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What You've Learned
California Content Standards
1.0 Identify and use the arithmetic properties of subsets of integers and
rational, irrational, and real numbers for the four basic arithmetic operations.
2.0 Understand and use such operations as taking the opposite and finding
the reciprocal.
4.0 Simplify expressions before solving linear equations and inequalities in one
variable.
I • for Help to the Lesson in green.
Converting Fractions to Decimals
{Skills Handbook page 599)
Write as a decimal.
8
2
7
1· 10
2. 6:s
Using the Order of Operations
3
7
4. 2
3. 1000
5. 11
{Lesson 1-2)
Simplify each expression.
6. (9
7. 5 + (0.3) 3
3 + 4)2
7
Evaluating Expressions
8. 3 - (1.5) 2
9. 64
7
24
{Lessons 1-4 to 1-6)
Evaluate each expression for a = -2 and b = 5.
10. (ab )2
11. (a - b)2
U. a 3 + b 2
13. b - (3a)2
16. 3x 2 + 12x2
17. rst - t2
Simplifying Algebraic Expressions {Lesson 1-7)
Simplify each expression.
14. 3(3b + 4)
15.
!(1 -~h)
-
11rst
Identifying Properties {Lesson 1-8)
Name the property that each equation illustrates.
18. s2 · 1 = s2
19. 34 · 1 = 1
34
Evaluating a Function Rule
20. 26 (8 - 1) = 26(8) - 26 (1)
{Lesson 4-3)
Find the range of each function with domain (- 2, 0, 3.5}.
21. f(x) = -2x 2
326
Chapter 7
22. g(x) = 10 - x 3
23. y = Sx- 1
•
What You'll Learn Next
Dust particles are usually less
than 2 x 1o- 6 meter in size
and can travel thousands of
kilometers on strong winds.
California Content Standards
2.0 Understand and use the rules of exponents.
10.0 Multiply and divide monomials. Solve multi-step problems, including
word problems, by using these techniques.
-~~
• scientific notation (p. 334)
ish Audio Online
Zero and Negative Exoonents
California Content Standards
2.0 Understand and use the rules of exponents. Develop
What You'll Learn
• To simplify expressions
with zero and
negative exponents
• To evaluate exponential
expressions
for Help
@ Check Skills You'll Need
Simplify each expression.
1
1. 23
2. 2
3. 42
-:-
22
4. ( -3) 3
5. -3 3
6. 62
-:-
12
4
Evaluate each expression for a
. . . And Why
To find the size of a
population, as in Example 4
Lessons 1-2 and 1-6
a
7• 2a
= 2, b = -1, and c = 0.5.
9• ab
be
8• be
c
Zero and Negative Exponents
CA Standards
lnvestiCJation Exponents
1. a. Copy the table below. Replace each blank with the value of the power in
simplest form.
2X
sx
10X
24 = .
54= .
104 = .
103 = I
102 = .
I
23 =
22 = .
I
53=
52= .
i
I
b. Look at the values that you used to replace the blanks. What pattern do you
see as you go down each column?
2. Copy the table below. Use the pattern you described in Question 1 to complete
the table.
sx
2X
10X
21 = .
51= .
101 = LJ
20 = .
so= .
100 = .
2-1 =
I
s- 1 =
2- 2 =
I
s-2 =
10- 1 =
10- 2 =
3. Critical Thinking What pattern do you notice in the row with 0
as an exponent?
4. Copy and complete each expression.
a.
328
Chapter 7
Exponents
2-l = _1._
211
b 2- 2 = _1._
•
2
c. 2- 3 = _1._
2
Consider 33, 32 , and 31. Decreasing the exponent by one is the same as
dividing by 3. Continuing the pattern, 3° equals 1 and 3- 1 equals ~-
Zero as an Exponent
Property
For every nonzero number a, a 0 = 1.
s0
Examples
= 1
( -2) 0 = 1
(~)0 = 1
(1.02) 0 = 1
Negative Exponent
Property
For every nonzero number a and integer n , a- n = \ .
a
6- 4 =
Examples
_1_
64
(- 8) - 1 = _ 1 _
( -8)1
Why can't you use 0 as a base? By the first property, 3° = 1, 2° = 1, and 1° = 1,
which implies 0° = 1. However, the pattern 0 3 = 0, 02 = 0, and 01 = 0 implies 0° = 0.
Since both 1 and 0 cannot be the answer, 0° is undefined. In the second property,
using 0 as a base results in division by zero, which you know is undefined.
Simplifying a Power
Simplify.
a. 4- 3 = \
Read 4- 3 as "four to the
negative three."
Use the definition of negative exponent.
4
l
= 4
Simplify.
b. ( -1.23) 0 = 1
(iff CA Standards Check
Use the definition of zero as an exponent.
G) Simplify each expression.
a. 3- 4
b.(- 7) 0
d.7-1
c. ( - 4) -3
e. -3 -2
An algebraic expression is in simplest form when it is written with only positive
exponents. If the expression is a fraction in simplest form , the only common factor
of the numerator and denominator is 1.
Simplifying an Exponential Expression
Simplify each expression.
a. 4yx- 3 =
4y(;
3)
4y
=3
Use the definition of negative exponent.
