. 3 6 3 6 3 6 31 1 1 1 1 1 1

5 5 5 5 5 5 125- - - +•• = = = =•

16. ( )( ) ( )

( )( )( )( )( )( )

2 2

2 4 2 4

2

2

2 2

0

1.5 1.5

1.5 1.5 1.5

1.5

1.5

1.5

1.5

1

- - +

-

=•

=

=

=

=

17. The negative sign goes with the exponent, not the base.

( ) 3

31 144 64

- = =

18. 3

4

3

4 3

7

1000 grams 1 gram of sand10 kilograms1 kilogram 10 gram

10 10

10 10

10

-

-

=

= •

=

There are about 710 , or 10,000,000, grains of sand.

19. For any nonzero number a, 0 1.a = So, 02 1= and 010 1.=

20. 44

66yy

- = 21. 7 7

2 728

8 64a a

a- • = =

22. ( )3

3 4 74

9 9 9cc c

c- -

- = = 23. ( )2

2 33

5 5 5bb b

b

-- - -

- = =

24. 3

3 9 69 6

8 8 442 2

xx x

x x- -= • = • =

25. 4 4 4 4

4 4

0

3 4 3 4

12

1212 112

d d d d

d

d

- -

- +

• = • • •

= •

= •= •=

26. 3

2 3 32 2

1 nm n n

m m- • = • =

27. 2 0 0 2 6 6

6 6 23 3 1 1

3 9k w w w

w w

- -

- -• • • •= = =

28. Sample answer:

44

1 122 16

- = = 22

1 144 16

- = =

29. ( )1

1 3 23

decimeter 10 m 10 10 100millimeter 10 m

-- - -

-= = = =

There are 100 millimeters in a decimeter.

30. ( )2

2 6 46

centimeter 10 m 10 10 10,000micrometer 10 m

-- - -

-= = = =

There are 10,000 micrometers in a centimeter.

31. ( )3

3 9 69

millimeter 10 m 10 10 1,000,000nanometer 10 m

-- - -

-= = = =

There are 1,000,000 nanometers in a millimeter.

32.

( )

6

0

6

0 6

6

meter 1 mmicrometer 10 m

101010

101,000,000

-

-

- -

=

=

=

==

There are 1,000,000 micrometers in a meter.

33. a. 1 10 11010,000 10,000 1000

• = = micrometer

66

9

1 1 1101000 1000 10

1 11000 1,000,000

11,000,000,000

0.000000001, or 10

-

-

• = •

= •

=

=

The length of the virus is 910- meter.

Chapter 10


313

b. 9

9

1 nanometer 10 meter1 meter

101 meter

1,000,000,0000.000000001 meter

-=

=

=

=

The answer to part (a) is equal to one nanometer.

34. a. 3 33

500 500 10 mm10- = •

The donation is 3500 10• cubic millimeters.

3 4 3 4

7

500 10 10 500 10

500 10500 10,000,0005,000,000,000

+• • = •

= •= •=

There are about five billion white blood cells in the donation.

b. 3 6 3 6

3 6

9

500 10 5 10 500 5 10 10

2500 10

2500 102500 1,000,000,0002,500,000,000,000

+

=

=

= = =

There are about two trillion five hundred billion red blood cells in the donation.

c. The ratio of red blood cells to white blood cells is

2,500,000,000,000 500 .5,000,000,000 1

= There are about 500

times more red blood cells than white blood cells.

35. Sample answer: Write the power as 1 divided by the

power and use a negative exponent. ( ) 1nnna a

a- -

-= =

36. If you substitute 0 for a, you get 1 .0n

Zero raised to any

nonzero exponent is 0. Because division by 0 is undefined, the rule for negative exponents does not apply when 0.a =

Fair Game Review

37. 3 6 3 6 910 10 10 10+• = =

38. 2 2 1 310 10 10 10+• = =

39. 8

8 4 44

10 10 1010

-= =

40. D; A stem-and-leaf plot orders numerical data and shows how they are distributed.

Study Help Available at BigIdeasMath.com.

Quiz 10.1�–10.4 1. ( ) ( ) ( ) ( )5 5 5 5- • - • - • -


So, ( ) ( ) ( ) ( ) ( )45 5 5 5 5 .- • - • - • - = -

2. 7 7 m m m• • • •

Because 7 is used as a factor 2 times, the exponent is 2. Because m is used as a factor 3 times, the exponent is 3.

So, 2 37 7 7 .m m m m• • • • =

3. 45 5 5 5 5 625= • • • =

4. ( ) ( ) ( ) ( ) ( ) ( ) ( )62 2 2 2 2 2 2 64- = - • - • - • - • - • - =

5. ( ) ( ) ( ) ( )9 9 9 9 04.8 4.8 4.8 4.8 1- - +- • - = - = - =

6. 4

4 7 37 3

5 1 15 55 5 125

- -= = = =

7. 8 8 1 93 3 3 3+• = =

8. ( )35 5 3 15a a a•= =

9. ( )4 4 4 43 3 81c c c= • =

10. 2 2

2 22 2 47 7 49

p p p - = - • =

11. 7

7 4 34

8 8 88

-= =

12. 3 7 3 7 10

10 2 82 2 2

6 6 6 6 6 66 6 6

+-• = = = =

13. 15 15 15

15 12 33 9 3 9 12p p p p p

p p p p-

+= = = =•

14. 13 8

13 5 8 6 8 2 8 2 105 6

t tt t t t t t

t t- - +• = • = • = =

15. 66

88dd

- =

16. 5

5 7 27 2

12 33 34

xx x

x x- -= = =

Chapter 10


17. a. 3

6 3 6 36 6

1000 101000 10 10 1010 10

- - -• = = = =

The length of the dinoflagellate is 310- meter.

b. 1000 mm 1 m1000 1000

11 mm m 0.001 m1000

=

= =

So, the length of the dinoflagellate is 1 millimeter.

18. 7

7 2 52

10 10 1010

-= =

An earthquake of magnitude 8.0 is 510 times stronger than an earthquake of magnitude 3.0.

Section 10.5 10.5 Activity (pp. 436�–437) 1. 2,000,000,000

3,000,000,0006,000,000,000,000,000,000

6.0 is a factor of 6,000,000,000,000,000,000 and E 18+ represents how many places the decimal point is from the number in standard form.

The calculator did not show the answer in standard form because the calculator screen is not large enough to display the number.

2. 0.000000002 0.0000000030.000000000000000006

6.0 is a factor of 0.000000000000000006 and E 18- represents how many places the decimal point is from the number in standard form.

3. A. This picture appears to have been taken about 1 centimeter or 0.01 meter away from the frog, so this picture matches with 210- meter.

B. This picture appears to have been taken about 100 meters away from the frog, so this picture matches with 210 meters.

C. This picture appears to have been taken very far away from the frog, about 100,000 meters. So, this picture matches with 510 meters.

D. This picture appears to have been taken about 10 centimeters or 0.1 meter away from the frog, so this picture matches with 110- meter.

E. This picture appears to have been taken very close to the frog, about 0.00001 meter away. So, this picture matches with 510- meter.

F. This picture appears to have been taken about 1 meter away from the frog. So, this picture matches with

010 meter.

