Download Chapter 10 - Waynesville R

Chapter 10
13.9
Chapter 10 Opener
9. 0.3 4.17  3 41.7
-3
11
-9
27
-27
0
Try It Yourself (p. 409)
8
 4
1. 15  + 2 2 - 3 • 7 = 15( 2) + 22 - 3 • 7
= 15 • 2 + 4 - 3 • 7
= 30 + 4 - 21
So, 4.17 ÷ 0.3 = 13.9.
= 34 - 21
24
= 13
10. 0.15 3.60  15 360
- 30
60
- 60
0
2. 52 • 2 ÷ 10 + 3 • 2 - 1 = 25 • 2 ÷ 10 + 3 • 2 - 1
= 50 ÷ 10 + 6 - 1
= 5+ 6 -1
So, 3.6 ÷ 0.15 = 24.
= 11 - 1
= 10
(
)
1800
(
3. 32 - 1 + 2 4(3 + 2) = 32 - 1 + 2 4(5)
)
11. 0.004 7.200  4 7200
-4
3200
- 3200
0
= 3 - 1 + 2( 20)
2
= 9 - 1 + 2 • 20
= 9 - 1 + 40
= 8 + 40
= 48
4.
1. a.
Power
Repeated
Multiplication Form
Value
( -3)
1
-3
-3
(-3)
2
(-3) • ( -3)
9
(-3)
3
(-3) • ( -3) • ( -3)
-27
(-3)
4
(-3) • (-3) • (-3) • ( -3)
81
(-3)
5
So, 7.03 • 4.3 = 30.229.
(-3)
6
0.894
 0.2
0.1788
(-3)
7
1.4
 0.6
0.84
So, 1.4 • 0.6 = 0.84.
6.
7.
Section 10.1
10.1 Activity (pp. 410 ––411)
1.75
 0.2
0.350
So, 1.75 • 0.2 = 0.35.
5.
So, 7.2 ÷ 0.004 = 1800.
7.03
 4.3
2109
2812
30.229
So, 0.894 • 0.2 = 0.1788.
60
8. 0.09 5.40  9 540
- 540
0
(-3) • ( -3) • ( -3)
• ( -3) • ( -3)
(-3) • ( -3) • ( -3) • ( -3)
• ( -3) • ( -3)
(-3) • ( -3) • ( -3) • ( -3)
• ( -3) • ( -3) • ( -3)
-243
729
-2187
b. The expression ( -3) means -3 raised to an exponent
n
of n. To find the value of ( -3) , multiply ( -3) as a
n
factor n times.
So, 5.40 ÷ 0.09 = 60.
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301
Chapter 10
2. a. The large cube is made up of 3 • 3 • 3 small cubes.
2. 0.3 • 0.3 • 0.3 • 0.3 • x • x
Because each small cube contains $3, the total amount
of money in the large cube is 3 • 3 • 3 • 3 = 34.
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, 0.3 • 0.3 • 0.3 • 0.3 • x • x = (0.3) x 2 .
4
b. 34 = 81
There is $81 in the large cube.
3. a. 1026 = 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.
3. -54 = -(5 • 5 • 5 • 5) = -625
3
1
 1
 1  1  1
= -  • -  • -  = 216
 6
 6  6  6
4.  - 
b. 1021 = 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. 1016 = 10,000,000,000,000,000
5.
-33 ÷ 27 = -27 ÷ 27 = -1 = 1
6. 9 - 25 • 0.5 = 9 - 32 • 0.5 = 9 - 16 = -7
7. The diameter is 1.8 meters, so the radius is 0.9 meter.
4 3
pr
3
4
3
= p (0.9)
3
4
= p (0.729)
3
= 0.972p
The diameter of the solar system is
10,000,000,000,000,000 meters.
Inner sphere: V =
This can be written as ten quadrillion meters.
d. 107 = 10,000,000
The diameter of Earth is 10,000,000 meters.
This can be written as ten million meters.
e. 106 = 1,000,000
Outer sphere:
The length of the Lake Erie shoreline is
1,000,000 meters.
The volume of the inflated space is
4.5p - 0.972p = 3.528p , or about 11.08 cubic meters.
This can be written as one million meters.
f. 105 = 100,000
10.1 Exercises (pp. 414 ––415)
The width of Lake Erie is 100,000 meters.
This can be written as one hundred thousand meters.
Vocabulary and Concept Check
1. -34 is the negative of 34 , so the base is 3, the exponent
is 4, and its value is -81. ( -3) has a base of -3, an
4
4. Wives: 71
exponent of 4, and a value of 81.
Sacks: 7 • 7 = 7 2
Cats: 7 • 7 • 7 = 73
Kits: 7 • 7 • 7 • 7 = 7
2. The second one does not belong because it is an incorrect
statement about the expression. The power is the entire
expression 53.
4
5. You can use exponents to write the product of repeated
factors.
Sample answer: The formula for the volume of a cube,
V = s 3 , 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
1.
• • • •
4 4 4 4 4
Because
9
p = 4.5p
2
Practice and Problem Solving
3. 3 • 3 • 3 • 3
Because 3 is used as a factor 4 times, the exponent is 4.
So, 3 • 3 • 3 • 3 = 34.
4.
( -6 ) • ( -6 )
Because -6 is used as a factor 2 times, the exponent is 2.
So, ( -6) • ( -6) = ( -6) .
2
1
is used as a factor 5 times, its exponent is 5.
4
5
So,
1 1 1 1 1
1
• • • • =  .
4 4 4 4 4
 4
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Chapter 10
 1
 2
 1
 2
17. The negative sign is not part of the base;
 1
 2
5.  -  •  -  •  - 
-62 = -(6 • 6) = -36.
1
Because - is used as a factor 3 times, the exponent is 3.
2
18.
675
25
3
 1  1  1  1
So,  -  •  -  •  -  =  -  .
 2  2  2  2
6.
5 5 3 3 3
The prime factorization of 675 is 5 • 5 • 3 • 3 • 3,
1 1 1
• •
3 3 3
Because
or 52 • 33.
1
is used as a factor 3 times, the exponent is 3.
3
1 1 1 1
• • • 
 4 4 4 4
19. - 
3
1 1 1
1
• • =  .
3 3 3
 3
So,
7.
Because
p •p •p • x• x• x• x
4
So, p • p • p • x • x • x • x = p 3 x 4 .
( - 4) • ( - 4) • ( - 4) •
1
is used as a factor 4 times, the exponent is 4.
4
1 1 1 1
1
So, -  • • •  = -   .
4
4
4
4


 4
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.
8.
27
5 5 9 3
20. The largest doll is 12 inches and the other 3 are
7
the
10
7
as a factor 3 times.
10
So, an expression for the height of the smallest doll is
y • y
height of the next larger doll. Use
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.
3
12 •
So, ( - 4) • ( - 4) • ( - 4) • y • y = ( - 4) y .
3
2
7
7
7
7
•
•
= 12 •   .
10 10 10
 10 
3
343
7
12 •   = 12 •
= 4.116
10
1000
 
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, 6.4 • 6.4 • 6.4 • 6.4 • b • b • b = (6.4) b3 .
The height of the smallest doll is 4.116 inches.
4
10.
( - t ) • ( -t ) • ( -t ) • ( -t ) • ( - t )
Because -t is used as a factor 5 times, the exponent is 5.
So, ( -t ) • ( -t ) • ( -t ) • ( -t ) • ( -t ) = ( -t ) .
5
21. 5 + 3 • 23 = 5 + 3 • 8 = 5 + 24 = 29
22. 2 + 7 • ( -3)
(
24.
3
13.
(-1)
= ( -1) • ( -1) • ( -1) • ( -1) • ( -1) • ( -1) = 1
6
6
1 1 1 1 1 1
1
1
= • • • • •
=
2 2 2 2 2 2
64
 2
14.  
2
1
 1
 1  1
15.  -  =  -  •  -  =
12
12
12
144



 

3
1
1
1 1 1
•  = 16. -   = - •
729
9
9 9 9
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= 2 + 7 • 9 = 2 + 63 = 65
)
23. 132 - 12 2 ÷ 5 = (169 - 144) ÷ 5 = 25 ÷ 5 = 5
11. 52 = 5 • 5 = 25
12. -11 = -(11 • 11 • 11) = -1331
2
25.
1 3
(4 - 6 • 32 ) = 12 (64 - 6 • 9)
2
1
= (64 - 54)
2
1
= (10)
2
= 5
1
(7 + 53 ) = 12 (7 + 125) = 12 (132) = 66 = 66
2
 1
 2
3
1
 4
26.  -  ÷  
2
= -
1
1
1 16
÷
= - •
= -2 = 2
8 16
8
1
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303
Chapter 10
27.
h
2h - 1
2h - 1
2
21 - 1 = 1
22 - 1 = 3
21-1 = 20 = 1
2 2 -1 = 21 = 2
3
4
23 - 1 = 7
24 - 1 = 15
h
2h - 1
2h - 1
Section 10.2
1
2 3 -1 = 2 2 = 4
2
4 -1
= 2 = 8
h
2 - 1 = 31
2h - 1
25 -1 = 24 = 16
2 -1
1. a.
Product
Repeated
Multiplication Form
Power
22 • 24
2•2•2•2•2•2
26
(-3) • ( -3) • ( -3)
• ( -3) • ( -3) • ( -3)
(-3)
73 • 7 2
7•7•7•7•7
75
5.11 • 5.16
5.1 • 5.1 • 5.1 • 5.1
• 5.1 • 5.1 • 5.1
5.17
( - 4) • ( - 4)
• ( - 4) • ( - 4)
(- 4)
10 • 10 • 10 • 10
• 10 • 10 • 10 • 10
108
1 1 1 1 1
• • • •
2 2 2 2 2
1 1 1 1 1
• • • • •
2 2 2 2 2
1
 
 2
(-3)
2
• ( -3)
4
3
5
h
10.2 Activity (pp. 416 ––417)
5
( - 4)
You should choose getting paid 2 h - 1 dollars because
when you work more than 1 hour, you will get paid more
than the other option.
28. a. C = 100(0.99988) = 100(0.99988)
t
4
 99.95
After 4 years, the amount of carbon-14 remaining is
about 99.95 grams.
b. percent remaining =
amount remaining
original amount
99.95
100
= 99.95%
29. a. To travel from A-440 to A, it takes 12 notes.
= 440(1.0595)
12
 880
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 • x = x • 8 represents the
Commutative Property of Multiplication.
31. The statement ( 2 • 10) x = 2(10 • x ) represents the
Associative Property of Multiplication.