Simplify.
X
b. ~ 4 = 1 -:- w- 4
w
= 1 -:- \
Use the definition of negative exponent.
= 1 · w4
Multiply by the reciprocal of~, which is w 4 .
w
Identity Property of Multiplication
w
= w4
(iff CA Standards Check
2}
Rewrite using a division symbol.
Simplify each expression.
a. 11m- 5
b. 7s- 4t2
2
c. ---=3
a
Lesson 7-1
d
n-5
• v2
Zero and Negative Exponents
329
1"--·r ··=···-~·····~~.~-~-"''""'··
····-·.. .. ·"· ..
~-~-~~-···
....,....,. .
~··
Evaluating Exponential Expressions
When you evaluate an exponential expression, you can write the expression with
positive exponents before substituting values.
Evaluating an Exponential Expression
Evaluate 3m 2 t~ 2 form = 2 and t = -3.
Method 1 Write with positive exponents first.
3m 2 t~ 2 = 3':/
Use the definition of negative exponent.
t
3(2) 2
Substitute 2 for m and -3 for t.
( -3)2
113
12 9 -
Simplify.
Method 2 Substitute first.
3m2t~2
{iffCA Standards Check
•
3(2)2 ( -3 )~ 2
Substitute 2 form and - 3 fort.
3(2) 2
( -3)2
Use the definition of negative exponent.
113
12 9 -
Q) Evaluate each expression for n
a. n
=
Simplify.
-2 and w = 5.
n- 1
-3 0
b. -
w
w2
wO
c. 4
n
d. _1
nw - 2
You can also evaluate exponential expressions that model real-world situations.
Application
A biologist is studying green peach aphids, like the one shown at the left. In the
lab, the population doubles every week. The expression 1000 · 2w models an initial
population of 1000 insects after w weeks of growth.
a. Evaluate the expression for w = 0. Then describe what the value of the
expression represents in the situation.
1000 · 2w = 1000 · 2° Substitute 0 for w.
= 1000 · 1
=
Simplify.
1000
The value of the expression represents the initial population of insects. This
makes sense because when w = 0, no time has passed.
During the months of June
and July, green peach aphids
in a field of potato plants can
double in population every
three days.
b. Evaluate the expression for w = -3. Then describe what the value of the
expression represents in the situation.
1000 · 2w = 1000 · 2- 3 Substitute -3 for w.
= 1000 ·
g
Simplify.
= 125
There were 125 aphids 3 weeks before the present population of 1000 insects.
{iffCA Standards Check
330
Chapter 7
@ A sample of bacteria triples each month. The expression 5400 · 3m models a
population of 5400 bacteria after m months of growth. Evaluate the expression
form = -2 and m = 0. Describe what each value of the expression represents.
Exponents
For more exercises, see Extra Skills and Word Problem Practice.
Practice by Example
Example 1
(page 329)
for
Help
Simplify each expression.
1. -(2.57) 0
2.4-2
3. ( -5) -2
4. -5-2
5. ( -4)- 2
6. -3 - 4
7. 2-6
8. -12-1
1
10. 78- 1
9. 0
11. ( -4) -3
12. -4-3
15. _a!_ - b3
16. 3xy• = 3x
y5
2
Example 2
Copy and complete each equation.
(page 329)
•
13. 4n • = _.1_
14.
n2
~·
_1
.
= 2x- 3y 4
3b. - 3
Simplify each expression.
1
18. 5x- 4
17. 3ab 0
19. --=7
20 _1
• c-1
X
5-2
21• - p
22. a- 4c0
23. 3xy-2
24 7ab -2
25 • X - 5y - 7
26. x- 5y 7
8
27. -----=3
28. _1L3
31. g0y7t-11
32. 7s0r -5
29. 6a-lg -3
30. 2-
d
Example 3
(page 330)
Example 4
(page 330)
Apply Your Skills
· 3"W
2c
3x 2 z- 7
5t -
2-1m2
Evaluate each expression for r = -3 and s = 5.
33. s -2
34. ,-2
35. _,-2
36. so
37. 3s-2
38. (2s) - 2
39. , - 4 s2
40.
41. s2,-3
42. ,os-2
43. 5r3s- 1
1
r
-4 2
s
44. 2- 4,3s- 2
45. Suppose your allowance doubles every week. This week you receive $2.56.
How much will your allowance be three weeks from now? How much was your
allowance three weeks ago?
I
Mental Math Is the value of each expression positive or negative?
46. -22
47. ( -2)2
48. 2- 2
49. (-2)3
50. ( -2)- 3
Write each number as a power of 10 using negative exponents.
1
1
1
1
51· 10
52· 100
54· 10,000
53· 1000
1
55. 100,000
Write each expression as a decimal.
56. 10- 3
57. 10- 6
58. 7 · 10- 1
59. 3 · 10- 2
60. 5 · 10- 4
61. a. Complete the pattern using powers of 5.
j__ C1
j__ _
j_ _ _
52 -
51 -
5o -
1
5-l = •
1
5-2 = •
b. Write ~ 4 using a positive exponent.