4. A. 02 10 2 =

The unit that is most appropriate for a door height of 2 is meters.

B. 41.6 10 16,000 =

The unit that is most appropriate for a volcano height of 16,000 is feet.

C. 21.4 10 140 =

The unit that is most appropriate for a pen length of 140 is millimeters.

D. 16.3 10 0.63- =

The unit that is most appropriate for a steel ball bearing diameter of 0.63 is centimeters.

E. 17.5 10 75 = The unit that is most appropriate for a beach ball

circumference of 75 is inches.

5. Numbers that are written in scientific notation are represented by the product of a factor that is at least 1 and less than 10 and a power of 10. This type of notation is called �“scientific notation�” because it is used in scientific fields of study. Scientific notation is important because you can use the notation to easily write very large or very small numbers.

10.5 On Your Own (p. 439)

1. The factor is greater than 10. So, the number 412 10 is not written in scientific notation.

2. 76 10 60,000,000 7

=

The number in standard form is 60,000,000.

3. 59.9 10 0.000099 5

- =

The number in standard form is 0.000099.

4. 41.285 10 12,8504

=

The number in standard form is 12,850.

5. Water: 31.0 10 1000 =

Lead: 41.14 10 11,400 =

Lead is denser than water, so it will sink.

6. 51.4 10 75 0.000014 75 0.00105- • = • = The fleas consume about 0.00105 liter, or 1.05 milliliters,

of blood per day.

Chapter 10


315

10.5 Exercises (pp. 440�–441)

Vocabulary and Concept Check 1. Scientific notation uses a factor of greater than or equal

to one but less than 10 multiplied by a power of 10. A number in standard form is written out with all the zeros and place values included.

2. The expression 1310 9.2- does not belong because it is not written in scientific notation (the factor is not less than 10, and the power is not a power of 10), whereas the other three expressions are written in scientific notation.

Practice and Problem Solving 3. 5,600,000,000,000

4. 0.00000000021

5. 87,300,000,000,000,000

6. The factor is greater than or equal to 1 and less than 10. The power of 10 has an integer exponent. So, the number

91.8 10 is written in scientific notation.



8. The factor is less than 1. So, the number 250.26 10- is not written in scientific notation.

9. The factor is greater than 10. So, the number 1210.5 10 is not written in scientific notation.

10. The factor is greater than 10. So, the number 1746 10- is not written in scientific notation.


195 10- is written in scientific notation.


367.814 10- is written in scientific notation.

13. The factor is less than 1. So, the number 420.999 10 is not written in scientific notation.



15. 77 10 70,000,000 7

=


16. 38 10 0.0083

- =


17. 25 10 5002

=

The number in standard form is 500.

18. 42.7 10 0.000274

- =


19. 54.4 10 0.0000445

- =


20. 32.1 10 21003

=

The number in standard form is 2100.

21. 91.66 10 1,660,000,000 9

=

The number in standard form is 1,660,000,000.

22. 83.85 10 0.00000003858

- =


23. 69.725 10 9,725,000 6

=


24. Because the exponent is negative, the decimal point should be moved to the left, not to the right.

64.1 10 0.00000416

- =


25. a. 82.7 10 3 270,000,000 3 810,000,000 • = • = There are 810,000,000 platelets in 3 milliliters of

blood.

b. 5 L 1000 mLL

5000 mL=

82.7 10 5000 270,000,000 50001,350,000,000,000

• = •=

There are about 1,350,000,000,000 platelets in an adult body.

Chapter 10


26. To write the number 1001.0 10 in standard form, move the decimal point 100 places to the right. So, there are 100 zeros in a googol.

27. a. Betelgeuse: 36.2 10 6200 62003

= =

Bellatrix: 43.8 10 38,000 38,000 4

= =

Sun: 41.1 10 11,000 11,000 4

= =

Aldebaran: 37.2 10 7200 72003

= =

Rigel: 42.2 10 22,000 22,000 4

= =

Bellatrix has the highest surface temperature with 38,000 F.°

b. Betelgeuse has the lowest surface temperature with 6200 F.°

28. The value of the number is 10 times greater.

29. 16.2% of 3 39.6 10 0.162 9.6 100.162 96001555.2

= • = •=

The area of the Florida Reef Tract is 1555.2 square kilometers.

30. ( ) ( ) ( )26 6 61.0 10 1.0 10 1.0 10

1,000,000 1,000,0001,000,000,000,000

= •

= •=

1 square gigameter 1,000,000,000,000= square kilometers

5 1,000,000,000,000 5,000,000,000,000• =

There are 5,000,000,000,000 or 125 10 square kilometers in 5 square gigameters.

31.

9

9

cubic kilometers of fresh waterpercent fresh watertotal cubic kilometers of water

0.0251.4 10

0.025 1.4 100.025 1,400,000,000

35,000,000

x

x

x

x

=

=

• =• =

=

There are about 35,000,000 cubic kilometers of fresh water on Earth.

32. a. Air: 86.7 10 mi

1 h1.6 km1 mi

• 1 h•

8

3600 sec

6.7 10 1.6 km 1 3600 sec

670,000,000 1.6 km 3600 sec

1,072,000,000 km 3600 sec

297,778 km sec

•=•

•=

=

Glass: 86.6 10 ft 1 mi

1 sec•

5280 ft1.6 km1 mi

•

86.6 10 1.6 km 1 5280 sec

660,000,000 1.6 km 5280 sec

1,056,000,000 km 5280 sec

200,000 km sec

•=•

•=

=

=

Ice: 52.3 10 km 230,000 km sec

1 sec =

Vacuum: 83.0 10 m 1 km

1 sec 1000 m•

83.0 10 1 km 1 1000 sec

300,000,000 km 1000 sec

300,000 km sec

•=•

=

=

Water: 102.3 10 cm 1 m

1 sec•

100 cm1 km

1000 m•

102.3 10 1 km 1 100 1000 sec23,000,000,000 km

100,000 sec 230,000 km sec

•=• •

=

=

Light travels the fastest through a vacuum. b. Light travels the slowest through glass.

Fair Game Review

33. 54 4 4 4 4 4• • • • =

34. 3 33 3 3 3y y y y• • • • • =

35. ( ) ( ) ( ) ( )32 2 2 2- • - • - = -

Chapter 10


317

36. B; 2 2 2

2 2 2

2

2

2

4 5

16 25

41

41

41

a b c

c

c

c

c

c

+ =+ =+ =

=

=

=

The length of the hypotenuse is 41 inches.

Section 10.6 10.6 Activity (pp. 442–443)

1. a. 20.01 1 102

-= �

The pH of lime juice is 2, and so it is an acid.

b. 80.00000001 1 108

-= �

The pH of an egg is 8, and so it is a base.

c. 70.0000001 1 10 7

-= �

The pH of distilled water is 7, and so it is neutral.

d. 110.00000000001 1 10 11

-= �

The pH of ammonia water is 11, and so it is a base.

e. 40.0001 1 104

-= �

The pH of tomato juice is 4, and so it is an acid.

f. 01 1 10= � The pH of hydrochloric acid is 0, and so it is an acid.