32. The statement 3( x • 1) = 3 x represents the Identity
Property of Multiplication.
x
24
=
33. B;
18
27
x
8
=
18
9
x = 16
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2
103 • 105
5
1
1
  • 
 2
 2
5
4
10
b. To find the product of two powers with the same base,
add their exponents.
c. 2 2 • 24 = 22 + 4 = 26
After 4 years, 99.95% of the carbon-14 remains.
n
• ( - 4)
am • an = am + n
=
b. F = 440(1.0595)
2
6
(-3)
2
• ( -3) = ( -3)
4
= ( -3)
2+ 4
6
73 • 7 2 = 73 + 2 = 75
5.11 • 5.16 = 5.11+ 6 = 5.17
( - 4)
2
• ( - 4) = ( - 4)
2
2+2
= ( - 4)
4
103 • 105 = 103 + 5 = 108
5
5
1
1
1
  •  =  
 2
 2
 2
5+ 5
10
1
=  
 2
Using the rule to simplify the products results in the
values in the third column of the table in part (a).
d. 26 = 64
(-3)
6
= 729
75 = 16,807
5.17  89,741.1
( - 4)
4
= 256
108 = 100,000,000
10
1
 
 2
=
1
1024
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Chapter 10
2. a.
(32 )
3
10.2 On Your Own (p. 419)
= (3 • 3)(3 • 3)(3 • 3) = 36
b.
(22 ) = (2 • 2)(2 • 2)(2 • 2)(2 • 2) = 28
c.
(7 3 )
d.
( y3 )
3
e.
( x4 )
2
4
2
1. 62 • 64 = 62 + 4 = 66
 1
 2
3
6
 1
 1
= - 
 2
 2
= (7 • 7 • 7)(7 • 7 • 7) = 7 6
2.  -  •  - 
= ( y • y • y )( y • y • y )( y • y • y ) = y 9
3. z • z12 = z1+12 = z13
= ( x • x • x • x )( x • x • x • x) = x8
4.
( 44 )
5.
( y2 )
6.
((- 4) )
7.
(5 y )
8.
(ab)
9.
(0.5mn)
General Rule: ( a m ) = a m • n
n
3. a.
(2 • 3)
b.
3
= ( 2 • 3)( 2 • 3)( 2 • 3) = 2 • 3
3
(2 • 5)
= ( 2 • 5)( 2 • 5) = 2 • 5
c.
( 5 • 4)
3
= (5 • 4)(5 • 4)(5 • 4) = 53 • 43
d.
(6a )
e.
(3 x )
2
4
= (6a)(6a)(6a)(6a ) = 64 • a 4
2
= (3 x)(3 x) = 3 • x
2
General Rule: ( a • b)
m
4
2
= am • bm
4. a. For (3, 5): 2 x • 2 y = 23 • 25 = 8 • 32 = 256
= y 2 • 4 = y8
10.
3 2
= ( - 4)
= ( - 4)
3• 2
4
= 54 • y 4 = 625 y 4
5
= a 5b5
2
6
= 0.52 • m 2 • n 2 = 0.25m 2 n 2
Total number
Number of bytes Number of
=
•
of bytes
in a gigabyte
gigabytes
= 230 • 16
b. 2 x • 2 y = 32
= 230 • 24
2 x + y = 32
2
= 230 + 4
= 2
= 234
5
The stacks that have 32 pennies are the locations in
which the sum x + y equals 5.
The computer has 234 bytes of free storage space.
10.2 Exercises (pp. 420 ––421)
x
y
x + y
1
4
5
2
3
5
3
2
5
2. no; The bases are not the same.
4
1
5
Practice and Problem Solving
Vocabulary and Concept Check
1. Use the Product of Powers Property to multiply powers
with the same base.
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).
2 • 2 = 2 • 2 = 256 • 256 = 65,536
x
y
8
9
= 44 • 3 = 412
There are 256 pennies in the stack.
x+ y
 1
= - 
 2
3
2
2
3
3+ 6
8
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
3. 32 • 32 = 32 + 2 = 34
4. 810 • 84 = 810 + 4 = 814
5.
( - 4)
5
• ( - 4) = ( - 4)
7
5+ 7
= ( - 4)
12
6. a 3 • a 3 = a 3 + 3 = a 6
7. h 6 • h = h 6 +1 = h 7
2
3
2
6
 2
 2
=  
 3
 3
8.   •  
2+6
8
 2
=  
 3
encounter, then you can use inductive reasoning to write
a general rule stating that the pattern is always true.
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305
Chapter 10
8
9
 5
 7
 5
 5
= - 
 7
 7
8+ 9
9.  -  •  - 
10.
(-2.9) • (-2.9)
11.
(54 )
12.
(b12 )
13.
(3.83 )
3
7
4
17
= ( -2.9)
1+ 7
26. 16 x 
(
= ( -2.9)
8
27. 52 53 • 52
= 54 • 3 = 512
3
= 52 + 5
4
= 57
= 78,125
= 3.83 • 4 = 3.812
 3  
 3
= - 
 4


28. highest altitude = 3 • lowest altitude
= 3 • 38
5• 2
14.   -  
 4  
10
= 31+ 8
 3
= - 
 4
= 39
The highest altitude of an altocumulus cloud is about
39 feet.
15. The bases should not be multiplied.
52 • 59 = 52 + 9 = 511
16. You should multiply the exponents instead of adding.
4
= r 6 • 4 = r 24
17.
(6 g )
18.
(-3v)
3
= 63 • g 3 = 216 g 3
= ( -3) • v5 = -243v5
5
5
2
The volume of the egg is 16p  50.27 cubic inches.
2
1 2
1 
1
=   • k2 =
k
25
5 
5
20. (1.2m)
(rt )
12
4
4
p abc
3
4
= p ( 2 • 2 • 3)
3
4
= p (12)
3
= 16p
29. a. V =
 50.27
19.  k 
21.
b. a 2 = 22
b 2 = 22
= 1.24 • m 4 = 2.0736m 4
c 2 = 32
V =
= r12t12
3
=
3
27
 3 
 3
p  =  -  • p3 = - p3
64
 4 
 4
22.  -
=
23. No. They are not equal.
32 • 33 = 35 = 243, but 32 + 33 = 9 + 27 = 36.
24. a. V = s 3 = (3w)
3
(3w)
3
= 33 • w3 = 27 w3
( )
25. 2 4 • 25 - 22
2
= 24 + 5 - 2 2 • 2
=
=
3
An expression for the volume of the case is (3w) .
b.
) = 52 (53 + 2 )
= 52 (55 )
= b12 • 3 = b36
5 2
(r 6 )
4
1 
1
1
= 16 •   • x 4 = 16 •   • x 4 = x 4
2 
 2
 16 
 5
= - 
 7
=
4
p abc
3
4
p • 22 • 22 • 32
3
4
p • 22 + 2 • 32
3
4
p • 24 • 9
3
4
p • 16 • 9
3
192p
 603.19
The volume of the egg is 192p  603.19 cubic inches.
Because 16p • 12 = 192p , the volume is 12 times
greater than the volume in part (a).
= 29 - 2 4
= 512 - 16
= 496
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Chapter 10
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.
1
V = b 2h
3
1
2
= (1.5b) ( h)
3
1
= • 1.52 • b 2 • h
3
1
= • 2.25 • b 2 • h
3
3
= 0.75b 2 h, or b 2 h
4
31.
Total pieces
Amount delivered Number of
=
•
of mail
each second
seconds
= (2 • 5
8
2
) • (2
•3 •5
8
4
= 2 •2 •3 •5 •5
8
8
8+8
= 2
4
•3 •5
4
2
2
Section 10.3
10.3 Activity (pp. 422––423)
1. a.
Quotient
Repeated Multiplication Form
24
22
2• 2•2•2
2 • 2
1
1
( - 4)
2
( - 4)
5
( -4 ) • ( -4 )
2 •2 = 2
1
1
1
x
1
 
 3
1
=  
 3
1
10
105
1
103
1
1
1
1
1
1
3 • 3 • 3 • 3 • 3•3
312
34
38
•3•3•3•3•3•3
3 • 3 • 3 • 3
1
34.
5•5• 5
25
=
= 25
1
5
1
1
1
2• 3
3
=
= 3
35.
1
2
1
8• 6• 6
6
=
= 6
36.
1
6• 8
1
1
n
1
1
1
37. B;
1
1
1
1
1
4• 4
4
=
= 4
1
4
- 2) • 180°
1
• 10 • 10 • 10
10 • 10 • 10 • 10 • 10
Fair Game Review
(n
1
1
1
1
Because 2 + 4 = 6, x = 4.
33.
8.53
10 • 10 • 10 • 10 • 10
8
1
1
1
1
1
=
729
 3
6
1
• 8.5 • 8.5
b.   •  
2+ x
1
• 8.5 • 8.5 • 8.5 • 8.5
8.5 • 8.5 • 8.5 • 8.5
Because 5 + 3 = 8, x = 3.
1
1
1
  •  =  
 3
 3
 3
74
1
1
1
6
1
1
25 + x = 28
x
1
8.5 • 8.5 • 8.5 • 8.5 • 8.5
8.59
8.56
8
2
1
•7•7•7•7
7 • 7 • 7
4
32. a. 25 • 2 x = 256
2
1
1
•3 •5
3
7 • 7 • 7
7
73
4
(- 4)
• ( - 4) • ( - 4)
1
The United States Postal Service delivers 2
pieces of mail in 6 days.
1
 3
1
7
16
x
1
1
)
2+2
22
( - 4 ) • ( - 4 ) • ( - 4)
= 216 • 34 • 54
5
1
1
2
Power
=
(8 - 2) • 180°
8
6 • 180°
=
8
1080°
=
8
= 135°
1
(-5)
5
(-5)
7
( -5)
1
1
1
( -5) • ( -5)
1
1
1
• ( -5) • ( -5) • ( -5)
• ( -5) • ( -5)
• ( -5) • ( -5) • ( -5) • ( -5)
1
1
1
(-5)
2
1
1
114
111
11 • 11 • 11 • 11
11
113
1
Each angle measures 135°.
Copyright © Big Ideas Learning, LLC
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Big Ideas Math Blue
Worked-Out Solutions
307
Chapter 10
b. To find the quotient of two powers with the same
10.3 On Your Own (pp. 424 ––425)
base, subtract their exponents.
am
= am-n
an
24
= 24 - 2 = 22
22
c.
( - 4)
2
( - 4)
5
= ( - 4)
= ( - 4)
5-2
1.
97
94
2.
4.26
= 4.26 - 5 = 4.21 = 4.2
4.25
3.
(-8)
4
(-8)
4.
x8
= x8 - 3 = x5
x3
5.