5
c. Rewrite
a
!n so that the power of a is in the numerator.
. h
. .
. 1
3x-2y3?
62. W h 1c expressiOn IS eqmva ent to ~.
3 -5
®_x_
y8
2
®xy-
9x y
(0 3xy2
3
Lesson 7-1
y8
®3x 5
Zero and Negative Exponents
331
Simplify each expression.
63. 45 . (0.5) 0
65. s - 2
10-3
64. 54 . 3- 2
Evaluate each expression for a = 3, b = 2, and c
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Web Code: bae-0701
7.
(-3) - 4
-3
= -4.
70. b - a
69. a- bb
68. cb
6
66. 4- 1
90
72. c-abab
71. be
73. Copy and complete the table below.
l:d :I: I ~ I!I
0:
•
74. a. Critical Thinking Simplify an · a-n.
b. What is the mathematical relationship of an and a-n? Justify your answer.
75. Which expressions equal i?
A. 4- 1
B. 2- 2
C. -41
0. 21
2
F. -2-2
E.14
76. Choose a fraction to use as a value for the variable a. Find the values of a-1, a 2 ,
and a- 2 .
77. Critical Thinking Are 3x- 2 and 3x2 reciprocals? Explain.
78. Writing Explain why the value of- 3° is negative but the value of (- 3) 0 is
positive.
Error Analysis Two students simplified the expressions as shown below. Find the
error in each of their work and simplify each expression.
79.
80.
81. Suppose you are the only person in your class who knows a certain story. After a
minute you tell a classmate. Every minute after that, every student who knows
the story tells another student (sometimes the person being told already will
30
have heard it). In a class of 30 students, the expression
1 predicts the
1 + 29 . 2
a
approximate number of people who will have heard the story after t minutes.
About how many students will have heard your story after 2 min? After 5 min?
After 10 min?
Challenge
Simplify each expression.
82. 23(5° - 6m 2)
85. (0.8) - 3
+ 19° - 2- 6
83. (- 5) 2 - (0.5) - 2
2r - 5y5 . y5
86. - 2 - 7 n
88. For what values of n is n - 3 = (~) 5 ?
3- 2 2
89. Simplify the expression ~.·
pq
332
Chapter 7
Exponents
2n
84 _Q_ + sm - 2
• m2
87• 2- 1
3- 3
-
- 13- 2
+
s(l)
22
~M
~·-.-,.li
·-n-t•
·' . _.'_, _ ·. '-··
o•ce
::: ~rac Ice;" , , .·
/u
· · -·; . .·._ ·-· .-._:,•"
._.-. r . . e· - '~. "'
-..
•:·;.--;
~.,\-~.
•:··-~----~--
'"!' ·. .,_. c- -, ... _-_,, ·. ;_,_ :-.").
:~·;:. ----~
·:·-..-.__,,.....
. ·---
~ ..' -~·-
.• ,
'·-·t··:·.-.> ..-"!"_;.·..;. ...
I' ....
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Alg1 2.0
90. Elisa is plotting points on the graph of f(x) = x1_ 3 . What is the value off(x)
2
whenx = 2?
®
Alg1 24.0
®
CD 4
16
91. Charlie made a stack of cans that is 4 cans high, as shown at the
right. If Eli makes a similar stack that is 6 cans high, how many
cans will be in Eli's stack?
®
®
Alg1 6.0
® ~
1
16
CD 91
® 729
36
61
92. Which graph represents the line y = x?
® ~ ® ~ ©* ®-fix
Alg1 24.3
93. What value for x can you use to show that x - 2 ::; x 2 is NOT always true?
®
Alg1 8.0
1
®
2
y =x - 3
y =x +3
®
CD y = -x- 3
®
y = -x + 3
~~~view
Lesson 6-6
Solve each system by graphing.
95. y > 3x + 4
y ::; - 3x + 1
96. y ::; - 2x + 1
y < 2x - 1
98. y < -3x
99. y- 1 >X
y:=;x-6
for
Help
®
CD 1
-2
94. Which equation represents a line that is perpendicular to the line y = x and has
a y-intercept of 3?
®
: MiJJeil
-2
Lesson 5-5
Lesson 5-2
97. y 2:: 0.5x
y:=;x+2
2
8
100. 3Y + 3 2:: X
-2<x
y
< -~x- 4
Write an equation for the line that passes through the given point and is parallel to
the given line.
101. y = 5x + 1; (0, 0)
102. y = 3x - 2; (0, 1)
103. y = -2x + 5; (4, 0)
104. y = 0.4x + 5; (2, - 3)
Write an equation of the line with the given slope and y-in terce pt.
105. m = -1, b = 4
106. m = 5, b = -2
107. m = ~, b = -3
108. m =
109. m = ~, b =
i
nline Lesson Quiz Visit: PHSchool.com, Web Code: baa-0701
-A,b= -17
110. m = 1.25, b = -3.79
333
Scientific Notation
S
California Content Standards
2.0 Understand and use the rules of exponents. Develop
• To write numbers in scientific
and standard notation
Lesson 7-1
Simplify each expression.