2. a. The distance is the second greatest distance, so the description matches Uranus.

91,800,000,000 mi 1.8 10 mi= �

The distance from Uranus to the Sun is 91.8 10�miles.

b. The distance is the seventh greatest distance, so the description matches Venus.

767,000,000 mi 6.7 10 mi= �

The distance from Venus to the Sun is 76.7 10�miles.

c. The distance is the third greatest, so the description matches Saturn.

8890,000,000 mi 8.9 10 mi= �

The distance from Saturn to the Sun is 88.9 10�miles.

d. The distance is the sixth greatest, so the description matches Earth.

793,000,000 mi 9.3 10 mi= �

The distance from Earth to the Sun is 79.3 10� miles. e. The distance is the fifth greatest, so the description

matches Mars.

8140,000,000 mi 1.4 10 mi= �

The distance from Mars to the Sun is 81.4 10� miles. f. The distance is the greatest, so the description matches

Neptune.

92,800,000,000 mi 2.8 10 mi= �

The distance from Neptune to the Sun is 92.8 10�miles.

g. The distance is the fourth greatest, so the description matches Jupiter.

8480,000,000 mi 4.8 10 mi= �

The distance from Jupiter to the Sun is 84.8 10�miles.

h. The distance is the least, so the description matches Mercury.

736,000,000 mi 3.6 10 mi= �

The distance from Mercury to the Sun is 73.6 10�miles.

Sample answer: scientific notation; In scientific notation, the greater the exponent of the power of 10, the greater distance the planet is from the Sun. When the powers of 10 are the same, compare the factors.

3. Sample answer: Let 1 inch represent 100,000,000 miles. Then the distances from the Sun will be: Mercury: 0.36 inch, Venus: 0.67 inch, Earth: 0.93 inch, Mars: 1.4 inches, Jupiter: 4.8 inches, Saturn: 8.9 inches, Uranus: 18 inches, and Neptune: 28 inches. Check scale drawings. Students should include a key for their scale.

4. Move the decimal point left or right so the number is at least 1 but less than 10. Then multiply by 10 raised to the number of places you moved the decimal point. If you moved the decimal point to the left, the exponent will be positive. If you moved the decimal point to the right, the exponent will be negative.

10.6 On Your Own (pp. 444–445)

1. 450,000 5 104

= �

The number in scientific notation is 45 10 .�

2. 725,000,000 2.5 107

= �

The number in scientific notation is 72.5 10 .�

Chapter 10


3. 2683 6.83 10 2

= �


4. 30.005 5 10 3

-= �

The number in scientific notation is 35 10 .-�

5. 70.00000033 3.3 10 7

-= �

The number in scientific notation is 73.3 10 .-�

6. 40.000506 5.06 104

-= �


7.

6

Remaining sales Sales required Current salesneeded for award for award total

10,000,000 955,0009,045,000

9.045 10

= -

= -== �

The album must sell 69.045 10� more copies to receive the award.

8. Because 7.0 6.55,> 7 77.0 10 6.55 10 .� > � So, the Tyrannosaurus rex lived before the Cenozoic era began. The Tyrannosaurus rex lived in the Mesozoic era.

10.6 Exercises (pp. 446–447)

Vocabulary and Concept Check 1. If the number written in standard form is greater than

or equal to 10, the exponent when written in scientific notation will be positive. If the number written in standard form is less than 1 and greater than 0, the exponent when written in scientific notation will be negative.

2. It is appropriate to use scientific notation when a number is either very large or very small.


3. 30.0021 2.1 10 3

-= �


4. 65,430,000 5.43 10 6

= �


5. 8321,000,000 3.21 10 8

= �


6. 60.00000625 6.25 106

-= �


7. 50.00004 4 10 5

-= �

The number in scientific notation is 54 10 .-�

8. 710,700,000 1.07 107

= �


9. 1045,600,000,000 4.56 10 10

= �


10. 120.000000000009256 9.256 10 12

-= �


11. 5840,000 8.4 105

= �


12. The decimal point was moved to the right. So, the exponent should be negative.

50.000036 3.6 10-= �

13. The factor is greater than 10. The decimal point needs to be moved one more place to the left.

772,500,000 7.25 10 7

= �

14. Because 1.12 1.19 1.2,< < 8 8 81.12 10 1.19 10 1.2 10 .� < � < �

From least to greatest, the order is 8 8 81.12 10 , 1.19 10 , and 1.2 10 .� � �

15. Because 6.09 6.78 6.8,< < 5 5 56.09 10 6.78 10 6.8 10 .- - -� < � < �

From least to greatest, the order is 5 5 56.09 10 , 6.78 10 , and 6.8 10 .- - -� � �

Chapter 10


319

16. Because 10 11 1210 10 10 ,< < 10 11 125.7 10 9.66 10 5.76 10 .� < � < �

From least to greatest, the order is 10 11 125.7 10 , 9.66 10 , and 5.76 10 .� � �

17. Because 8 6 510 10 10 ,- - -< < 8 6 54.8 10 4.8 10 4.8 10 .- - -� < � < �


18. Because 7.6 9.9,< 15 157.6 10 9.9 10 .- -� < � Because 15 1410 10 ,- -< 15 149.9 10 1.01 10 .- -� < �


19. Because 5.78 5.82,< 23 235.78 10 5.82 10 .� < � Because 23 2310 10 ,- < 23 236.88 10 5.78 10 .-� < �

From least to greatest, the order is 23 23 236.88 10 , 5.78 10 , and 5.82 10 .-� � �

20. 50.000099 9.9 10 5

-= �

The diameter of a human hair is 59.9 10-� meter.

21. 740,100,000 4.01 107

= �

The circumference of Earth is about 74.01 10� meters.

22. Sample answer: kilometers or miles; They are both larger units of length, so the number would be smaller.

23. 3

2

3

68,500 6850 6.85 1010

680 6.8 10

6.8 10

= = �

= ��

Because 6.8 6.85,< 3 36.8 10 6.85 10 .� < � Because 2 310 10 ,< 2 36.8 10 6.8 10 .� < �

The order from least to greatest is 680, 36.8 10 ,�

and 68,500.10

24. 2

2

2

5 0.0207 2.07 102410.02 2.0 10

2.1 10

-

-

-

� = �

= ��

Because 2.0 2.07 2.1,< < 2 2 22.0 10 2.07 10 2.1 10 .- - -� < � < �

The order from least to greatest is 0.02, 5 ,241

and

22.1 10 .-�

25. 2

3

0

1

6.3% 0.063 6.3 10

6.25 1016 6.25 6.25 104

0.625 6.25 10

-

-

-

= = ��

= = �

= �

Because 3 2 1 010 10 10 10 ,- - -< < < 3 2 1 06.25 10 6.3 10 6.25 10 6.25 10 .- - -� < � < � < �

The order from least to greatest is 36.25 10 ,-� 6.3%,

0.625, and 16 .4

26. 3

0

3

2

3033.4 3.0334 10

305% 3.05 3.05 1010,000 3333.3 3.3333 10

33.3 10

= �= = �

� = �

�

Because 3.0334 3.3333,< 3 33.0334 10 3.3333 10 .� < �

Because 0 2 310 10 10 ,< < 0 2 3 33.05 10 3.3 10 3.0334 10 3.3333 10 .� < � < � < �

The order from least to greatest is 305%, 23.3 10 ,�

3033.4, and 10,000.3

27. 101.174 10 11,740,000,000� =

9

Power from Total Power from solidmain engines power rocket boosters

11,740,000,000 9,750,000,0001,990,000,000

1.99 10

= -

= -== �

The power from the main engines is 91.99 10� watts.