215
215
215
= 3 + 5 = 8 = 215 - 8 = 27
5
2 •2
2
2
6.
d5 d9
• 8 = d 5 -1 • d 9 - 8 = d 4 • d 1 = d 4 + 1 = d 5
d
d
7.
59 55
•
= 59 - 4 • 55 - 2 = 55 • 53 = 58
54 5 2
8.
People per
Population in 2030
=
square kilometer
Land area
8
3
77
= 77 - 3 = 74
73
8.59
= 8.59 - 6 = 8.53
8.56
8
10
= 108 - 5 = 103
105
312
= 312 - 4 = 38
34
(-5)
5
(-5)
7
= ( -5)
7-5
= ( -5)
= 9 7 - 4 = 93
2
114
= 114 -1 = 113
111
= ( -8)
Volume
of Smaller
Cube
a.
4
( 4 2 ) = 46
b.
33
(32 )
c.
63
(62 )
d.
3
(10 )
3
10
Larger
Volume
Smaller
Volume
46
43
Volume
of Larger
Cube
3
2
3
3
3
36
33
= 36
2.25 • 221
217
221
= 2.25 • 17
2
= 2.25 • 221 -17
=
= 10
Answer
6
63
6
106
103
= 36
4
33
63
25
= 25 - 3 = 22
23
79
= 7 9 -1 = 7 8
71
308 Big Ideas Math Blue
Worked-Out Solutions
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.
10
3. To divide two powers that have the same base, subtract
Sample answer:
There will be about 36 people per square kilometer in
Alabama in 2030.
3
The volume of the smaller cube equals the number of
smaller cubes that will fit inside the larger cube.
their exponents.
= 2.25 • 24
3
6
= 66
4
3
Using the rule to simplify the quotients results in the
values in the third column of the table in part (a).
2.
= ( -8)
8- 4
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.
Practice and Problem Solving
3.
610
= 610 - 4 = 66
64
5.
(-3)
1
( -3)
7.
59
= 59 - 3 = 56
53
4
= ( -3)
4 -1
= ( -3)
3
4.
89
= 89 - 7 = 82
87
6.
4.55
= 4.55 - 3 = 4.52
4.53
Copyright © Big Ideas Learning, LLC
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Chapter 10
8.
644
= 644 - 3 = 641 = 64
643
9.
(-17)
2
(-17)
5
= ( -17)
3
= ( -7.9)
11.
(-6.4)
6
(-6.4)
= ( -6.4)
10
10 - 4
8-6
12.
p 11
= p 11- 7 = p 4
p7
14.
n18
= n18 - 7 = n11
n7
= ( -7.9)
a 3 • b 4 • 54
b 4 54
= a3 • 2 •
2
b •5
b
5
= ( -6.4)
= a 3 • b 2 • 53
6
= 125a 3b 2
26.
2
13.
615
= 615 - 5 = 610
65
75 • 73
75 + 3
78
= 2 = 2 = 78 - 2 = 7 6
2
7
7
7
17.
219 • 25
219 + 5
2 24
= 12 + 3 = 15 = 224 -15 = 29
12
3
2 •2
2
2
18.
(-8.3)
7
(-8.3)
•
(-8.3)
3
(-8.3)
27.
x15 y 9
x15
y9
= 8 • 3 = x15 - 8 • y 9 - 3 = x 7 y 6
8 3
x y
x
y
28.
m10 n 7
m10 n 7
= 1 • 6 = m10 -1 • n 7 - 6 = m9 n1 = m9 n
1 6
mn
m
n
= ( -8.3)
8-7
1
• ( -8.3)
4-3
1
= ( -8.3)
1 +1
= ( -8.3)
2
p
p
p
= 18 + 4 = 22 = p 30 - 22 = p 8
p 18 • p 4
p
p
30
24
= 24 - 2 = 22 = 4
22
MP3 Player D has 22 = 4 times more memory than
MP3 Player B.
= ( -8.3) • ( -8.3)
30
= 125cd 2
29. a.
16.
4
= 53 • c1 • d 2
b 24
= b 24 -11 = b13
b11
dividing them.
8
512 • c10 • d 2
512 c10
=
• 9 • d2
59 • c9
59
c
= 512 - 9 • c10 - 9 • d 2
15. When dividing powers, you subtract exponents instead of
30
c 22
c 22
c 22
20. 8
= 8 + 9 = 17 = c 22 -17 = c 5
9
c •c
c
c
13
25.
17
21.
k
k
• 11 = k 13 - 5 • k 17 -11 = k 8 • k 6 = k 8 + 6 = k 14
k5
k
22.
1014
= 1014 - 6 = 108
106
b.
270
240
Price (dollars)
(-7.9)
4
( -7.9)
19.
63 • w
63
= 2 • w = 63 - 2 • w = 61 w = 6 w
2
6
6
= a 3 • b 4 - 2 • 5 4 -1
10.
8
= ( -17)
5-2
24.
y
210
180
150
120
90
60
30
0
0 4 8 12 16 20 24 28 32 36 x
M emory (GB)
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 m = 4 and n = 2, the
difference m - n = 4 - 2 is 2.
b. infinitely many solutions; Any two numbers that
satisfy the equation m - n = 2 are solutions.
The sound of a jet at takeoff is 108 times more intense
than the sound of a normal conversation.
23.
x • 48
48
= x • 5 = x • 48 - 5 = x • 43 = 64 x
5
4
4
Copyright © Big Ideas Learning, LLC
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Big Ideas Math Blue
Worked-Out Solutions
309
Chapter 10
31.
Number of
Number of stars in the Universe
=
galaxies
Number of stars in the Milky Way Galaxy
1024
=
10 • 1010
Section 10.4
10.4 Activity (pp. 428––429)
1. a.
24
10
= 1+10
10
24
10
= 11
10
= 1024 -11
= 1013
83 x
= 89
82 x + 1
8 x -1 = 89
+1
9
+1
53 - 3
50
62
62
62 - 2
60
(-3)
4
(-3)
(-3)
( -4)
5
( -4)
(-4)
4-4
5-5
( -3)
0
(-4)
0
62
36
53
125
(-3) = 81 = 1;
=
= 1;
=
=
1;
2
4
3
6
36
5
125
81
(-3)
( - 4)
5
( - 4)
5
=
-1024
= 1; They are all equal to 1.
-1024
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 a 0 = 1, where a  0.
x = 10
So, x = 10.
Fair Game Review
53
53
4
b.
For the equation to be true, the value of x - 1 must
equal 9.
x -1=
Power
5
83 x - (2 x + 1) = 89
83 x - 2 x -1 = 89
Quotient of
Powers Property
4
There are about 1013 galaxies in the Universe.
32.
Quotient
2. a.
33. - 4 - 5 = - 4 + ( -5) = -9
Product
Product of
Powers Property
Power
34. -23 - ( -15) = -23 + 15 = -8
30 • 3 4
30 + 4
34
35. 33 - ( -28) = 33 + 28 = 61
82 • 80
82 + 0
82
36. 18 - 22 = 18 + ( -22) = - 4
37. B; 2 x° + x° = 90°
3 x = 90
x = 30
(-2)
3
0
• ( -2)
( -2)
0
 1
 1
-  • - 
 3
 3
5
3+ 0
 1
- 
 3
0+5
( -2 )
3
 1
- 
 3
5
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 a 0 = 1,
where a  0.
310 Big Ideas Math Blue
Worked-Out Solutions
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Chapter 10
3. a.
Product
Product of
Powers Property
Power
5 -3 • 53
5-3 + 3
50
62 • 6-2
62 + (-2)
60
(-3)
4
( -4 )
• ( -3)
-5
-4
• ( -4 )
(-3)
4 + ( -4)
( -4)
5
-5 + 5
6.
45 • 4-3
45 + (-3)
42
=
=
= 4 2 - 2 = 40 = 1
42
42
42
7. 8x -2 =
8
x2
8. b 0 • b -10 = 1 •
( -3)
0
(-4)
0
b. They are all equal to 1.
9.
z6
1
1
1
=
• z6-9 =
• z -3 =
15 z 9
15
15
15 z 3
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,
1
a - n is equal to the reciprocal of a n , or n .
a
=
=
4. a. The exponents decrease by 1.
=
0
tenths: 10
1
55
1
3600 •
3125
3600
3125
19
1
125
1.152
10. 3600 • 5-5 = 3600 •
c. From the Multiplicative Inverse Property, you know
ones: 10
1
1
= 10
b10
b
So, 1.152 liters of water leak from the faucet in 1 hour.
-1
10.4 Exercises (pp. 432––433)
hundredths: 10- 2
Vocabulary and Concept Check
thousandths: 10-3
1. no; For any nonzero number a, the value of a 0 = 1.
b. 3452.867
2. Rewrite 10-3 as
= 3 • 103 + 4 • 102 + 5 • 101 + 2 • 100
+ 8 • 10-1 + 6 • 10-2 + 7 • 10-3
3. The numbers, in order from least to greatest, are 5-5 , 50 ,
= 3 • 1000 + 4 • 100 + 5 • 10 + 2 • 1
1
1
1
+ 6•
+ 7•
10
100
1000
= 3000 + 400 + 50 + 2 + 0.8 + 0.06 + 0.007
and 54.
+8•
4. The last statement does not belong.
(-3) • ( -3) • ( -3)
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.
1. 4
3. 6
-8
1
1
= 2 =
4
16
•6 = 6
8
-8 + 8
2.
(-2)
-5
=
1
(-2)
5
3
1
1
1
= 3-3 =   = 3
3•3•3
3
3
 
1
= 32
5.
= 6 =1
7.
(-3)
6
(-3)
5.
1
1
1•1
1
1
1
• -4 = 7
= 7 + ( - 4) = 3 =
7
-4
5
5
5 •5
5
125
5
= ( -3)
5-6
= ( -3)
-1
=
3
87
= 87 - 7 = 80 = 1
87
6. 50 • 53 = 50 + 3 = 53 = 125
0
4.
5
= ( -3)
Practice and Problem Solving
10.4 On Your Own (pp. 430 ––431)
-2
1
1
, or
.
103
1000
1
(-3)
Copyright © Big Ideas Learning, LLC
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1
= -
1
3
( -2 )
-8
• ( -2 ) = ( -2 )
8
-8 + 8
= ( -2 ) = 1
0
4 + -4
8. 94 • 9-4 = 9 ( ) = 90 = 1
9. 6 -2 =
1
1
=
62
36
10. 1580 = 1
Big Ideas Math Blue
Worked-Out Solutions
311
Chapter 10
11.
12.
43
1
1
= 4 3 - 5 = 4 -2 = 2 =
45
4
16
-3
(-3)
= ( -3)
= ( -3)
1- 2
2
-1
=
24.