1. 6 . 10 4
• To use scientific notation
2. 7 · 10- 2
5. 3.4 . 101
4. 3 . 10- 3
... And Why
7. Simplify 3
To order planets based on their
masses, as in Example 4
for Help
@ Check Skills You'll Need
What You'll Learn
X
10 2 + 6
·>~ New Vocabulary •
X
101 + 7
X
100 + 8
3. 8.2. 10 5
6. 5.24. 10 2
X
10-1.
scientific notation
The planet Jupiter has an average radius of 69,111 km. What is Jupiter's volume?
Since Jupiter is a sphere, to answer this question you use the formula for the
volume of a sphere.
V
=
=
1nr
3
1n( 69,111 ) 3
Substitute 69,111 for r.
= 1,382,706,933,000,000
Simplify.
The number 1,382,706,933,000,000 is written in standard notation. In scientific
notation, you write the number as 1.382706933 X 1015 . Scientific notation is a
shorthand way to write very large or very small numbers.
Definition
Scientific Notation
A number in scientific notation is written as the product of two factors in
the form ax 10n, where n is an integer and 1 ~ a < 10.
Examples 3.4 x 106
5.43
X
1013
2.1 x 10-10
Recognizing Scientific Notation
Is each number written in scientific notation? If not, explain.
a. 56.29
@ CA Standards Check
334
Chapter 7
•
X
10 12
No; 56.29 is greater than 10.
b. 0.84 x
10- 3
No; 0.84 is less than 1.
c. 6.11
10 5
yes
X
(!) Is each number written in scientific notation? If not, explain.
b. 52 X 10 4
c. 0.04
a. 3.42 X 10- 7
Exponents
X
10- 5
In scientific notation, you use positive exponents to write a number greater than 1.
You use negative exponents to write a number between 0 and 1.
Writing a Number in Scientific Notation
Write each number in scientific notation.
a. 56,900,000
56,900,000. = 5.69 X 107 Move the decimal point 7 places to the left and use 7 as
an exponent. Drop the zeros after the 9.
~
b. 0.00985
Move the decimal point 3 places to the right and use
-3 as an exponent. Drop the zeros before the 9.
o.00985 = 9.85 x Io -3
\...J(
(i/ CA Standards Check
Q) Write each number in scientific notation.
b. 46,205 ,000
c. 0.0000325
d. 0.000000009
9
e. Critical Thinking You express 1 billion as 10 . Explain why you express 436 billion
as 4.36 X 10 11 in scientific notation.
a. 267,000
Writing a Number in Standard Notation
Write each number in standard notation.
a. temperature at the sun's core: 1.55
1.55 X 106 = 1\550009.
X
10 6 kelvins
A positive exponent indicates a number greater than
10. Move the decimal point 6 places to the right.
'-----"'
= 1,550,000
b. lowest temperature recorded in a lab: 2 x 10- 11 kelvin
2 X 10- 11 = 0 .QOOOOOOOOO;
·~
-
(i/ CA Standards Check
A negative exponent indicates a number between 0
and 1. Move the decimal point 11 places to the left.
= 0.00000000002
Q) Write each number in standard notation.
a. 3.2
,..........-;
X
10 12
---~---·~
,._.
b. 5.07
___ .
-
-
X
10 4
-
Using Scientific Notation
Masses of Planets
(kilograms)
c. 5.6
X
10- 4
d. 8.3 x 10- 2
--·--·--------
You can compare and order numbers in scientific notation. First compare the
powers of 10, and then compare the decimals.
Application
List the planets in order from least to greatest mass.
Order the powers of 10. Arrange the decimals with the same power of 10 in order.
8.7 X 10 25
Uranus
1.0 X 10 26
Neptune
5.7 X 10 26
Saturn
Lesson 7-2
3.7 X 10 27
Jupiter
Scientific Notation
335
You can write numbers like 815
815
X
10 5
81 ,500,000
=
8.15
=
X
10 5 and 0.078 x 10- 2 in scientific notation.
X
10 7
0.078
X
10- 2
=
0.00078
=
7.8
X
10- 4
The examples above show this pattern: When you move a decimal n places left, the
exponent of 10 increases by n ; when you move a decimal point n places right, the
exponent of 10 decreases by n.
Using Scientific Notation to Order Numbers
Order 0.052
X
10 7 , 5.12
X
10 5 , 53.2 x 10, and 534 from least to greatest.
Write each number in scientific notation.
0.052
Remember that
53.2
X
10 is 53.2
X 10 1 .
5.2
X
10 7
5.12
X
10 5
53.2
10
X
534
t
t
t
t
X 10 5
X 10 5
X 10 2
5.12
5.32
5.34
X
10 2
Order the powers of 10. Arrange the decimals with the same power of 10 in order.
5.32
X
10 2
5.34
X
10 2
5.12
105 5.2
X
X
10 5
Write the original numbers in order.
53.2
@'CA Standards Check
X
10 534
tj) Order 60.2 x
5.12
X
10 5
0.052
10 7
X
10- 5 , 63 x 10 4 , 0.067 x 10 3 , and 61 x 10- 2 from least to greatest.
You can multiply a number that is in scientific notation by another number. If the
product is less than one or greater than 10, rewrite the product in scientific notation.