28. Sample answer: Enter E E1.174 10 9.75 9.-

Chapter 10


29. 2324

2021

1 1.2 108.3 10

1 6.02 101.66 10

-

-

� ��

� ��

So, 1 carat 231.2 10� � atomic mass units and 1 milligram 206.02 10� � atomic mass units. Because

23 201.2 10 6.02 10 ,� > � a carat is greater than a milligram.

30. a. Paleozoic era: 8 8

8

5.42 10 2.51 10 542,000,000 251,000,000 291,000,000

2.91 10 years

� - �= -== �

Mesozoic era: 8 7

8

2.51 10 6.55 10 251,000,000 65,500,000 185,500,000

1.855 10 years

� - �= -== �

Cenozoic era: 76.55 10 years�

Because 1.855 2.91,< 8 81.855 10 2.91 10 .� < � Because 7 810 10 ,<

7 8 86.55 10 1.855 10 2.91 10 .� < � < � The order of the lengths from least to greatest is

76.55 10 years,� 81.855 10 years,� and 82.91 10 years.�

b.

c. The statement “the older the era, the longer it is” holds true for the three eras (Phanerozoic eon). The table below shows the seven eras leading up to the Paleozoic era, and when each era began.

Find the length of each era.

9 8

8

Neoproterozoic era 1.0 10 5.42 101,000,000,000 542,000,000458,000,000

4.58 10

= � - �= -== �

9 9

9

8

Mesoproterozoic era 1.6 10 1.0 10

0.6 10

6.0 10

= � - �= �= �

9 9

9

8

Paleoproterozoic era 2.5 10 1.6 10

0.9 10

9.0 10

= � - �= �= �

9 9

9

8

Neoarchean era 2.8 10 2.5 10

0.3 10

3.0 10

= � - �= �= �

9 9

9

8

Mesoarchean era 3.2 10 2.8 10

0.4 10

4.0 10

= � - �= �= �

9 9

9

8

Paleoarchean era 3.6 10 3.2 10

0.4 10

4.0 10

= � - �= �= �

9 9

9

8

Eoarchean era 4.0 10 3.6 10

0.4 10

4.0 10

= � - �= �= �

So, the statement “the older the era, the longer it is” also holds true for the eras in the Proterozoic eon but not for the eras in the Archean eon.

Era Began

Neoproterozoic 91.0 10 years ago�

Mesoproterozoic 91.6 10 years ago�

Paleoproterozoic 92.5 10 years ago�

Neoarchean 92.8 10 years ago�

Mesoarchean 93.2 10 years ago�

Paleoarchean 93.6 10 years ago�

Eoarchean 94.0 10 years ago�

0Cenozoic

Eras

Mesozoic

Paleozoic

100

200

300

400

500

Millions of years

Chapter 10


321

Fair Game Review 31. The number 15 is a natural number, a whole number, an

integer, and a rational number.

32. Because 3 8 2,- = - the number 3 8- is an integer and a rational number.

33. The number 73 is not a perfect square, so, 73 is irrational.

34. D;

( )( ) ( )( ) ( )( )2 2 22 2.5 2 2 2.5 1 2 2 1

10 5 419

S w h wh= + += + += + +=

� �

The surface area of the prism is 19 square inches.

Section 10.7 10.7 Activity (pp. 448–449) 1. a. First evaluate factors with exponents. Next, complete

all multiplication from left to right. Then, add the terms together.

3 32.4 10 7.1 10 2.4 1000 7.1 10002400 71009500

� + � = � + �= +=

b. 310 ; ( ) ( ) ( )3 3 32.4 10 7.1 10 2.4 7.1 10� + � = + �

c. ( ) 3 32.4 7.1 10 9.5 10

9.5 10009500

+ � = �= �=

The results are the same. d. Add the factors.

( ) ( ) ( )5 5 5

5

4.9 10 1.8 10 4.9 1.8 10

6.7 10

� + � = + �

= �

( ) ( ) ( )4 4 4

4

3.85 10 5.72 10 3.85 5.72 10

9.57 10

� + � = + �

= �

2. a. First evaluate factors with exponents. Next, complete all multiplication from left to right. Then, add the terms together.

3 42.4 10 7.1 10 2.4 1000 7.1 10,0002400 71,00073,400

� + � = � + �= +=

b. The powers of 10 are not the same. c. Sample answer: There is no common factor to factor

out.

d. Move the decimal of the factor to adjust the exponent of the power of 10 to match the other number.

3 1 3 42.4 10 0.24 10 10 0.24 10� = � � = �

( ) ( ) ( )4 4 4

4

0.24 10 7.1 10 0.24 7.1 10

7.34 1073,400

� + � = + �

= �=

The results are the same. e. yes; Sample answer: You can subtract the factors

when the powers of 10 are the same. When the powers of 10 are different, rewrite one of the numbers.

( ) ( ) ( )5 5 5

5

8.2 10 4.6 10 8.2 4.6 10

3.6 10

� - � = - �

= �

Rewrite 41.5 10� so that it has the same power of 10 as 54.6 10 .�

4 1 4 51.5 10 0.15 10 10 0.15 10� = � � = � Subtract the factors.

( ) ( ) ( )5 5 5

5

5.88 10 0.15 10 5.88 0.15 10

5.73 10

� - � = - �

= �

3. 1. D 2. F 3. E 4. C 5. B 6. A

yes; ( ) ( )( ) ( )

2 5 2 5

2 5

7

1.9 10 2.3 10 1.9 2.3 10 10

1.9 2.3 10 10

4.37 1043,700,000

� � � = � � �

= � � �

= �=

( ) ( )( ) ( )

6 4 6 4

6 4

2

1 2

3

8.4 10 5.7 10 8.4 5.7 10 10

8.4 5.7 10 10

47.88 10

4.788 10 10

4.788 104788

- -

-

� � � = � � �

= � � �

= �= � �= �=

Chapter 10


4. Sample answer:

( ) ( )( ) ( )

3 4 3 4

3 4

7

1 7

8

9 10 3 10 9 3 10 10

9 3 10 10

27 10

2.7 10 10

2.7 10270,000,000

� � � = � � �

= � � �

= �= � �= �=

A person breathes about 270,000,000 liters of air over a lifetime.

5. Sample answer: To add or subtract numbers written in scientific notation, add or subtract the factors when the powers of 10 are the same. When the powers of 10 are different, first use properties of exponents to rewrite the numbers so the powers are the same.

To multiply or divide numbers written in scientific notation, use the properties of multiplication to rewrite the expressions. Then use the Product of Powers Property and the Quotient of Powers Property to simplify.