1
(-3)
1
= -
1
3
8 x3
8
4
=
• x 3 - 9 = 4 • x -6 = 6
x
2 x9
2
25. 3d -4 • 4d 4 = 3 • 4 • d -4 • d 4
= 12 • d -4 + 4
= 12 • d 0
1
+5
24
1
= 4•
+5
16
1
=
+5
4
1
= 5
4
13. 4 • 2-4 + 5 = 4 •
14. 3-3 • 3-2 = 3-3 - 2 = 3-5 =
15.
= 12 • 1
= 12
26. m -2 • n 3 =
27.
1
1
=
35
243
2 -4 =
1
1
1•1
1
1
1
•
= -3
= -3 + 6 = 3 =
5 -3 5 6
5 • 56
5
5
125
(1.5)
=
(1.5)
1.5
( )-2 + 4
=
(1.5)2
(1.5)2
2
(1.5)-2 • (1.5)4
2
= (1.5)
2-2
= (1.5)
0
30.
31.
1
1
=
43
64
18. 10 kilograms 
centimeter
10 -2 m
=
= 10 -2 - (-6) = 10 4 = 10,000
micrometer
10 -6 m
millimeter
10 -3 m
=
= 10 -3 - (-9) = 10 6 = 1,000,000
nanometer
10 -9 m
There are 1,000,000 nanometers in a millimeter.
32.
meter
1m
= -6
micrometer
10 m
1000 grams 1 gram of sand

1 kilogram
10-3 gram
=
= 106
= 1,000,000
There are 1,000,000 micrometers in a meter.
7
There are about 107 , or 10,000,000, grains of sand.
19. For any nonzero number a, a 0 = 1. So, 20 = 1 and
100 = 1.
6
y4
21. 8-2 • a 7 =
3
22.
100
10-6
= 100 - (-6)
= 104 • 103
20. 6 y -4 =
1
1
=
2
4
16
decimeter
10-1 m
=
= 10 -1- (-3) = 10 2 = 100
millimeter
10 -3 m
104
= -3
10
= 10
4 -2 =
There are 10,000 micrometers in a centimeter.
17. The negative sign goes with the exponent, not the base.
=
1
1
=
4
2
16
There are 100 millimeters in a decimeter.
=1
( 4 ) -3
3-2 • k 0 • w0
3 -2 • 1 • 1
w6
w6
=
= 2 =
-6
-6
3
9
w
w
28. Sample answer:
29.
16.
1
n3
• n3 = 2
2
m
m
9c
= 9 c 3 - ( -4 ) = 9 c 7
c -4
23.
-2
a7
a7
=
2
8
64
5b
= 5b -2 - (-3) = 5b
b -3
33. a. 10 •
1
10
1
micrometer
=
=
10,000
10,000
1000
1
1
1
• 10-6 =
•
1000
1000 106
1
1
=
•
1000 1,000,000
=
1
1,000,000,000
= 0.000000001, or 10-9
The length of the virus is 10-9 meter.
312 Big Ideas Math Blue
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Chapter 10
Study Help
b. 1 nanometer = 10-9 meter
Available at BigIdeasMath.com.
1
meter
109
1
meter
=
1,000,000,000
=
Quiz 10.1––10.4
1.
= 0.000000001 meter
Because -5 is used as a factor 4 times, the exponent is 4.
The answer to part (a) is equal to one nanometer.
34. a.
So, ( -5) • ( -5) • ( -5) • ( -5) = ( -5) .
4
500
= 500 • 103 mm 3
10-3
2. 7 • 7 • m • m • m
The donation is 500 • 103 cubic millimeters.
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.
500 • 103 • 104 = 500 • 103 + 4
So, 7 • 7 • m • m • m = 7 2 m3 .
= 500 • 10
7
3. 54 = 5 • 5 • 5 • 5 = 625
= 500 • 10,000,000
= 5,000,000,000
There are about five billion white blood cells in the
donation.
b. 500  10  5  10
3
6
= 500  5  10  10
3
6
= 2500  103 + 6
= 2500  109
= 2500  1,000,000,000
4.
( -2 )
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
=
. There are about 500
5,000,000,000
1
times more red blood cells than white blood cells.
5.
(- 4.8)
6.
54
1
1
= 5 4 - 7 = 5 -3 = 3 =
57
5
125
8.
( a5 )
9.
(3c)
-9
• ( - 4.8) = ( - 4.8)
9
-9 + 9
= ( - 4.8) = 1
0
1
a-n
1
. Zero raised to any
0n
nonzero exponent is 0. Because division by 0 is
undefined, the rule for negative exponents does not
apply when a = 0.
36. If you substitute 0 for a, you get
Fair Game Review
37. 103 • 106 = 103 + 6 = 109
38. 102 • 10 = 102 +1 = 103
108
= 108 - 4 = 10 4
10 4
3
= a5 • 3 = a15
4
= 34 • c 4 = 81c 4
2
2
4 2
 2 
 2
p  =  -  • p2 =
p
49
 7 
 7
10.  -
35. Sample answer: Write the power as 1 divided by the
power and use a negative exponent. a n = a -(- n) =
= ( -2) • ( -2) • ( -2) • ( -2) • ( -2) • ( -2) = 64
6
7. 38 • 3 = 38 +1 = 39
= 2,500,000,000,000
39.
( -5) • (-5) • (-5) • (-5)
11.
87
= 87 - 4 = 83
84
12.
63 • 6 7
63 + 7
610
=
= 2 = 610 - 2 = 68
2
2
6
6
6
13.
p 15
p 15
p 15
= 3 + 9 = 12 = p 15 -12 = p 3
9
p •p
p
p
14.
t13 t 8
• 6 = t13 - 5 • t 8 - 6 = t 8 • t 2 = t 8 + 2 = t10
t5
t
3
15. 8d -6 =
16.
8
d6
12 x 5
3
= 3 x 5 - 7 = 3 x -2 = 2
x
4 x7
40. D; A stem-and-leaf plot orders numerical data and shows
how they are distributed.
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313
Chapter 10
17. a. 1000 • 10 -6 =
1000
103
=
= 103 - 6 = 10 -3
6
10
106
The length of the dinoflagellate is 10-3 meter.
b.
1000 mm
1m
=
1000
1000
1
1 mm =
m = 0.001 m
1000
So, the length of the dinoflagellate is 1 millimeter.
18.
107
= 107 - 2 = 105
10 2
An earthquake of magnitude 8.0 is 105 times stronger
than an earthquake of magnitude 3.0.
Section 10.5
10.5 Activity (pp. 436 ––437)
1.
2,000,000,000

3,000,000,000
6,000,000,000,000,000,000
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

0.000000003
0.000000000000000006
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 10-2 meter.
B. This picture appears to have been taken about
100 meters away from the frog, so this picture
matches with 102 meters.
C. This picture appears to have been taken very far away
from the frog, about 100,000 meters. So, this picture
matches with 105 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 10-1 meter.
E. This picture appears to have been taken very close to
the frog, about 0.00001 meter away. So, this picture
matches with 10-5 meter.
4. A. 2  100 = 2
The unit that is most appropriate for a door height of 2
is meters.
B. 1.6  104 = 16,000
The unit that is most appropriate for a volcano height
of 16,000 is feet.
C. 1.4  102 = 140
The unit that is most appropriate for a pen length of
140 is millimeters.
D. 6.3  10-1 = 0.63
The unit that is most appropriate for a steel ball
bearing diameter of 0.63 is centimeters.
E. 7.5  101 = 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 12  104
is not written in scientific notation.
2. 6  107 = 60,000,000
7
The number in standard form is 60,000,000.
3. 9.9  10 -5 = 0.000099
5
The number in standard form is 0.000099.
4. 1.285  10 4 = 12,850
4
The number in standard form is 12,850.
5. Water: 1.0  103 = 1000
Lead: 1.14  104 = 11,400
Lead is denser than water, so it will sink.
6. 1.4  10-5 • 75 = 0.000014 • 75 = 0.00105
The fleas consume about 0.00105 liter, or 1.05 milliliters,
of blood per day.
F. This picture appears to have been taken about 1 meter
away from the frog. So, this picture matches with
100 meter.
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Chapter 10
10.5 Exercises (pp. 440 ––441)
16. 8  10 -3 = 0.008
3
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 10  9.2-13 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
The number in standard form is 0.008.
17. 5  102 = 500
2
The number in standard form is 500.
18. 2.7  10 -4 = 0.00027
4
The number in standard form is 0.00027.
19. 4.4  10 -5 = 0.000044
5
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
1.8  109 is written in scientific notation.
7. The factor is greater than or equal to 1 and less than 10.
The power of 10 has an integer exponent. So, the number
3.45  1014 is written in scientific notation.
8. The factor is less than 1. So, the number 0.26  10-25 is
not written in scientific notation.
9. The factor is greater than 10. So, the number 10.5  1012
is not written in scientific notation.
10. The factor is greater than 10. So, the number 46  10-17
The number in standard form is 0.000044.
20. 2.1  103 = 2100
3
The number in standard form is 2100.
21. 1.66  109 = 1,660,000,000
9
The number in standard form is 1,660,000,000.
22. 3.85  10-8 = 0.0000000385
8
The number in standard form is 0.0000000385.
23. 9.725  106 = 9,725,000
6
is not written in scientific notation.
11. The factor is greater than or equal to 1 and less than 10.
The power of 10 has an integer exponent. So, the number
5  10-19 is written in scientific notation.
12. The factor is greater than or equal to 1 and less than 10.
The power of 10 has an integer exponent. So, the number
7.814  10-36 is written in scientific notation.
13. The factor is less than 1. So, the number 0.999  1042 is
not written in scientific notation.
14. The factor is greater than or equal to 1 and less than 10.
The power of 10 has an integer exponent. So, the number
6.022  1023 is written in scientific notation.
15. 7  107 = 70,000,000
7
The number in standard form is 70,000,000.
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The number in standard form is 9,725,000.
24. Because the exponent is negative, the decimal point
should be moved to the left, not to the right.
4.1  10-6 = 0.0000041
6
The number in standard form is 0.0000041.
25. a. 2.7  108 • 3 = 270,000,000 • 3 = 810,000,000
There are 810,000,000 platelets in 3 milliliters of
blood.
1000 mL
b. 5 L 
= 5000 mL
L
2.7  108 • 5000 = 270,000,000 • 5000
= 1,350,000,000,000
There are about 1,350,000,000,000 platelets in an
adult body.
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315
Chapter 10
26. To write the number 1.0  10100 in standard form, move
the decimal point 100 places to the right. So, there are
100 zeros in a googol.