Multiplying a Number in Scientific Notation
Simplify. Write each answer using scientific notation.
Use the Associative Property of
Multiplication.
a. 7 ( 4 X 10 5 ) = (7 · 4) X 10 5
= 28 X 10 5
= 2.8 X 10 6
b. 0.5 (1.2
X
Simplify inside the parentheses.
Write the product in scientific notation.
10- 3 ) = (0.5 · 1.2)
X
10- 3
= o.6 x 10- 3
= 6 x 10-
@'CA Standards Check
Use the Associative Property of
Multiplication.
Simplify inside the parentheses.
4
Write the product in scientific notation.
6} Simplify. Write each answer using scientific notation.
a. 2.5(6 X 10 3 )
b. 0.4(2
X
10- 9)
For more exercises, see Extra Skills and Word Problem Practice.
Practice by Example
for
Example 1
Help
\
336
Chapter 7
(page 334)
Exponents
Is each number written in scientific notation? H not, explain.
1. 55 X 10 4
2. 3.2
4. 7.3 X 10- 5
5. 1.12
X
10 5
X
101
3. o.9 x 10- 2
6. 46 X 10 7
Example 2
(page 335)
Write each number in scientific notation.
7. 9,040,000,000
11. 0.00325
Example 3
(page 335)
(pages 335, 336)
10. 21,700
9. 9.3 million
12. 8,003,000
13. 0.00092
14. 0.0156
Write each number in standard notation.
15. 5
X
10 2
19. 8.97 x 10- 1
Examples 4, 5
8. 0.02
16.
s x 10- 2
20. 1.3
X
17. 2.04
X
10 3
18. 7.2 X 10 5
21. 2.74 x 1o- 5
100
22. 4.8
X
10 - 3
Order the numbers in each list from least to greatest.
23. 10 5, 1o- 3, 10°, 10-1, 10 1
24. 9 x 1o- 7 , 8 x 1o- 8, 7 x 1o- 6 , 6 x 10- 10
25. 50.1 X 10- 3, 4.8 X 10-1, 0.52 X 10- 3, 56 X 10- 2
26. 0.53 X 10 7, 5300 X 10- 1, 5.3 X 10 5, 530 X 108
27. Measuring instruments may have different degrees of precision. Instrument A
is precise to 10- 2 em, Instrument B is precise to 5 x 10- 2 em, and Instrument C
is precise to 8 X 10- 3 em. Order the instruments from most precise (least
possible error) to least precise (greatest possible error).
Example 6
(page 336)
Apply Your Skills
Simplify. Write each answer using scientific notation.
28. 8(7 X 10- 3)
29. 8(3 X 10 14)
30. 0.2(3
31. 6(5.3 x 1o- 4)
32. 0.3(8.2
33. 0.5(6.8
X
10-3)
X
10 2)
105)
X
For Exercises 34-39, find the missing value.
Selected Masses (kilograms)
Elephant calves weigh from
1.0 X 102 to 1.45 X 102
kilograms.
34.
1 Elephant
35.
~ult human I
70
•
10
36.
Dog
37.
Golf ball
0.046
38.
Paper clip
u
39.
Oxygen atom I
I
s x 10- 4
0.00000000000000000000000003
40. Critical Thinking Is the number 10 5 in scientific notation? Explain.
41. Writing Explain how to write 48 million and 48 millionths in scientific notation.
42. In 2010, the population in the United States will be about 3.09 X 108. Spending for
health care will be about $8754 per person. About how much will the United
States spend on health care in 2010? Use scientific notation.
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43. A computer can perform 4.66 X 10 8 instructions per second. How many
instructions is that per minute? Per hour? Use scientific notation.
44. The national debt at the beginning of 2005 was about $7,600,000,000,000. Write
this number in scientific notation.
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337
.....
Challenge
45. The world population in 2025 may reach 7.84 X 10 9 persons. This is about
3 times the world population in 1950. What was the world population in 1950?
46. Use a calculator to find the volume of each planet with the given radius.
a. Mercury: 2439 km
b. Earth: 6378 km
c. Saturn: 60,268 km
6
47. Write 3 0 using scientific notation.
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Alg1 2.0
48. Florence Griffith-Joyner set the world record for the women's 100-meter sprint
in 1988. Which expression equals her winning time of 1.748 x 10-1 minutes?
®
Alg1 6.0
0.1748
CD 1748
17.48
X
10- 3
®
1748
X
103
49. What is the equation of the line with an x-intercept of 3 and a y-intercept of 2?
®
Alg1 5.0
®
2y = 3x
®
CD 3x + 2y = 6
3y = 2x
®
2x
+ 3y =
6
50. Abby will receive 1 free football ticket for every 30 tickets she sells for a raffle.
How many raffle tickets must she sell to get 5 free football tickets?
®
for
Help
Lesson 7-1
®
6
Lesson 3-5
®
305
Simplify each expression.
51. 4(1.8) 0
Lesson 6-5
CD 150
35
52. 12 . 2- 2
Graph each linear inequality.
1
56. y < -4 X + 2
57. y
54 43
• 72
53.6. 3-2
2:
32 X
5 3-2
5. - 0
9
58. y < 3x- 4
Solve each compound inequality.