6. a. ( ) ( )4 4 41.5 10 6.3 10 7.8 10 78,000� + � = � =

b. ( ) ( )5 3 57.2 10 2.2 10 7.178 10 717,800� - � = � =

c. ( ) ( )3 3 54.1 10 4.3 10 1.763 10

0.00001763

- - -� � � = �

=

d. ( ) ( )6 7 14.75 10 1.34 10 6.365 10 63.65-� � � = � =

10.7 On Your Own (pp. 450–451) 1. The numbers do not have the same power of 10. Rewrite

13.41 10-� so it has the same power of 10 as 28.2 10 .�

1 3 1 23.41 10 0.00341 10 10 0.00341 10- -� = � � = � Add the factors.

( ) ( ) ( )2 2 2

2

8.2 10 0.00341 10 8.2 0.00341 10

8.20341 10

� + � = + �

= �

2. ( ) ( ) ( )5 5 5

5

7.8 10 4.5 10 7.8 4.5 10

3.3 10

- - -

-

� - � = - �

= �

3. ( ) ( )5 5

5

1 5

4

6 8 10 6 8 10

48 104.8 10 104.8 10

- -

-

-

-

� � = � �

= �= � �= �

4. ( ) ( )( ) ( )

2 5 2 5

2 5

7

1 7

8

7 10 3 10 7 3 10 10

7 3 10 10

21 10

2.1 10 10

2.1 10

� � � = � � �

= � � �

= �= � �= �

5. 12

12

12

12

9.2 109.2 10 4.64.6

9.2 104.62 10

�� =

= �

= �

6. ( ) ( )3

3 22

3

2

3

2

5

1 5

6

1.5 101.5 10 7.5 107.5 101.5 107.5 10

100.210

0.2 10

2 10 10

2 10

--

-

-

-

- -

-

�� =�

= �

= �

= �= � �= �

7. The diameter of the Sun is 1,400,000 kilometers, so the radius is 5700,000 7 10 kilometers.= �

The diameter of Earth is 41.28 10 kilometers,� so the

radius is 4

4

4

1 4

3

1.28 10 1.28 102 2

0.64 10

6.4 10 10

6.4 10 kilometers.

-

� = �

= �= � �= �

Find ( ) ( )5 37 10 6.4 10 .� - �

The numbers do not have the same power of 10. Rewrite 36.4 10� so it has the same power of 10 as 57 10 .�

3 2 3 56.4 10 0.064 10 10 0.064 10� = � � = � Subtract the factors. ( ) ( ) ( ) ( )

( )

5 3 5 5

5

5

7 10 6.4 10 7 10 0.064 10

7 0.064 10

6.936 10693,600

� - � = � - �

= - �= �=

The radius of the Sun is about 693,600 more kilometers than the radius of Earth.

Expression Answer wr itten in scientific notation

Number of minutes in a day 60 24� 31.5 10� (estimate)

Liter s of air breathed in

one day ( )36 1.5 10� 39.0 10�

Number of days in 80 years 80 365� 43.0 10� (estimate)

Chapter 10


323

10.7 Exercises (pp. 452–453)

Vocabulary and Concept Check 1. Use the Distributive Property to group the factors

together. Then subtract the factors and write it with the power of 10. The number may need to be rewritten so that it is still in scientific notation.

2. no; You can use the Commutative and Associative Properties of Multiplication to group the factors and the powers of 10. Then, you multiply the factors and multiply the powers of 10.

Practice and Problem Solving 3. Method 1:

( ) ( )7 7

7

2.74 10 5.6 10 27,400,000 56,000,000

83,400,000

8.34 10

� + � = +

== �

Method 2:

( ) ( ) ( )7 7 7

7

2.74 10 5.6 10 2.74 5.6 10

8.34 10

� + � = + �

= �

4. Method 1:

( ) ( )6 5

6

8.3 10 3.4 10 8,300,000 340,000

8,640,000

8.64 10

� + � = +

== �

Method 2: The numbers do not have the same power of 10. Rewrite

53.4 10� so it has the same power of 10 as 68.3 10 .�

5 1 5 63.4 10 0.34 10 10 0.34 10� = � � = � Add the factors.

( ) ( ) ( ) ( )( )

6 5 6 6

6

6

8.3 10 3.4 10 8.3 10 0.34 10

8.3 0.34 10

8.64 10

� + � = � + �

= + �

= �

5. Method 1:

( ) ( )5 5

11

5.1 10 9.7 10 510,000 970,000

494,700,000,000

4.947 10

� � � = �

== �

Method 2:

( ) ( )( ) ( )

5 5 5 5

5 5

10

1 10

11

5.1 10 9.7 10 5.1 9.7 10 10

5.1 9.7 10 10

49.47 10

4.947 10 10

4.947 10

� � � = � � �

= � � �

= �= � �= �

6. Method 1:

( ) ( )4 3

8

4.5 10 6.2 10 45,000 6200

279,000,000

2.79 10

� � � = �

== �

Method 2:

( ) ( )( ) ( )

4 3 4 3

4 3

7

1 7

8

4.5 10 6.2 10 4.5 6.2 10 10

4.5 6.2 10 10

27.9 10

2.79 10 10

2.79 10

� � � = � � �

= � � �

= �= � �= �

7. ( ) ( ) ( )5 5 5

5

2 10 3.8 10 2 3.8 10

5.8 10

� + � = + �

= �

8. ( ) ( ) ( )9 9 9

9

6.33 10 4.5 10 6.33 4.5 10

1.83 10

- - -

-

� - � = - �

= �

9. ( ) ( ) ( )8 8 8

8

9.2 10 4 10 9.2 4 10

5.2 10

� - � = - �

= �

10. ( ) ( ) ( )6 6 6

6

1 6

5

7.2 10 5.44 10 7.2 5.44 10

12.64 101.264 10 101.264 10

- - -

-

-

-

� + � = + �

= �= � �= �

11. The numbers do not have the same power of 10. Rewrite 62.45 10� so it has the same power of 10 as 77.8 10 .�


( ) ( ) ( ) ( )( )

7 6 7 7

7

7

7.8 10 2.45 10 7.8 10 0.245 10

7.8 0.245 10

7.555 10

� - � = � - �

= - �

= �

12. The numbers do not have the same power of 10. Rewrite 55 10-� so it has the same power of 10 as 32.46 10 .-�

5 2 5 35 10 0.05 10 10 0.05 10- - -� = � � = � Add the factors.

( ) ( )( ) ( )( )

5 3

3 3

3

3

5 10 2.46 10

0.05 10 2.46 10

0.05 2.46 10

2.51 10

- -

- -

-

-

� + �

= � + �

= + �= �

Chapter 10



5 1 5 66.7 10 0.67 10 10 0.67 10� = � � = � Add the factors.

( ) ( ) ( ) ( )( )

6 5 6 6

6

6

1 6

7

9.7 10 6.7 10 9.7 10 0.67 10

9.7 0.67 10

10.37 101.037 10 101.037 10

� + � = � + �

= + �= �= � �= �

14. The numbers do not have the same power of 10. Rewrite 25.5 10-� so it has the same power of 10 as 12.4 10 .-�

2 1 2 15.5 10 0.55 10 10 0.55 10- - -� = � � = � Subtract the factors.