32. a. Air:
6.7  108 mi 1.6 km
1 h
•
•
3600 sec
1 mi
1 h
6.7  108 • 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
=
27. a. Betelgeuse: 6.2  103 = 6200 = 6200
3
Bellatrix: 3.8  104 = 38,000 = 38,000
4
Sun: 1.1  104 = 11,000 = 11,000
4
Glass:
Aldebaran: 7.2  103 = 7200 = 7200
6.6  108 ft
1 mi
1.6 km
•
•
1 sec
5280 ft
1 mi
6.6  108 • 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
=
3
Rigel: 2.2  104 = 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.
Ice:
2.3  105 km
= 230,000 km sec
1 sec
28. The value of the number is 10 times greater.
Vacuum:
29. 16.2% of 9.6  10 = 0.162 • 9.6  10
3
3
3.0  108 m
1 km
•
1 sec
1000 m
3.0  108 • 1 km
1 • 1000 sec
300,000,000 km
=
1000 sec
= 300,000 km sec
= 0.162 • 9600
=
= 1555.2
The area of the Florida Reef Tract is 1555.2 square
kilometers.
(
30. 1.0  106
)
2
= (1.0  106 ) • (1.0  106 )
Water:
= 1,000,000 • 1,000,000
= 1,000,000,000,000
1 square gigameter = 1,000,000,000,000 square
kilometers
5 • 1,000,000,000,000 = 5,000,000,000,000
2.3  1010 • 1 km
1 • 100 • 1000 sec
23,000,000,000 km
=
100,000 sec
=
There are 5,000,000,000,000 or 5  1012 square
kilometers in 5 square gigameters.
31.
cubic kilometers of fresh water
total cubic kilometers of water
x
0.025 =
1.4  109
= 230,000 km sec
Light travels the fastest through a vacuum.
b. Light travels the slowest through glass.
percent fresh water =
0.025 • 1.4  109 = x
0.025 • 1,400,000,000 = x
35,000,000 = x
2.3  1010 cm
1 m
1 km
•
•
1 sec
100 cm 1000 m
Fair Game Review
33. 4 • 4 • 4 • 4 • 4 = 45
34. 3 • 3 • 3 • y • y • y = 33 y 3
35.
( - 2 ) • ( - 2) • ( - 2)
= ( - 2)
3
There are about 35,000,000 cubic kilometers of fresh
water on Earth.
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Chapter 10
d. The distance is the sixth greatest, so the description
36. B; a 2 + b 2 = c 2
2
2
4 +5 = c
matches Earth.
2
93,000,000 mi = 9.3
16 + 25 = c 2
The distance from Earth to the Sun is 9.3
41 = c 2
matches Mars.
41 = c
108 mi
140,000,000 mi = 1.4
The length of the hypotenuse is
41 inches.
The distance from Mars to the Sun is 1.4
108 miles.
f. The distance is the greatest, so the description matches
Section 10.6
Neptune.
10.6 Activity (pp. 442–443)
1. a. 0.01 = 1
107 miles.
e. The distance is the fifth greatest, so the description
c2
41 =
107 mi
2,800,000,000 mi = 2.8
10- 2
109 mi
The distance from Neptune to the Sun is 2.8
miles.
2
The pH of lime juice is 2, and so it is an acid.
g. The distance is the fourth greatest, so the description
10- 8
b. 0.00000001 = 1
matches Jupiter.
8
480,000,000 mi = 4.8
The pH of an egg is 8, and so it is a base.
c. 0.0000001 = 1
10
108 mi
The distance from Jupiter to the Sun is 4.8
miles.
-7
108
h. The distance is the least, so the description matches
7
The pH of distilled water is 7, and so it is neutral.
d. 0.00000000001 = 1
10- 11
11
The pH of ammonia water is 11, and so it is a base.
e. 0.0001 = 1
10- 4
4
The pH of tomato juice is 4, and so it is an acid.
f. 1 = 1
109
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.
109 mi
The distance from Uranus to the Sun is 1.8
miles.
109
b. The distance is the seventh greatest distance, so the
description matches Venus.
67,000,000 mi = 6.7
107 mi
The distance from Venus to the Sun is 6.7
miles.
107
107 mi
The distance from Mercury to the Sun is 3.6
miles.
107
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.
matches Saturn.
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)
104
4
The number in scientific notation is 5
108 mi
The distance from Saturn to the Sun is 8.9
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.
1. 50,000 = 5
c. The distance is the third greatest, so the description
890,000,000 mi = 8.9
36,000,000 mi = 3.6
3. Sample answer: Let 1 inch represent 100,000,000 miles.
100
1,800,000,000 mi = 1.8
Mercury.
108
2. 25,000,000 = 2.5
107
7
The number in scientific notation is 2.5
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104.
107.
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Chapter 10
102
3. 683 = 6.83
5. 321,000,000 = 3.21
2
8
The number in scientific notation is 6.83
4. 0.005 = 5
102.
10- 3
The number in scientific notation is 3.21
6. 0.00000625 = 6.25
3
108.
10- 6
6
The number in scientific notation is 5
10- 3.
10- 7
5. 0.00000033 = 3.3
The number in scientific notation is 6.25
7. 0.00004 = 4
7
10- 6.
10- 5
5
The number in scientific notation is 3.3
6. 0.000506 = 5.06
10- 7.
10- 4
The number in scientific notation is 4
10- 5.
107
8. 10,700,000 = 1.07
4
7
The number in scientific notation is 5.06
7.
108
10- 4.
Remaining sales
Sales required Current sales
=
needed for award
for award
total
= 10,000,000 - 955,000
= 9,045,000
= 9.045
The album must sell 9.045
the award.
106
The number in scientific notation is 1.07
9. 45,600,000,000 = 4.56
1010
10
The number in scientific notation is 4.56
10. 0.000000000009256 = 9.256
106 more copies to receive
107 > 6.55 107. So, the
Tyrannosaurus rex lived before the Cenozoic era began.
The Tyrannosaurus rex lived in the Mesozoic era.
12
11. 840,000 = 8.4
Vocabulary and Concept Check
10- 12.
105
5
The number in scientific notation is 8.4
10.6 Exercises (pp. 446 –447)
1010.
10- 12
The number in scientific notation is 9.256
8. Because 7.0 > 6.55, 7.0
107.
105.
12. The decimal point was moved to the right. So, the
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.
exponent should be negative.
0.000036 = 3.6
10- 5
13. The factor is greater than 10. The decimal point needs to
be moved one more place to the left.
72,500,000 = 7.25
107
7
2. It is appropriate to use scientific notation when a number
is either very large or very small.
14. Because 1.12 < 1.19 < 1.2,
Practice and Problem Solving
3. 0.0021 = 2.1
1.12
10- 3
108 < 1.2
108.
From least to greatest, the order is
1.12 108 , 1.19 108 , and 1.2 108.
3
The number in scientific notation is 2.1
4. 5,430,000 = 5.43
108 < 1.19
10
10- 3.
6
6.09
6
The number in scientific notation is 5.43
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15. Because 6.09 < 6.78 < 6.8,
106.
10- 5 < 6.78
10- 5 < 6.8
10- 5.
From least to greatest, the order is
6.09 10- 5 , 6.78 10- 5 , and 6.8 10- 5.
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Chapter 10
16. Because 1010 < 1011 < 1012 ,
10
5.7
10
11
< 9.66
10
24.
12
< 5.76
10 .
From least to greatest, the order is
5.7 1010 , 9.66 1011 , and 5.76 1012.
5
0.0207 = 2.07
241
0.02 = 2.0 10- 2
2.1
10- 2
10- 2
Because 2.0 < 2.07 < 2.1,
17. Because 10- 8 < 10- 6 < 10- 5 ,
4.8
10
-8
< 4.8
10
-6
2.0
< 4.8
10 .
2.1
10- 15 < 9.9
Because 10- 15 < 10- 14 , 9.9
10- 14.
Because 10
< 10 , 6.88
10
- 23
20. 0.000099 = 9.9
1023.
23
< 5.78
From least to greatest, the order is
6.88 10- 23 , 5.78 1023 , and 5.82
10 .
10- 3
6.25
Because 10- 3 < 10- 2 < 10- 1 < 100 ,
10- 3 < 6.3
6.25
10- 5 meter.
305% = 3.05 = 3.05
10,000
3
107
7
3.3
107 meters.
The circumference of Earth is about 4.01
units of length, so the number would be smaller.
68,500
= 6850 = 6.85
10
680 = 6.8
6.8
103
10- 3 , 6.3%,
100
3333.3 = 3.3333
103
Because 3.0334 < 3.3333,
103 < 3.3333
0
2
103.
3
Because 10 < 10 < 10 ,
3.05
100 < 3.3
102 < 3.0334
103 < 3.3333
The order from least to greatest is 305%, 3.3
10,000
3033.4, and
.
3
102
100.
102
3.0334
22. Sample answer: kilometers or miles; They are both larger
23.
10- 1 < 6.25
103
26. 3033.4 = 3.0334
5
21. 40,100,000 = 4.01
10- 2 < 6.25
The order from least to greatest is 6.25
1
0.625, and 6 .
4
1023.
10- 5
The diameter of a human hair is 9.9
10- 2
6
1023 < 5.82
23
5
, and
241
1
= 6.25 = 6.25 100
4
0.625 = 6.25 10- 1
From least to greatest, the order is
7.6 10- 15 , 9.9 10 - 15 , and 1.01 10- 14.
- 23
10- 2.
10- 2.
25. 6.3% = 0.063 = 6.3
10- 15.
10- 15 < 1.01
19. Because 5.78 < 5.82, 5.78
10- 2 < 2.1
The order from least to greatest is 0.02,
From least to greatest, the order is
4.8 10- 8 , 4.8 10- 6 , and 4.8 10- 5.
18. Because 7.6 < 9.9, 7.6
10- 2 < 2.07
-5
103.
102 ,
103
Because 6.8 < 6.85, 6.8
Because 102 < 103 , 6.8
103 < 6.85
102 < 6.8
103.
1010 = 11,740,000,000
Power from
Total
Power from solid
=
main engines
power rocket boosters
103.
The order from least to greatest is 680, 6.8
68,500
and
.
10
27. 1.174
103 ,
= 11,740,000,000 - 9,750,000,000
= 1,990,000,000
= 1.99
109
The power from the main engines is 1.99
109 watts.
28. Sample answer: Enter 1.174E10 - 9.75E9.
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319
Chapter 10
29.
1
10 - 24
1
1.66 10 - 21
So, 1 carat
1 milligram
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.