59. 5
~
2a < 13
60. -7
62. y + 5 < -1 or 3y + 1 > -2
~
b
+ 8 ~ 23
63.
61. -16 <4m-2< -3
-!(p -2) <~or 3p ~ -9
Simplify each expression.
,-s s - 4
1. 5- 1 (3 - 2)
2.
5. a2boc-3
6. (3 2)( 4m 2 )
3. 2x53y - 12
mn- 4
4. o-=2
pq
7 2t0
• 3s- 2
8. 5- 315n6
9. A certain bacteria population doubles in size every day. Suppose a sample
starts with 500 bacteria. The expression 500 · 2x models the number of bacteria
in the sample after x days. Evaluate the expression for x = 0, 2, 5.
10. Saturn is about 1,433,500,000 km from the sun. Mars is located about
22.79
338
Chapter 7
Exponents
X
107 km from the sun. Which planet is farther from the sun?
Multiplication Properties
of Exponents
California Content Standards
2.0 Understand and use the rules of exponents. Develop
10.0 Multiply monomials. Solve multi-step problems, including word problems, by using this
technique. Develop
What You'll Learn
• To multiply powers
• To work with scientific
notation
... And Why
To find the number of red
blood cells in the human body,
as in Example 5
{i! Check Skills You'll Need
Rewrite each expression using exponents.
2. (6 - m)(6 - m)(6 - m)
4. 5 . 5 . 5 . s . s . s
l.t·t·t·t·t·t·t
3. (r + 5)(r + 5)(r + 5)(r + 5)(r + 5)
Simplify.
s. (- s)- 4
7. ( - 5) 0
6. ( -5) 4
5. -5 4
r -Mult:iplying Powers
CA Standards
Investigation
Exponents With the Same Base and Multiplication
1. Copy and complete the table.
Product Using
Factors Repeated Factors
2. What is true for 22 22 and
23 . 21?
o
3. Patterns What relationship do
you see between the sum of the
exponents of 2 in the first column
and the exponent of 2 in the third
column?
Power
of 2
21.21
2·2
22
22.22
•
•
•
•
•
•
•
•
23.21
24.22
23 23
0
You can write the product of powers with the same base, like 24 • 22 , using
one exponent.
24 ° 22 = (2 ° 2 ° 2 ° 2) (2 . 2) = 26
0
Notice that the sum of the exponents of the expression 24 22 equals the
exponent of 26 .
o
Multiplying Powers With the Same Base
Property
For every nonzero number a and integers m and n, am
Examples 35
o
34
=
35
+4
=
39
lesson 7-3
o
an = am + n.
h2 . h9 = h 2 + 9 = hll
Multiplication Properties of Exponents
339
~
.....
Multiplying Powers
Rewrite each expression using each base only once.
a. 11 4 • 11 3 = 11 4 +
b. 5 -2 . 5 2
•
@CA Standards Check
1 .~
= 11 7
= 5 -2 +
3
Add exponents of powers with the same base.
Simplify the sum of the exponents.
2
Add exponents of powers with the same base.
=s o
Simplify the sum of the exponents.
= 1
Use the definition of zero as an exponent.
Rewrite each expression using each base only once.
a. 5 3 • 5 6
b. 2 4 • 2 - 3
c. 7-3 . 72 . 76
When variable factors have more than one base, be careful to combine only those
powers with the same base.
Multiplying Powers in an Algebraic Expression
Simplify each expression.
a. 2n 5 • 3n - 2 = (2 · 3)(n 5
•
n - 2)
Commutative Property of Multiplication
= 6(n5 + ( -2))
=
Add exponents of powers with the same base.
6n 3
Simplify.
Commu~a~ive .and Associative Properties
of Multtphcatton
b. Sx . 2y4 . 3x 8 = (5 . 2 . 3)(x . x 8)(y4)
= 30(x 1 • x 8)(y 4 )
Remember that x
=
x 1.
@CA Standards Check
30(x 1 +
= 30x 9y 4
=
•
Multiply the coefficients. Write x as x1.
8)(y 4 )
Add exponents of powers with the same base.
Simplify.
Q) Simplify each expression.
a. n 2 • n 3 • 7n
b. 2y 3 • 7x 2
•
2y 4
c. m 2 • n - 2 • 7m
' 'worldng With- scientific Notation
You can use the property for multiplying powers with the same base to write
numbers and to multiply numbers in scientific notation.
Multiplying Numbers in Scientific Notation
Simplify (7
(7
X
10 2 )(4
X
X
10 2 )( 4
X
10 5). Write the answer in scientific notation.
10 5) = (7 · 4)(10 2 • 10 5)
•
@CA Standards Check
10 7
= 28
X
= 2.8
X 10 1 •
= 2.8
X
10 1 +
= 2.8
X
10 8
Commutative and Associative Properties
of Multiplication
Simplify.
10 7
7
Write 28 in scientific notation.
Add exponents of powers with the
same base.
Simplify the sum of the exponents.
Q) Simplify each expression. Write each answer in scientific notation.
a. ( 2.5
340
Chapter 7 Exponents
X
10 8 )( 6
X
10 3 )
b.