( ) ( ) ( ) ( )( )

1 2 1 1

1

1

2.4 10 5.5 10 2.4 10 0.55 10

2.4 0.55 10

1.85 10

- - - -

-

-

� - � = � - �

= - �

= �


8 1 8 95.3 10 0.53 10 10 0.53 10� = � � = � Add the factors. ( ) ( ) ( ) ( )

( )9 8 9 9

9

9

2.5 10 5.3 10 2.5 10 0.53 10

2.5 0.53 10

3.03 10

� + � = � + �

= + �= �

16. ( ) ( )7 7

7

1 7

8

5 7 10 5 7 10

35 103.5 10 103.5 10

� � = � �

= �= � �= �

17. ( ) ( )6

6 33

6

3

6

3

3

5.8 105.8 10 2 102 10

5.8 102 10

102.910

2.9 10

-- -

-

-

-

-

-

-

�� =�

= �

= �

= �

18. ( )5

5

5

5

1 5

6

1.2 101.2 10 44

1.2 104

0.3 103 10 103 10

--

-

-

- -

-

�� =

= �

= �= � �= �

19. ( ) ( )( ) ( )

7 6 7 6

7 6

1

1 1

0

5 10 3 10 5 3 10 10

5 3 10 10

15 10

1.5 10 10

1.5 10

- -

-

-

-

� � � = � � �

= � � �

= �= � �= �

20. ( ) ( )7

7 77

7

7

7

7

0

1 0

1

3.6 103.6 10 7.2 107.2 103.6 107.2 10

100.510

0.5 10

5 10 10

5 10

-

-

�� =�

= �

= �

= �= � �= �

21. ( ) ( )( ) ( )

1 7 1 7

1 7

8

1 8

7

7.2 10 4 10 7.2 4 10 10

7.2 4 10 10

28.8 10

2.88 10 10

2.88 10

- - - -

- -

-

-

-

� � � = � � �

= � � �

= �= � �= �

22. ( ) ( )( ) ( )

8 5 8 5

8 5

3

6.5 10 1.4 10 6.5 1.4 10 10

6.5 1.4 10 10

9.1 10

- -

-

� � � = � � �

= � � �

= �

23. ( ) ( )4

4 66

4

6

4

6

2

2.8 102.8 10 2.5 102.5 102.8 102.5 10

101.1210

1.12 10-

�� =�

= �

= �

= �

24. Divide the thickness of a dime by the thickness of a dollar bill.

1

2 2

1

2

1

0.135 1.35 101.0922 10 1.0922 10

1.35 101.0922 101.236 1012.36

-

- -

-

-

�=� �

= �

� �=

The thickness of a dime is about 12 times greater than the thickness of a dollar bill.

Chapter 10


325

25. ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( )( )

2 8

6 2 8

6 2 8

8 8

8

8

1 8

9

5,200,000 8.3 10 3.1 10

5.2 10 8.3 10 3.1 10

5.2 8.3 10 10 3.1 10

43.16 10 3.1 10

43.16 3.1 10

40.06 104.006 10 104.006 10

� � - �

= � � � - �

= � � � - �

= � - �

= - �= �= � �= �

26. ( ) ( )( ) ( ) ( )( )

( )

( )( ) ( )( ) ( )( ) ( )( )

3 5

3 5 3

53

3

53

3

53

3

3 2

1 3 2

2 2

2

2

9 10 2.4 10 0.0012

9 10 2.4 10 1.2 10

2.4 109 101.2 102.4 109 101.2 10

109 10 210

9 10 2 10

0.9 10 10 2 10

0.9 10 2 10

0.9 2 10

2.9 10

- -

- - -

--

-

--

-

--

-

- -

- -

- -

-

-

� + � �

= � + � � �

�= � +�

� �= � + �� = � + ��

= � + �

= � � + �

= � + �

= + �= �

27.

( )14 7

14

7

14

7

14

7

7

1 7

6

5.612 10 9.2 10

5.612 109.2 10

5.612 109.2 10

100.6110

0.61 106.1 10 10

6.1 10

A w

w

w

w

w

www

-

= �� = � �

� =�

� =

� =

� =� � =

� =

�

The width is 66.1 10� centimeters.

( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( )

7 6

7 6

7 6

1 7 2 6

8 8

8

8

2 2

2 9.2 10 2 6.1 10

2 9.2 10 2 6.1 10

18.4 10 12.2 10

1.84 10 10 0.122 10 10

1.84 10 0.122 10

1.84 0.122 10

1.962 10

P w= += � + �

= � � + � �

= � + �

= � � + � �

= � + �

= + �= �

�

The perimeter is 81.962 10� centimeters.

28.

( )7

72 beats 60 min 24 hr 365 days1 min 1 hr 1 day 1 yr

37,843,200 beatsyr

3.78432 10 beatsyr

� � �

=

�=

( ) ( )

( ) ( )

2 7

2 7

2 7

5

1 5

6

7 10 L 3.78432 10 beatsbeat yr

7 3.78432 10 10

7 3.78432 10 10

26.49024 10

2.649024 10 10

2.649024 10 Lyr

-

-

-

� ��

= � � �

= � � �

= �= � �

�=

A human heart pumps about 62.65 10� liters of blood in one year.

( ) ( )

( ) ( )

6

6 1

6 1

6 1

7

1 7

8

2.649024 10 L 70 yryr

2.649024 10 7 10

2.649024 7 10 10

2.649024 7 10 10

18.543168 10

1.8543168 10 10

1.8543168 10

� �

= � � �

= � � �

= � � �

= �= � �= �

In 70 years, the human heart pumps about 81.85 10�liters of blood.

Sample answer: For estimation, use the average human heart rate as 70 beats per minute.

( ) ( )

( ) ( )

2 1

2 1

2 1

1

1 1

0

7 10 L 7 10 beatsbeat min

7 7 10 10

7 7 10 10

49 10

4.9 10 10

4.9 10 Lmin

-

-

-

-

-

� ��

= � � �

= � � �

= �= � �

�=

So, the heart beats about 5 liters per minute.

Chapter 10


( )6

5 L 60 min 24 hr 365 days1 min 1 hr 1 day 1 yr

2,628,000 Lyr

2.628 10 Lyr

� � �

=

�=

So, a human heart pumps about 62.63 10� liters of blood in one year.

( ) ( )

( ) ( )

6

6 1

6 1

6 1

7

1 7

8

2.63 10 L 70 yryr

2.63 10 7 10

2.63 7 10 10

2.63 7 10 10

18.41 10

1.841 10 10

1.841 10

� �

= � � �

= � � �

= � � �

= �= � �= �

In 70 years, the human heart pumps about 81.84 10�liters of blood.

The answers are justified because the estimations are close to the solutions.

29. Number Length NumberDiameter of ridges of ridge of valleysLength Diameter ofof valley center of disc

= � +

� +

( ) ( ) ( )( )

( ) ( )

4 5 4

5

4 5

4 5

1 1

73,000 0.000032 73,000 0.000074 4.26

7.3 10 3.2 10 7.3 10

7.4 10 4.26

7.3 3.2 10 10 7.3 7.4

10 10 4.26

23.36 10 54.02 10 4.26

2.336 5.402 4.2611.99812

-

-

-

-

- -

= � +� +

= � � � + �� +

= � � � + �

� � +

= � + � +

= + +=�

The diameter of the DVD is about 12 centimeters.