1020
6.02
1.2
c. The statement “the older the era, the longer it is” holds
1023
1.2
8.3
Er a
1023 atomic mass units and
6.02
10
23
20
atomic mass units. Because
10 , a carat is greater than a
30. a. Paleozoic era: 5.42
8
10 - 2.51
10
= 291,000,000
= 2.91
Mesozoic era: 2.51
108 years
108 - 6.55
107
= 251,000,000 - 65,500,000
109 years ago
Mesoproterozoic
1.6
109 years ago
Paleoproterozoic
2.5
109 years ago
Neoarchean
2.8
109 years ago
Mesoarchean
3.2
109 years ago
Paleoarchean
3.6
109 years ago
Eoarchean
4.0
109 years ago
Find the length of each era.
= 185,500,000
= 1.855
Cenozoic era: 6.55
= 458,000,000
8
10 < 2.91
8
10 .
= 4.58
Mesoproterozoic era = 1.6
108 < 2.91
108.
The order of the lengths from least to greatest is
6.55 107 years, 1.855 108 years, and
108 years.
Cenozoic
200
M esozoic
300
400
Paleozoic
500
M illions of years
= 6.0
108
109
= 9.0
108
109 - 2.5
109
= 3.0
108
Mesoarchean era = 3.2
109 - 2.8
= 0.4
109
= 4.0
108
Paleoarchean era = 3.6
109 - 3.2
= 0.4
109
= 4.0
108
109 - 3.6
= 0.4
109
= 4.0
108
109
109 - 1.6
= 0.9
= 0.3
Eoarchean era = 4.0
109 - 1.0
109
Neoarchean era = 2.8
100
108
= 0.6
Paleoproterozoic era = 2.5
Eras
0
108
= 1,000,000,000 - 542,000,000
107 years
Because 107 < 108 ,
107 < 1.855
109 - 5.42
Neoproterozoic era = 1.0
108 years
Because 1.855 < 2.91, 1.855
b.
1.0
8
= 542,000,000 - 251,000,000
2.91
Neoproterozoic
20
1.2 10 > 6.02
milligram.
6.55
Began
109
109
109
109
109
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.
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d. Move the decimal of the factor to adjust the exponent
Fair Game Review
31. The number 15 is a natural number, a whole number, an
integer, and a rational number.
3
32. Because
of the power of 10 to match the other number.
103 = 0.24
2.4
- 8 = - 2, the number
3
(0.24
- 8 is an integer
101
104 ) + (7.1
103 = 0.24
104 ) = (0.24 + 7.1)
104
104
= 7.34
and a rational number.
33. The number 73 is not a perfect square, so,
104
= 73,400
73 is
The results are the same.
irrational.
e. yes; Sample answer : You can subtract the factors
34. D; S = 2 w + 2 h + 2 wh
when the powers of 10 are the same. When the powers
of 10 are different, rewrite one of the numbers.
= 2( 2.5)( 2) + 2( 2.5)(1) + 2( 2)(1)
(8.2
= 10 + 5 + 4
105 ) - (4.6
105 ) = (8.2 - 4.6)
= 19
= 3.6
The surface area of the prism is 19 square inches.
Rewrite 1.5
Section 10.7
as 4.6
10.7 Activity (pp. 448–449)
1.5
1. a. First evaluate factors with exponents. Next, complete
103 + 7.1
2.4
103 = 2.4
1000 + 7.1
103 ) + (7.1
c.
(2.4 + 7.1)
= 9.5
10
10
= 5.73
102 )
(2.3
105 ) = 1.9
2.3 10 2
1000
= 43,700,000
(8.4
106 )
(5.7
10- 4 ) = 8.4
= (8.4
= 4.788 101
105
= 9.57
10
105 )
10- 4
(106
10- 4 )
= 47.88 102
105
) = (3.85 + 5.72)
5.7 106
5.7)
105
(102
107
4
105
105
= 4.37
105 ) = ( 4.9 + 1.8)
) + (5.72
105 ) = (5.88 - 0.15)
103
= 6.7
(3.85
105
2.3)
d. Add the factors.
4
104 = 0.15
= (1.9
= 9500
105 ) + (1.8
105 ) - (0.15
103
The results are the same.
(4.9
101
Subtract the factors.
yes; (1.9
103 ) = ( 2.4 + 7.1)
103 = 9.5
104 = 0.15
3. 1. D 2. F 3. E 4. C 5. B 6. A
= 9500
(
104 so that it has the same power of 10
1000
= 2400 + 7100
b. 103 ; 2.4
105
105.
(5.88
all multiplication from left to right. Then, add the
terms together.
105
10
4
102
= 4.788 103
= 4788
4
2. a. First evaluate factors with exponents. Next, complete
all multiplication from left to right. Then, add the
terms together.
2.4
103 + 7.1
104 = 2.4
1000 + 7.1
10,000
= 2400 + 71,000
= 73,400
b. The powers of 10 are not the same.
c. Sample answer : There is no common factor to factor
out.
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Big Ideas Math Blue
Worked-Out Solutions
321
Chapter 10
4. Sample answer :
3. 6
60
24
6(1.5
Number of days
in 80 year s
80
(9
103 )
(3
103 (estimate)
1.5
4.
Liter s of air
br eathed in
one day
103 )
103
= (9
10- 5
= 4.8
101
= 4.8
10- 4
(3
3
10
4
1012
5. 9.2
)
10
= 2.7
101
= 2.7
108
105
(102
3)
= 21
107
= 2.1
101
= 2.1
108
107
1012
4.6
9.2
1012
4.6
= 2 1012
107
(
10- 3 )
6. 1.5
(7.5
102 ) =
A person breathes about 270,000,000 liters of air over a
lifetime.
1.5 10- 3
7.5 102
1.5
7.5
10- 3
102
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.
= 0.2
10- 3
102
= 0.2
10- 5
= 2
10- 1
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.
= 2
10- 6
=
5. Sample answer : To add or subtract numbers written in
6. a.
b.
c.
(1.5
(7.2
(4.1
104 ) + (6.3
104 ) = 7.8
105 ) - ( 2.2
103 ) = 7.178
10- 3 )
10- 3 ) = 1.763
(4.3
(4.75
10- 6 )
(1.34
105 = 717,800
radius is
1.28
10
4
101 = 63.65
(
Find 7
10.7 On Your Own (pp. 450 –451)
1. The numbers do not have the same power of 10. Rewrite
3.41
10- 1 so it has the same power of 10 as 8.2
102.
3.41
10- 1 = 0.00341
102
103
10- 1 = 0.00341
Add the factors.
(8.2
102 ) + (0.00341
102 ) = (8.2 + 0.00341)
= 8.20341
(7.8
10- 5 ) - ( 4.5
10- 5 ) = (7.8 - 4.5)
= 3.3
322 Big Ideas Math Blue
Worked-Out Solutions
10- 5
102
10- 5
102
1.28
2
= 0.64
105 ) - (6.4
104 kilometers, so the
104
=
2
10- 5
107 ) = 6.365
105 kilometers.
radius is 700,000 = 7
The diameter of Earth is 1.28
104 = 78,000
10- 5
7. The diameter of the Sun is 1,400,000 kilometers, so the
= 0.00001763
d.
105 )
=
= 270,000,000
2.
102
3
= (7
9.2
4.6 =
7
= 27
10- 5
105 ) = 7
104
(10
3)
102 )
10- 5
= 48
10 4 (estimate)
3.0
3
(7
8)
103
9.0
365
104 ) = 9
10- 5 ) = (6
Answer wr itten in
scientific notation
Expr ession
Number of
minutes in a day
(8
104
= 6.4
10- 1
= 6.4
103 kilometers.
104
103 ).
The numbers do not have the same power of 10. Rewrite
6.4 103 so it has the same power of 10 as 7 105.
6.4
103 = 0.064
102
103 = 0.064
105
Subtract the factors.
(7
105 ) - (6.4
103 ) = (7
105 ) - (0.064
= (7 - 0.064)
105 )
105
= 6.936 105
= 693,600
The radius of the Sun is about 693,600 more kilometers
than the radius of Earth.
Copyright © Big Ideas Learning, LLC
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Chapter 10
6. Method 1:
10.7 Exercises (pp. 452–453)
(4.5
Vocabulary and Concept Check
104 )
(6.2
103 ) = 45,000
1. Use the Distributive Property to group the factors
= 279,000,000
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.
= 2.79
(4.5
104 )
(6.2
103 ) = 4.5
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:
107 ) + (5.6
107 ) = 27,400,000 + 56,000,000
7.
(2
105 ) + (3.8
107
(6.33
10- 9 ) - ( 4.5
107 ) + (5.6
107 ) = ( 2.74 + 5.6)
107
9.
107
106 ) + (3.4
(9.2
108 ) - ( 4
10.
105 ) = 8,300,000 + 340,000
= 8.64
(7.2
10- 6 ) + (5.44
106
105 = 0.34
101
105 = 0.34
106
106 ) + (3.4
105 ) = (8.3
106 ) + (0.34
= (8.3 + 0.34)
= 8.64
106 )
105
105
10- 9
10- 9
108
108
= 12.64
10- 6
= 1.264
101
10
10- 6
10- 6
-5
11. The numbers do not have the same power of 10. Rewrite
2.45
106 so it has the same power of 10 as 7.8
2.45
106 = 0.245
101
(7.8
106 = 0.245
107.
107
107 ) - (2.45 106 ) = (7.8 107 ) - (0.245 107 )
= (7.8 - 0.245) 107
106
= 7.555 107
106
12. The numbers do not have the same power of 10. Rewrite
5. Method 1:
(9.7
105 ) = 510,000
970,000
= 494,700,000,000
= 4.947
(9.7
105 ) = 5.1
= (5.1
9.7
105
9.7)
= 49.47
1010
= 4.947
101
= 4.947
1011
Copyright © Big Ideas Learning, LLC
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5
10- 5 so it has the same power of 10 as 2.46
5
10- 5 = 0.05
102
10- 5 = 0.05
10- 3.
10- 3
Add the factors.
1011
Method 2:
105 )
108
Subtract the factors.
Add the factors.
(5.1
= 2.79
103 )
107
= 1.264
The numbers do not have the same power of 10. Rewrite
3.4 105 so it has the same power of 10 as 8.3 106.
105 )
101
(104
10- 6 ) = (7.2 + 5.44)
Method 2:
(5.1
= 2.79
108 ) = (9.2 - 4)
= 8,640,000
(8.3
107
= 5.2
4. Method 1:
3.4
= 27.9
= 1.83
= 8.34
(8.3
6.2)
103
10- 9 ) = (6.33 - 4.5)
Method 2:
(2.74
= ( 4.5
= 5.8
8.