(1.5
X
10- 2 )(3
X
10 4 )
c.
(9
X
10- 6 )(7
X
10- 9 )
Application
A human body contains about 3.2 X 10 4 p.L (microliters) of blood for each pound
of body weight. Each microliter of blood contains about 5 X 10 6 red blood cells.
Find the approximate number of red blood cells in the body of a 125-lb person.
red blood cells = pounds · microliters ·
pound
= 125 lb · (3.2
X
= (125 · 3.2 · 5)
(2000)
=
= 2000
= 2 X
There are about 2
Example 1
(page 340}
for
Help
Example 2
(page 340}
Example 3
(page 340}
Example 4
(page 341}
X
X
106) c~~s
X
Use dimensional analysis.
Substitute.
Commutative and
Associative Properties of
Multiplication
(10 4 · 106)
(10 4 + 6)
Simplify.
1010
Add exponents.
Write 2000 in scientific
notation.
1013
Add the exponents.
10 13 red blood cells in a 125-lb person.
@ About how many red blood cells are in the body of a 160-lb soccer player?
For more exercises, see Extra Skills and Word Problem Practice.
.
--Practice by Example
10 4) ~[; · (5
= 2 X 103 . 1010
The blood cells shown in the
photo above are magnified
5 x 10 3 times their actual size.
(iJ CA Standards Check
X
X
. cell~
microliter
---
--·--
Rewrite each expression using each base only once.
1. 26 . 2 4
2. 5- 13 · 55
4. (0.99) 3 . (0.99) 0
5. 66 . 6- 2
•
•
3. 10- 6 · 10 5
22
.
101
6. (1.025) 2 (1.025) - 2
65
Simplify each expression.
7. c- 2
•
9. 5t- 2 · 2t- 5
8. 3r · r 4
c7
10. (7x 5)(8x)
11. 3x 2 · x 2
12. ( -2.4n 4)(2n - 1)
13. b - 2 . b 4 . b
14. ( -2m 3)(3.5m -3)
15. (15a 3)(- 3a)
16. (x 5y 2)(x- 6y)
17. (5x 5 )(3y6)(3x2)
18. (4c 4)(ac 3)(3a 5c)
19. x6 . y2 . x4
20. a6b3 • a2b- 2
21. - m 2 · 4r 3 • 12r- 4 · 5m
Simplify each expression. Write each answer in scientific notation.
22. (2
X
10 3)(3
25. (1
X
10 3)(3.4
X
10 2)
X
10- 8)
23. (2
X
10 6)(3
26. (8
X
10- 5)(7
X
10 3)
X
10- 3)
24. (4
X
106) · 10- 3
27. (5
X
10 7)(3
X
101 4)
Write each answer in scientific notation.
28. The distance light travels in one year is about 5.88 X 10 12 miles. This distance
is a light-year. The closest star to Earth (other than the sun) is Alpha Centauri,
which is 4.35light-years from Earth. About how many miles from Earth is
Alpha Centauri?
Lesson 7-3
Multiplication Properties of Exponents
341
Write each answer in scientific notation.
29. Earth's crust contains about 120 trillion metric tons of gold. One metric ton of
gold is worth about $9 million. What is the value of the gold in Earth's crust?
30. Light travels through space at a constant speed of about 3 X 105 km/s. Sunlight
reflecting from the moon takes about 1.28
distance from the moon to Earth.
Apply Your Skills
X
10° s to reach Earth. Find the
Complete each equation.
31. 5 2 . 5
= 5 11
34. c- 5 . c
=
c6
35. m
33. 2• . 2 4 = 21
· m-4 = m- 9
36. a · a · a 3 = a
38. a 12 · a
· a4 = 1
37. a
= 53
32. 57 • 5
= a 12
39. x 3y • · x • = y 2
Error Analysis Correct each error.
41.
40.
42.
~
43.
~
t
Find the area of each figure.
~.
J2x
3x2
y3
~4
45. 0
;2
6;4 +
2
=
2x 2
+x
~- /~ 7
Visit: PHSchool.com
Web Code: bae-0703
=
47. ~
4c
1
I
+2
2c3
Simplify each expression. Write each answer in scientific notation.
48.
(9
X
107)(3
X
10-16)
50. (0.7
X
10- 12 )( 0.3
52. (0.2
X
10 5 )(4
X
X
108)
10-12)
49. (8 X 10- 3)(0.1 X 109)
51. (0.4
X
10°)(3
X
10- 4)
53. (0.5 X 10 13 )( 0.3 X 10- 4)
54. The term mole can be used in chemistry to refer to 6.02 X 10 23 atoms
of a substance. The mass of a single hydrogen atom is approximately
1.67 x 10- 24 gram. What is the mass of 1 mole of hydrogen atoms?
55. a. Write y 8 as a product of two powers with the same base in four different
ways. Use only positive exponents.
b. Write y 8 as a product of two powers with the same base in four different
ways using negative or zero exponents in each.
c. Reasoning How many ways are there to write y 8 as the product of two
powers? Explain your reasoning.