30. a. Sample answer:

b. 9 9

36 6

1.189 10 1.189 10 0.937 10 9371.269 10 1.269 10

� = � � � =�

India has approximately 937 people per square mile.

9 9

36 6

1.337 10 1.337 10 0.361 10 3613.705 10 3.705 10

� = � � � =�

China has approximately 361 people per square mile.

7 7

16 6

4.2 10 4.2 10 3.9 10 391.068 10 1.068 10

� = � � � =�

Argentina has approximately 39 people per square mile.

8 8

26 6

3.13 10 3.13 10 0.84 10 843.719 10 3.719 10

� = � � � =�

The United States has approximately 84 people per square mile.

7 7

25 5

8.2 10 8.2 10 2.12 10 2123.87 10 3.87 10

� = � � � =�

Egypt has approximately 212 people per square mile. So, the population densities from least to greatest are

Argentina, the United States, Egypt, China, and India.

31.

( )

( )

( )( )( )

2

11 27 2

112

27

112

27

11 27 2

16 2

8 8 2

28 2

8

15.066 10 1.674 10

15.066 101.674 10

15.066 101.674 109 10

9 10

3 10 3 10

3 10

3 10

E mc

c

c

c

c

c

c

c

c

- -

-

-

-

-

- - -

=

� = �

� =�

� =

� �� =

� � =

� =

� =

The speed of light is approximately 83 10� meters per second.

Population in 2011 (Nearest million)

Area (Nearest square mile)

India 91.189 10� 61.269 10�

China 91.337 10� 63.705 10�

Argentina 74.2 10� 61.068 10�

United States 83.13 10� 63.719 10�

Egypt 78.2 10� 53.87 10�

Chapter 10


327

Fair Game Review

32. Because ( ) ( )3 33 3729 9 , 729 9 9.- = - - = - = -

33. Because 3 3

331 1 1 1 1, .

512 8 512 8 8� � � �= = =� � � ��

34. Because 3 3

33125 5 125 5 5, .343 7 343 7 7

� � � �- = - - = - = -� � � ��

35. C; ( ) ( )2 31 1 4 9 48 cm3 3

V Bh p p= = =

Quiz 10.5–10.7 1. The factor is greater than 10. So, the number 923 10�

is not written in scientific notation.

2. The factor is less than 1. So, the number 70.6 10-� is not written in scientific notation.

3. 68 10 8,000,0006

� =


4. 21.6 10 0.0162

-� =


5. 30.00524 5.24 103

-= �


6. 8892,000,000 8.92 108

= �


7. ( ) ( ) ( )4 4 4

4

1 4

5

7.26 10 3.4 10 7.26 3.4 10

10.66 10

1.066 10 10

1.066 10

� + � = + �

= �= � �= �

8. Rewrite 61.6 10-� so that it has the same power of 10 as 52.8 10 .-�

6 1 6 51.6 10 0.16 10 10 0.16 10- - -� = � � = � Subtract the factors.

( ) ( ) ( )5 5 5

5

2.8 10 0.16 10 2.8 0.16 10

2.64 10

- - -

-

� - � = - �

= �

9. ( ) ( )( ) ( )

( )

4 6 4 6

4 6

4 6

2

2.4 10 3.8 10 2.4 3.8 10 10

2.4 3.8 10 10

9.12 10

9.12 10

- -

-

+ -

-

� � � = � � �

= � � �

= �= �

10. ( ) ( )

( )

33 12

12

3

12

3

12

3 12

3 12

9

5.2 105.2 10 1.3 101.3 105.2 101.3 10

10410

4 10

4 10

4 10

-- -

-

-

-

-

-

- - -

- +

�� =�

= �

= �

= �= �= �

11. a. Mercury: 32.44 10 2440 km� =

Venus: 36.05 10 6050 km� =

Earth: 36.38 10 6380 km� =

Mars: 33.4 10 3400 km� =

Jupiter: 47.15 10 71,500 km� =

Saturn: 46.03 10 60,300 km� =

Uranus: 42.56 10 25,600 km� =

Neptune: 42.48 10 24,800 km� =

The planet that has the second-smallest equatorial radius is Mars.

b. The planet that has the second-largest equatorial radius is Saturn.

12. ( ) ( )( ) ( )

5 8 5 8

5 8

5 8

13

2 10 1.5 10 2 1.5 10 10

2 1.5 10 10

3 10

3 10

+

� � � = � � �

= � � �

= �= �

The Oort Cloud is 133 10� kilometers from the Sun.

13. 30.0015 1.5 103

-= �

The number in scientific notation is 31.5 10 meter.-�

14. 8 8

62 2

2.3 10 2.3 10 0.92 10 920,0002.5 10 2.5 10

� = � = � =�

Pluto will orbit 920,000 times around the Sun while the Sun completes one orbit around the Milky Way.

Chapter 10


Chapter 10 Review 1. ( ) ( ) ( ) ( ) ( )9 9 9 9 9- � - � - � - � -


So, ( ) ( ) ( ) ( ) ( ) ( )59 9 9 9 9 9 .- � - � - � - � - = -

2. 2 2 2 n n� � � � Because 2 is used as a factor 3 times, the exponent is 3.

Because n is used as a factor 2 times, the exponent is 2.

So, 3 22 2 2 2 .n n n� � � � =

3. 36 6 6 6 216= � � =

4. 41 1 1 1 1 1

2 2 2 2 2 16� � � �- = - � � � = -� � � ��

5. ( ) ( )

( )

31 116 6 16 2162 2

1 2002100

100

- = -

= -

= -=

6. 5 2 5 2 7p p p p+� = =

7. ( )211 11 2 22n n n�= =

8. ( )3 3 3

3

5 5

125

y y

y

= �

=

9. ( ) ( )4 4 4

4

2 2

16

k k

k

- = - �

=

10. 8

8 3 53

8 8 88

-= =

11. 2 9 2 9 11

11 1 105 5 5 5 5 55 5 5

+-� = = = =

12. 8 5

8 7 5 2 1 3 1 3 47 2

w w w w w w w ww w

- - +� = � = � = =

13. 2 5 2 5 7

7 3 43 3 3

2 2 2 2 2 2 162 2 2

+-� = = = = =

14. ( )3 3 3

3 3 1

2

6 6

6

216

c cc c

c

c

-

�=

= �=

15. 8 10

8 6 10 9 2 1 2 1 36 9

m m m m m m m mm m

- - +� = � = � = =

16. 44

1 122 16

- = = 17. 095 1=

18. 2

2 4 24 2

8 1 18 88 8 64

- -= = = =

19. ( ) ( ) ( ) ( )7 7 7 7 012 12 12 12 1- - +- � - = - = - =

20. 9 6 9 6 9 6 31 1 1 1 1 17 7 7 7 7 7 343- - -� = = = =

�

21. 4 2 4 2 2

2 2 02 2 2

9 9 9 9 9 9 19 9 9

- --� = = = = =

22. 72 10 20,000,000 7

� =


23. 23.4 10 0.0342

-� =


24. 91.5 10 0.0000000015 9

-� =


25. 105.9 10 59,000,000,000 10

� =

The number in standard form is 59,000,000,000.