104
6.2
105 ) = ( 2 + 3.8)
= 83,400,000
= 8.34
108
Method 2:
2. no; You can use the Commutative and Associative
(2.74
6200
105
(105
105 )
(5 10- 5 ) + (2.46 10- 3 )
= (0.05 10- 3 ) + ( 2.46 10- 3 )
= (0.05 + 2.46)
= 2.51
10- 3
10- 3
1010
Big Ideas Math Blue
Worked-Out Solutions
323
Chapter 10
13. The numbers do not have the same power of 10. Rewrite
5
6.7
10 so it has the same power of 10 as 9.7
5
1
10 = 0.67
6.7
5
10 = 0.67
10
10
19.
6
10 .
(5
10- 7 )
(3
106 ) = 5
6
Add the factors.
(9.7
106 ) + (6.7
105 ) = (9.7
106 ) + (0.67
= (9.7 + 0.67)
= 10.37
106
= 1.037
101
= 1.037
7
10
106 )
106
20.
106
(3.6
107 )
(7.2
10- 2 so it has the same power of 10 as 2.4
3)
= 15
10- 1
= 1.5
101
= 1.5
100
5.5
10- 2 = 0.55
10- 2 = 0.55
101
3.6
7.2
107
107
=
3.6
7.2
107
107
= 0.5
107
107
= 0.5
100
10- 1
Subtract the factors.
= 5
10- 1
(2.4
= 5
10- 1
10- 1 ) - (5.5 10- 2 ) = ( 2.4 10- 1 ) - (0.55 10- 1 )
= ( 2.4 - 0.55) 10- 1
= 1.85 10
21.
(7.2
10- 1 )
(4
10- 7 ) = 7.2
-1
15. The numbers do not have the same power of 10. Rewrite
5.3
108 so it has the same power of 10 as 2.5
5.3
108 = 0.53
101
108 = 0.53
109.
109
Add the factors.
(2.5
109 ) + (5.3
108 ) = ( 2.5
109 ) + (0.53
= ( 2.5 + 0.53)
(7
17.
(5.8
107 ) = (5
10- 6 )
7)
107
= 35
107
= 3.5
101
= 3.5
108
(2
10- 3 ) =
4 =
1.2
1.2
4
= 0.3
=
(1.4
104 )
(2.5
5.8
2
= 3
10- 6
324 Big Ideas Math Blue
Worked-Out Solutions
101
= 2.88
10- 7
1.4
1.4)
103
106 ) =
2.8
2.5
104
106
=
2.8
2.5
104
106
= 1.12
104
106
= 1.12
10- 2
10- 7 )
10- 8
108
(108
10- 5
10- 5 )
dollar bill.
0.135
1.35 10- 1
=
-2
1.0922 10
1.0922 10- 2
1.35
1.0922
1.236
10- 1
10- 2
101
= 12.36
The thickness of a dime is about 12 times greater than the
thickness of a dollar bill.
10- 5
10- 1
= 2.88
(10- 1
24. Divide the thickness of a dime by the thickness of a
=
10- 5
= 3
10- 8
10- 7
-6
10
10- 3
10- 6
10- 3
10- 3
10- 5
4
= 28.8
= 9.1
5.8 10
2 10- 3
10 - 1
4)
= (6.5
(2.8
4
100
= (7.2
10- 5 ) = 6.5
-6
= 2.9
10- 5 )
108 )
109
23.
= 2.9
(
(6.5
107
=
18. 1.2
22.
109
= 3.03
16. 5
109 )
106 )
10- 1
107 ) =
10- 1.
106
(10- 7
= (5
14. The numbers do not have the same power of 10. Rewrite
5.5
10- 7
3
10- 5
Copyright © Big Ideas Learning, LLC
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Chapter 10
(8.3 102 ) - (3.1 108 )
= (5.2 106 ) (8.3 102 ) - (3.1 108 )
= (5.2 8.3) (106 102 ) - (3.1 108 )
= ( 43.16 108 ) - (3.1 108 )
25. 5,200,000
= ( 43.16 - 3.1)
= 40.06
108
= 4.006
1
26.
10
10- 3 )
= (9
= (9
10
10
10- 3
10
101
= (0.9
10- 2 ) + ( 2
)
10- 2 )
10- 2 )
10- 2
10- 2
= 2.9
27.
A=
w
1014 = (9.2
5.612
5.612 1014
9.2 107
5.612 1014
9.2
107
1014
0.61
107
0.61 107
107 )
w
)
(
107 + 2 6.1
106
)
107 + ( 2
6.1)
) (
106
107 + 12.2
= 2.649024
7
(7
106
101 )
101
(106
7)
= 18.543168
107
= 1.8543168
101
= 1.8543168
108
10- 2 ) L
101 )
107
108
(7
101 ) beats
106
) (
108
10
7)
= 49
10- 1
= 4.9
101
4.9
-2
(10
1
10
-2
101 )
10- 1
100 L
min
So, the heart beats about 5 liters per minute.
)
) (
108 + 0.122
7
min
= (7
=
107 + 0.122
= (1.84 + 0.122)
= 1.962
106 )
= 7
P = 2 + 2w
101
= ( 2.649024
= ( 2.649024
106 liters of blood in
70 yr
beat
106 centimeters.
The width is 6.1
(
= (1.84
= (1.84
2.649024 106 L
yr
(7
= w
106 = w
= 18.4
105
Sample answer: For estimation, use the average human
heart rate as 70 beats per minute.
= w
9.2)
101
= w
6.1
= (2
= 2.649024
In 70 years, the human heart pumps about 1.85
liters of blood.
107 = w
(
105
107 )
= w
10- 1
= 2 9.2
107
(10- 2
3.78432)
2.649024 106 L
yr
10- 3 ) + ( 2
= (0.9 + 2)
-2
A human heart pumps about 2.65
one year.
-5
-2
10
= 26.49024
=
107 ) beats
yr
3.78432
= (7
2.4 10- 5
1.2 10- 3
2.4 10- 5
+
1.2 10- 3
= (0.9
(3.78432
beat
10- 3 )
365 days
1 yr
107 ) beats
10- 2 ) L
= 7
) + (2
24 hr
1 day
yr
10
9
10- 3 ) + 2
-3
(3.78432
(7
10- 3 ) +
= (9
6.1
=
60 min
1 hr
37,843,200 beats
yr
8
(9 10- 3 ) + (2.4 10- 5 ) 0.0012
= (9 10- 3 ) + ( 2.4 10- 5 ) (1.2
= (9
72 beats
1 min
=
108
10
= 4.006
28.
102
106
)
)
8
10
8
10
The perimeter is 1.962
108 centimeters.
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Big Ideas Math Blue
Worked-Out Solutions
325
Chapter 10
5L
1 min
60 min
1 hr
24 hr
1 day
30. a. Sample answer:
365 days
1 yr
Population in 2011
(Near est million)
2,628,000 L
=
yr
=
(2.628
India
1.189
109
1.269
106
China
1.337
109
3.705
106
Ar gentina
4.2
107
1.068
106
United States
3.13
108
3.719
106
Egypt
8.2
107
3.87
105
106 ) L
yr
So, a human heart pumps about 2.63
in one year.
2.63
= 2.63
6
10 liters of blood
6
10 L
yr
= ( 2.63
Ar ea (Near est
squar e mile)
10
6
7
70 yr
) (7
10
6
1
10
10
(106
= ( 2.63
7)
= 18.41
107
= 1.841
101
= 1.841
8
)
1
101 )
b.
107
In 70 years, the human heart pumps about 1.84
liters of blood.
Number
of ridges
= 73,000
108
4.2 107
4.2
=
1.068 106
1.068
(10
= ( 23.36
107
106
101 = 39
3.9
3.13 108
3.13
=
6
3.719 10
3.719
3.2
10- 5 + 7.3
(104
8.2 107
8.2
=
3.87 105
3.87
104
0.84
10- 1 ) + 4.26
107
105
2.12
E = mc 2
31.
10- 11 = (1.674
15.066
= 11.998
15.066 10- 11
= c2
1.674 10- 27
12
The diameter of the DVD is about 12 centimeters.
10- 11- (- 27)
c2
16
= c2
9
(3
10- 27 )c 2
10- 11
= c2
10- 27
15.066
1.674
9
108 )(3
10
108 ) = c 2
2
(3
108 ) = c 2
3
108 = c
The speed of light is approximately 3
second.
Worked-Out Solutions
102 = 212
So, the population densities from least to greatest are
Argentina, the United States, Egypt, China, and India.
= 2.336 + 5.402 + 4.26
326 Big Ideas Math Blue
102 = 84
Egypt has approximately 212 people per square mile.
7.4)
-5
-1
108
106
The United States has approximately 84 people per
square mile.
10- 5 ) + (7.3
) + 4.26
10 ) + (54.02
10
103 = 361
Length
Diameter of
+
of valley center of disc
3.2)
4
0.361
Argentina has approximately 39 people per square
mile.
10- 5 + 4.26
= (7.3
109
106
Length
Number
+
of ridge
of valleys
0.000074 + 4.26
7.4
103 = 937
0.937
China has approximately 361 people per square mile.
0.000032 + 73,000
104
109
1.337
=
106
3.705
1.337
3.705
The answers are justified because the estimations are
close to the solutions.
= 7.3
109
106
India has approximately 937 people per square mile.
10
29. Diameter =
109
1.189
=
6
10
1.269
1.189
1.269
108 meters per
Copyright © Big Ideas Learning, LLC
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Chapter 10
Fair Game Review
9.
3
32. Because - 729 = ( - 9) ,
3
3
1
1
=
,
512
8
33. Because
3
- 729 =
1
=
512
3
34. Because -
125
5
= ,
343
7
3
-
3
3
(- 9)
5
7
-
(3.8
10- 6 ) = 2.4
1
.
8
=
3
104 )
= - 9.
3
1
8
125
=
343
3
(2.4
3
5
= - .
7
10.
(5.2
10- 3 )
(1.3
3.8)
= 9.12
104 + (- 6)
= 9.12
10- 2
is not written in scientific notation.
2. The factor is less than 1. So, the number 0.6
10- 7 is
Mars: 3.4
The number in standard form is 8,000,000.
10- 2 = 0.016
4. 1.6
2
5. 0.00524 = 5.24
The number in scientific notation is 5.24
8. Rewrite 1.6
2.8
Saturn: 6.03
104 = 60,300 km
104 = 25,600 km
104 = 24,800 km
radius is Saturn.
12.
(2
105 )
(1.5
108 ) = 2
1.6
108.
104 ) = (7.26 + 3.4)
= 10.66
104
= 1.066
101
= 1.066
105
104
104
13. 0.0015 = 1.5
10
= 0.16
10
10
-6
= 0.16
(2.8
10
) - (0.16
10
-5
) = (2.8 = 2.64
10
Copyright © Big Ideas Learning, LLC
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0.16)
10
1.5)
= 3
105 + 8
= 3
1013
(105
108 )
10- 3
The number in scientific notation is 1.5
14.