342
Chapter 7
Exponents
56. Medical X-rays, with a wavelength of about 10- 10 meter, can penetrate your skin.
a. Ultraviolet rays, which cause sunburn by penetrating only the top layers of
skin, have a wavelength about 1000 times the wavelength of an X-ray. Find
the wavelength of ultraviolet rays.
b. Critical Thinking The wavelengths of visible light are between
4 x 10- 7 meter and 7.5 x 10- 7 meter. Are these wavelengths longer or
shorter than those of ultraviolet rays? Explain.
57. Writing Explain why x 3 • y 5 cannot be written with fewer bases.
58. A CD-ROM stores about 650 megabytes (6.5 X 10 8 bytes) of information
along a spiral track. Each byte uses about 9 micrometers (9
space along the track. Find the length of the track.
X
10- 6 m) of
Simplify each expression. Write each answer in scientific notation.
Light used to "see" an object
cannot have a wavelength
that is larger than the object.
Only gamma rays are smaller
than X-rays. They have
lengths that are as small as
1Q-15 meter.
59. (6.12 X 10 5 )(12.5 X 108)
60. (1.9 X 10- 3)(2.04 X 10 11 )
61. (9.55 X 10 7)(7.3 X 10- 15 )
62. ( 6.9 x 1o- 9 )(2.5 x 1o- 4 )
63. There are about 3.35 X 10 25 molecules in a liter of water. Clear Lake, in
California, has about 1.4 X 10 12 liters of water. About how many molecules of
water are in Clear Lake?
64. About 8.4 X 10 11 drops of water flow over Niagara Falls each minute. Each
)..
drop of water contains about 1.7 X 10 21 molecules of water. About how many
molecules of water flow over the falls each minute?
Simplify each expression.
65.
1
---;2~-----,:
x ·x
68. 2a 2(3a
Challenge
+ 5)
66. -3-1 -
5
67. -----=4
c. c
a ·a
69.
8m 3 (m 2
70. -4x 3 (2x 2 - 9x)
+ 7)
Simplify.
71. 3x . 32 - x . 32
74. (a
+ b) 2(a + b) -3
72. 2 n • 2 n +
75. (t
2 •
73. 3x · 2Y · 32
2
•
2x
76. 5x + 1 . 51 - x
+ 3)\t + 3) - 5
77. a. Find the volume of a rectangular prism with length 1.3 X 10- 3 km,
width 1.5 x 10- 3 km, and height 9.4 X 10- 4 km. Write your answer
in scientific notation.
b. What is the volume of the prism in cubic meters?
78. A protozoan is 1.1 x 10- 4 meter long. A diagram of a protozoan in a science
book is 7.7 centimeters long. The diagram is how many times greater than the
protozoan in length?
·~ · M~l:fjple .~lioice-pracfice
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Alg1 2.0
79. Which expression is equivalent to 2(3a 2b )(a- 2b )?
®
Alg117.0
5b 2
®
6b 2
CO 6ab 2
®
6a - 4 b
®
1
80. Which value is NOT in the range of the functionf(x) = x 2 - 1?
®
-2
®
nline Lesson Quiz Visit: PHSchool.com, Web Code: baa-0703
-1
co
0
343
Alg1 5.0
81. Which problem is best represented by the number sentence 5x
+ 3(x + 2) = 26?
ClD Mia spent $5 on a pizza and $3 on 2 drinks. How much did she
®
©
®
Alg117.0
spend in all?
Devin spent 5 hours studying for a math test and 2 more hours
studying for 3 other tests. How many tests did he study for in 26 hours?
Angela bought 5 notebooks and 3 binders for $26. If a binder
costs $2 more than a notebook, how much does a notebook cost?
Steve's class has 2 more boys than girls. There are 26 students
in the class. If 5 girls and 3 boys play a sport, how many girls do NOT
play a sport?
82. Which graph below has a range that includes only negative real numbers?
ClD '
.tY
I
'
I
I
...
©
tY
4
2
-4 1-2
01
2
4x
I
4x
-2
t
r
®
I
:
4r ; !t
~
t
T
ll..
l
r
~
~
®
4
\
0
-2
I
4x
2
tY
"
2
-4
-4
-4 /-2 01
-4
•--· MIJ.C:eCIReview · ·
for
Help
Lesson 7-2
Write each number in scientific notation.
84. 0.0035
83. 1,280,000
85. 0.00009
86. 6.2 million
89. 9.1 X 1011
90. 2.9 x 10- 4
Write each number in standard form.
88. 1.052 X 10- 3
87. 8.76 X 108
Lesson 6-6
Solve each system by graphing.
< 3x + 2
2x+y2::4
92. y <X+ 6
X - 3y ::S 6
91. y
Lesson 4-7
98. A(n) = 8
Exponents
::S
6
(n - 1)(4)
95. A(n) = -5
+ (n - 1)(2)
+ (n - 1)( -4)
97. A(n) = 1.2
+ (n - 1)( - 4)
= 10 +
96. A(n) = 12
Chapter 7
X+ 2y
Find the third, seventh, and tenth terms of each sequence.
94. A(n)
344
93. y >X+ 4
+ (n - 1)(3)
99. A(n) = 8
+ (n - 1)( -3)