26. 34.8 10 0.00483

-� =


27. 56.25 10 625,0005

� =

The number in standard form is 625,000.

28. 40.00036 3.6 104

-= �


29. 5800,000 8 10 5

= �


Chapter 10


329

30. 779,200,000 7.92 107

= �


31. Rewrite 84.2 10� so that it has the same power of 10 as 95.9 10 .�

8 1 8 94.2 10 0.42 10 10 0.42 10� = � � = � Add the factors.

( ) ( ) ( ) ( )( )

8 9 9 9

9

9

4.2 10 5.9 10 0.42 10 5.9 10

0.42 5.9 10

6.32 10

� + � = � + �

= + �

= �

32. ( ) ( ) ( )4 4 4

4

5.9 10 1.8 10 5.9 1.8 10

4.1 10

- - -

-

� - � = - �

= �

33. ( ) ( )( ) ( )

( )

8 5 8 5

8 5

8 5

3

1 3

1 3

4

7.7 10 4.9 10 7.7 4.9 10 10

7.7 4.9 10 10

37.73 10

37.73 10

3.773 10 10

3.773 10

3.773 10

- -

-

+ -

+

� � � = � � �

= � � �

= �= �= � �= �= �

34. ( ) ( )5

5 99

5

9

5

9

5 9

4

3.6 103.6 10 1.8 101.8 103.6 101.8 10

10210

2 10

2 10

-

-

�� =�

= �

= �

= �= �

Chapter 10 Test 1. ( ) ( ) ( )15 15 15- � - � -


So, ( ) ( ) ( ) ( )315 15 15 15 .- � - � - = -

2. 1 1 1 1 112 12 12 12 12

� � � � � � � � � ��

Because 112


So, 51 1 1 1 1 1 .

12 12 12 12 12 12� � � � � � � � � � � �� =� � � � � � � � � � � ��

3. ( )32 2 2 2 8- = - � � = -

4. 310 3 9 10 27 9 10 3 13+ � = + � = + =

5. 10 10 1 119 9 9 9+� = =

6. ( )56 6 5 306 6 6�= =

7. ( )7 7 72 10 2 10� = �

8. ( )( ) ( ) ( )

1313 9 4

93.5

3.5 3.53.5

-- = - = --

9. 2 2 2 2 05 5 5 5 1- - +� = = =

10. ( )

( ) ( )( )

1 3 23 2

8 1 18 8648 8

- -- = - = - = =- -

11. 73 10 30,000,000 7

� =


12. 39.05 10 0.009053

-� =


13. ( ) ( ) ( )7 7 7

7

1 7

8

7.8 10 9.9 10 7.8 9.9 10

17.7 10

1.77 10 10

1.77 10

� + � = + �

= �= � �= �

14. Rewrite 45.4 10� so that it has the same power of 10 as 56.4 10 .�


( ) ( ) ( )5 5 5

5

6.4 10 0.54 10 6.4 0.54 10

5.86 10

� - � = - �

= �

15. ( ) ( )( ) ( )

( )

6 2 6 2

6 2

6 2

4

3.1 10 2.7 10 3.1 2.7 10 10

3.1 2.7 10 10

8.37 10

8.37 10

- -

-

+ -

� � � = � � �

= � � �

= �= �

Chapter 10


16. ( ) ( )

( )

77 4

4

7

4

7

4

7 4

7 4

11

9.6 109.6 10 1.2 101.2 109.6 101.2 10

10810

8 10

8 10

8 10

--

-

-

- -

+

�� =�

= �

= �

= �= �= �

17. no;

( ) ( )3 32 3 2 3 2 3 3 6xy x y x y x y�= = =

( ) ( )2 23 2 3 2 3 2 2 6xy x y x y x y�= = =

18. 9

9 3 63

3 3 3 7293

-= = =

There are about 729 grains of rice in one scoop.

19. 410,000 1 104

= �


20. new amount original amountpercent decreaseoriginal amount

-=

Let x represent the new amount of lead allowed in the air.

( )

( ) ( )( ) ( )

( ) ( )( )

6

6

6 6

6 6

6 6

6

6

1 6

7

1.5 100.9

1.5 100.9 1.5 10 1.5 10

1.35 10 1.5 10

1.35 10 1.5 10

1.35 1.5 10

0.15 10

1.5 10 10

1.5 10

x

x

x

x

x

x

x

x

-

-

- -

- -

- -

-

-

- -

-

- �- =

�- � = - �

- � = - �

- � + � =

- + � =

� =� � =

� =

The new amount of lead allowed in the air is 71.5 10-�gram per cubic meter.

Chapter 10 Standards Assessment 1. C; 75.79 10 57,900,000

7� =

Mercury’s distance to the Sun is approximately 57,900,000 kilometers.

2. I; Jim stopped when he found the value of x. However, the question asks for the angle measure of the largest angle. So, Jim should evaluate the largest angle when

15.x =

3. D; 4 3 4 3 72 2 2 2 128+= = =

4. 40; In similar triangles, the corresponding angles are equal. So, A� is equal to ,D� which means that

40.x =

5. F; 2 5000n �

6. B; 13

33

V Bh

V BhV hB

=

=

=

7. Part A: The GDP per person is GDP divided by the population.

GDPGDP per personPopulation15,100,000,000,000

312,000,00048,397.44

=

=

�

So, the GDP per person for the United States is about $48,397.44 per person.

Part B:

Population: 8312,000,000 3.12 108

= �

GDP: 1315,100,000,000,000 1.51 1013

= �

The population is 83.12 10� and the GDP is 131.51 10�dollars.

Part C:

13

8

13

8

13 8

5

1 5

1 5

4

GDPGDP per personPopulation

1.51 103.12 101.51 103.12 100.484 10

0.484 10

4.84 10 10

4.84 10

4.84 10

-

-

- +

=

�=�

= �

� �= �= � �= �= �

So, the GDP per person for the United States is about44.84 10� dollars.

Chapter 10


331

8. F; rise 4 2 2sloperun 3 3 6 3

- 1= = = = -- - -

The line crosses the y-axis at ( )0, 3 . So, the y-intercept

is 3. Because the slope is 13

- and the y-intercept is 3, the

equation of the line is 1 3.3

y x= - +

9. C; The diameter is 6 centimeters, so the radius is 3 centimeters.

( ) ( )23 5

4545 3.14141.3

V Bh

pp

=

==� �=

The volume is about 141.3 cubic centimeters.

10. 0.16 or 4 ;25

( )( )

( ) ( )

22

12.52.5

12.5 2.5

16.25

40.16, or 25

-- =-

=- � -

=

=

11. G; Because the lines have different slopes, the lines meet at exactly one point.

12. B; A circle graph is best suited for showing the data because circle graphs represent data as parts of a whole. So, a circle graph will clearly show the majority of the money is used for research.

mscc8 ws 1000a.docx - blue solutions 10 · 2015. 3. 25. · ! " # $ % & ' ( ) * + * ,...

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