-5
Subtract the factors.
-5
= (2
108
3
10- 6 so that it has the same power of 10 as
1
105
1.5
1013 kilometers from the Sun.
The Oort Cloud is 3
10- 5.
-6
109
b. The planet that has the second-largest equatorial
10- 3.
108
The number in scientific notation is 8.92
104 ) + (3.4
= 4
104 = 71,500 km
8
(7.26
10- 3 +12
The planet that has the second-smallest equatorial
radius is Mars.
3
7.
= 4
Jupiter: 7.15
Neptune: 2.48
10- 3
6. 892,000,000 = 8.92
10- 3 - (- 12)
103 = 3400 km
Uranus: 2.56
The number in standard form is 0.016.
= 4
103 = 6380 km
Earth: 6.38
6
10- 3
10- 12
103 = 6050 km
Venus: 6.05
106 = 8,000,000
3. 8
= 4
103 = 2440 km
11. a. Mercury: 2.44
not written in scientific notation.
10- 3
10- 12
5.2
1.3
=
109
1. The factor is greater than 10. So, the number 23
10- 6 )
5.2 10- 3
1.3 10- 12
10- 12 ) =
Quiz 10.5–10.7
10- 6
(104
= ( 2.4
1
1
2
Bh = p ( 4) (9) = 48p cm3
3
3
35. C; V =
104
3.8
10
-5
2.3
2.5
108
2.3
=
102
2.5
108
= 0.92
102
10- 3 meter.
106 = 920,000
Pluto will orbit 920,000 times around the Sun while the
Sun completes one orbit around the Milky Way.
-5
Big Ideas Math Blue
Worked-Out Solutions
327
Chapter 10
Chapter 10 Review
1.
15.
(- 9) (- 9) (- 9) ( - 9) ( - 9)
Because - 9 is used as a factor 5 times, the exponent is 5.
So, ( - 9)
2. 2
2
(- 9) (- 9) (- 9) (- 9)
2
n
5
= ( - 9) .
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, 2
2
3. 63 = 6
5.
= -
1
2
1
2
1
1
= 2
16
1
1
=
4
2
16
19.
(- 12)
-7
20.
1
79
1
1
1
1
1
= 9
= 9- 6 = 3 =
-6
-6
7
7
7
7
7
343
(- 12)
9- 2
94
9
2
=
(n )
8.
(5 y)
3
= n
= 53
0
= ( - 12) = 1
107 = 20,000,000
10- 2 = 0.034
23. 3.4
2
The number in standard form is 0.034.
10- 9 = 0.0000000015
24. 1.5
= n
- 7 +7
The number in standard form is 20,000,000.
p 2 = p5 + 2 = p 7
7.
= ( - 12)
94 - 2
92
= 2 = 9 2 - 2 = 90 = 1
2
9
9
= 100
11 2
7
7
= - 100
2
17. 950 = 1
82
1
1
= 82 - 4 = 8- 2 = 2 =
84
8
64
22. 2
1
= ( - 200)
2
11
m1 = m2 +1 = m3
18.
21.
1
(16 - 63 ) = 12 (16 - 216)
2
6. p 5
m10 - 9 = m2
n = 2 n .
6 = 216
1
2
m10
= m8 - 6
m9
16. 2 - 4 =
3 2
n
6
4
1
2
4. -
2
m8
m6
22
9
The number in standard form is 0.0000000015.
y3
1010 = 59,000,000,000
25. 5.9
= 125 y3
10
9.
( - 2k )
4
= ( - 2)
= 16k
10.
4
k
4
The number in standard form is 59,000,000,000.
4
10- 3 = 0.0048
26. 4.8
88
= 88 - 3 = 85
83
3
The number in standard form is 0.0048.
11.
52
59
5
52 + 9
511
=
=
= 511- 1 = 510
5
5
27. 6.25
105 = 625,000
5
w8
12.
w7
13.
14.
w5
= w8 - 7
w2
22
25
23
(6c)3
c
5- 2
w
1
= w
3
1+ 3
w = w
22 + 5
27
= 3 = 3 = 27 - 3 = 24 = 16
2
2
4
= w
The number in standard form is 625,000.
28. 0.00036 = 3.6
10- 4
4
The number in scientific notation is 3.6
=
63
10- 4.
c3
29. 800,000 = 8
c
= 63
c3 - 1
= 216c
2
328 Big Ideas Math Blue
Worked-Out Solutions
105
5
The number in scientific notation is 8
105.
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Chapter 10
30. 79,200,000 = 7.92
3. - 23 = - ( 2
107
2) = - 8
2
7
31. Rewrite 4.2
5.9
109.
4.2
8
4. 10 + 33
107.
The number in scientific notation is 7.92
108 so that it has the same power of 10 as
10 = 0.42
1
8
10 = 0.42
10
10
108 ) + (5.9
109 ) = (0.42
109 ) + (5.9
= (0.42 + 5.9)
32.
(5.9
10- 4 ) - (1.8
33.
(7.7
8
10
) (4.9
10
-5
)
105 )
(1.8
10
4.9)
(2
8.
(- 3.5)
(- 3.5)9
= 66
5
= 630
10
(108
7
10) = 27
-5
10- 5 )
10
= 3.773
101
= 3.773
101 + 3
= 3.773
104
13 - 9
= ( - 3.5)
= ( - 3.5)
(- 8)
1- 3
3
= ( - 8)
= ( - 8)
12. 9.05
3.6
1.8
10
109
=
3.6
1.8
105
109
Because
So,
1
12
1
12
= 2
10
109
= 2
105 - 9
= 2
10- 4
10- 3 = 0.00905
13.
(7.8
107 ) + (9.9
107 ) = (7.8 + 9.9)
14. Rewrite 5.4
= 17.7
107
= 1.77
101
= 1.77
108
107
107
104 so that it has the same power of 10 as
6.4
105.
5.4
104 = 0.54
101
104 = 0.54
105
Subtract the factors.
(6.4
15.
1
12
1
12
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105 ) - (0.54
105 ) = (6.4 - 0.54)
= 5.86
3
= ( - 15) .
1
12
1
12
1
64
The number in standard form is 0.00905.
1
is used as a factor 5 times, the exponent is 5.
12
1
12
=
5
109 ) =
(- 15) (- 15)
1
12
(- 8)
2
3
(- 15) (- 15) (- 15)
1
12
1
The number in standard form is 30,000,000.
103
Because - 15 is used as a factor 3 times, the exponent
is 3.
2.
=
7
Chapter 10 Test
So, ( - 15)
-2
107 = 30,000,000
5
1.
4
5 2 = 5 - 2 + 2 = 50 = 1
-8
11. 3
3
= 37.73
107
13
108 + (- 5)
= 37.73
(3.6
7.
10.
8
4.9
= (7.7
34.
( 66 )
9. 5- 2
10- 4
10- 4
= 7.7
5
6.
109 )
109
10- 4 ) = (5.9 - 1.8)
= 4.1
9 = 910 +1 = 911
5. 910
109
= 6.32
9 = 10 + 3 = 13
9
Add the factors.
(4.2
9 = 10 + 27
(3.1
106 )
(2.7
10- 2 ) = 3.1
105
105
2.7
106
(106
= (3.1
2.7)
= 8.37
106 + (- 2)
= 8.37
104
10- 2
10- 2 )
5
1
1
=
.
12
12
Big Ideas Math Blue
Worked-Out Solutions
329
Chapter 10
16.
(9.6
107 )
(1.2
9.6 107
1.2 10- 4
10- 4 ) =
4. 40; In similar triangles, the corresponding angles are
107
10- 4
9.6
1.2
=
3. D; 2 4 23 = 24 + 3 = 27 = 128
= 8
107
10- 4
= 8
107 - (- 4)
= 8
107 + 4
= 8
1011
equal. So,
3
= x3 ( y2 ) = x3 y2
( xy3 )
18.
3
2
= x2 ( y3 ) = x2 y3
2
3
= x3 y6
2
= x2 y6
5. F; 2n
6. B;
5000
1
Bh
3
3V = Bh
V =
3V
= h
B
7. Part A:
The GDP per person is GDP divided by the population.
GDP
GDP per person =
Population
39
= 39 - 3 = 36 = 729
33
=
There are about 729 grains of rice in one scoop.
19. 10,000 = 1
The number in scientific notation is 1
So, the GDP per person for the United States is about
$48,397.44 per person.
Part B:
104.
new amount - original amount
20. percent decrease =
original amount
Let x represent the new amount of lead allowed in the air.
- 0.9 =
x - (1.5
1.5
10- 6 )
(- 1.35
10- 6 ) + (1.5
10- 6 ) = x - (1.5
10- 6 )
(- 1.35 + 1.5)
0.15
10
-1
1.5
10
)
= x
-6
= x
-6
10
GDP: 15,100,000,000,000 = 1.51
10
= x
10
-7
= x
GDP
Population
=
1.51 1013
3.12 108
=
1.51
3.12
1013
108
0.484
1013 - 8
= 0.484
105
= 4.84
10- 1
Chapter 10 Standards Assessment
= 4.84
10- 1 + 5
107 = 57,900,000
= 4.84
104
The new amount of lead allowed in the air is 1.5
gram per cubic meter.
1. C; 5.79
1013
108 and the GDP is 1.51
GDP per person =
10- 6 = x
-6
108
8
The population is 3.12
dollars.
Part C:
10- 6
10- 6 ) = x - (1.5
Population: 312,000,000 = 3.12
13
10- 6 )
- 0.9(1.5
1.5
15,100,000,000,000
312,000,000
48,397.44
104
4
(- 1.35
D, which means that
x = 40.
17. no;
( xy2 )
A is equal to
10- 7
7
Mercury’s distance to the Sun is approximately
57,900,000 kilometers.
1013
105
So, the GDP per person for the United States is about
4.84 104 dollars.
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
x = 15.
330 Big Ideas Math Blue
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Chapter 10
8. F; slope =
rise
4- 2
2
1
=
=
= run
-3- 3
-6
3
The line crosses the y-axis at (0, 3). So, the y-intercept
1
and the y-intercept is 3, the
3
1
equation of the line is y = - x + 3.
3
is 3. Because the slope is -
9. C; The diameter is 6 centimeters, so the radius is
3 centimeters.
V = Bh
2
= p (3) (5)
= 45p
45
3.14
= 141.3
The volume is about 141.3 cubic centimeters.
10. 0.16 or
1
4
-2
; ( - 2.5) =
25
(- 2.5)2
=
=
1
(- 2.5) (- 2.5)
1
6.25
= 0.16, or
4
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.
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Big Ideas Math Blue
Worked-Out Solutions
331