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**Unformatted text preview: **Exponents and
Polynomials
7A Exponents
7-1 Integer Exponents 7-2 Powers of 10 and Scientific
Notation Lab Explore Properties of
Exponents 7-3 Multiplication Properties of
Exponents 7-4 Division Properties of
Exponents 7-5 Rational Exponents 7B Polynomials
7-6 Polynomials Lab Model Polynomial Addition
and Subtraction 7-7 Adding and Subtracting
Polynomials Lab Model Polynomial
Multiplication 7-8 Multiplying Polynomials 7-9 Special Products of Binomials • Use exponents and scientific notation
to describe numbers.
• Use laws of exponents to simplify
monomials.
• Perform operations with polynomials. Every Second Counts
How many seconds until you
graduate? The concepts in this
chapter will help you find and use
large numbers such as this one. KEYWORD: MA7 ChProj 456 Chapter 7 Vocabulary
Match each term on the left with a definition on the right.
A. a number that is raised to a power
1. Associative Property
2. coefficient B. a number that is multiplied by a variable 3. Commutative
Property C. a property of addition and multiplication that states you can
add or multiply numbers in any order 4. exponent D. the number of times a base is used as a factor 5. like terms E. terms that contain the same variables raised to the same
powers
F. a property of addition and multiplication that states you can
group the numbers in any order Exponents
Write each expression using a base and an exponent.
6. 4 · 4 · 4 · 4 · 4 · 4 · 4
7. 5 · 5
9. x · x · x 8. (-10)(-10)(-10)(-10) 10. k · k · k · k · k 11. 9 Evaluate each expression.
12. 3 4 13. -12 2 14. 5 3 15. 2 5 16. 4 3 17. (-1)6 19. 25.25 × 100 20. 2.4 × 6.5 Evaluate Powers Multiply Decimals
Multiply.
18. 0.006 × 10 Combine Like Terms
Simplify each expression.
21. 6 + 3p + 14 + 9p 22. 8y - 4x + 2y + 7x - x 23. (12 + 3w - 5) + 6w - 3 - 5w 24. 6n - 14 + 5n Squares and Square Roots
Tell whether each number is a perfect square. If so, identify its positive square root.
25. 42
26. 81
27. 36
28. 50
29. 100 30. 4 31. 1 32. 12 Exponents and Polynomials 457 Key
Vocabulary/Vocabulario
Previously, you • wrote and evaluated
• exponential expressions.
simplified algebraic
expressions by combining like
terms. You will study • properties of exponents.
• powers of 10 and scientific
• notation.
how to add, subtract, and
multiply polynomials by using
properties of exponents and
combining like terms. binomial binomio degree of a monomial grado de un monomio degree of a polynomial grado de un polinomio leading coefficient coeficiente principal monomial monomio perfect-square
trinomial trinomio cuadrado
perfecto polynomial polinomio scientific notation notación científica standard form of a
polynomial forma estándar de un
polinomio trinomial trinomio Vocabulary Connections
To become familiar with some of the
vocabulary terms in the chapter, consider
the following. You may refer to the chapter,
the glossary, or a dictionary if you like. You can use the skills
in this chapter • to model area, perimeter, and
• • volume in geometry.
to express very small or very
large quantities in science
classes such as Chemistry,
Physics, and Biology.
in the real world to model
business profits and
population growth or
decline. 1. Very large and very small numbers are
often encountered in the sciences. If
notation means a method of writing
something, what might scientific
notation mean?
2. A polynomial written in standard form
may have more than one algebraic term.
What do you think the leading
coefficient of a polynomial is?
3. A simple definition of monomial is “an
expression with exactly one term.” If the
prefix mono- means “one” and the prefix
bi- means “two,” define the word
binomial .
4. What words do you know that begin with
the prefix tri-? What do they all have in
common? Define the word trinomial
based on the prefix tri- and the
information given in Problem 3. 458 Chapter 7 Reading Strategy: Read and Understand the Problem
Follow this strategy when solving word problems.
• Read the problem through once.
• Identify exactly what the problem asks you to do.
• Read the problem again, slowly and carefully, to break it into parts.
• Highlight or underline the key information.
• Make a plan to solve the problem.
From Lesson 6-6
29. Multi-Step Linda works at a pharmacy for $15 an hour. She also baby-sits for
$10 an hour. Linda needs to earn at least $90 per week, but she does not want
to work more than 20 hours per week. Show and describe the number of hours
Linda could work at each job to meet her goals. List two possible solutions. Step 1 Identify exactly what
the problem asks you
to do. • Show and describe the number of hours Linda
can work at each job and earn at least $90
per week, without working more than 20 hours
per week.
• List two possible solutions of the system. Step 2 Step 3 Break the problem into
parts. Highlight or
underline the key
information. Make a plan to solve
the problem. • Linda has two jobs. She makes $15 per hour at
one job and $10 per hour at the other job.
• She wants to earn at least $90 per week.
• She does not want to work more than 20 hours
per week.
• Write a system of inequalities.
• Solve the system.
• Identify two possible solutions of the system. Try This
For the problem below,
a. identify exactly what the problem asks you to do.
b. break the problem into parts. Highlight or underline the key information.
c. make a plan to solve the problem.
1. The difference between the length and the width of a rectangle is 14 units. The
area is 120 square units. Write and solve a system of equations to determine
the length and the width of the rectangle. (Hint: The formula for the area of a
rectangle is A = w.) Exponents and Polynomials 459 7-1 Integer Exponents Objectives
Evaluate expressions
containing zero and
integer exponents. Who uses this?
Manufacturers can use negative exponents
to express very small measurements. Simplify expressions
containing zero and
integer exponents. In 1930, the Model A Ford was one of the first
cars to boast precise craftsmanship in mass
production. The car’s pistons had a diameter
of 3 __78 inches; this measurement could vary
by at most 10 -3 inch.
You have seen positive exponents. Recall
that to simplify 3 2, use 3 as a factor 2 times:
3 2 = 3 · 3 = 9.
But what does it mean for an exponent to
be negative or 0? You can use a table and look for a pattern
to figure it out. Base
x4
Exponent Power 55 54 53 52 51 Value 3125 625 125 25 5 ÷5 ÷5 ÷5 50 5 -1 5 -2 ÷5 When the exponent decreases by one, the value of the power is divided by 5.
Continue the pattern of dividing by 5:
5 =1
50 = _
5 1
1 ÷5=_
1 =_
5 -2 = _
5
25 5 2 1 =_
1
5 -1 = _
5 51 Integer Exponents
WORDS
Zero exponent—Any
nonzero number raised to
the zero power is 1. 2 -4 is read “2 to
the negative fourth
power.” Negative exponent—A
nonzero number raised to a
negative exponent is equal
to 1 divided by that number
raised to the opposite
(positive) exponent. NUMBERS
3 0 = 1 123 0 = 1 (-16) 0 = 1 ALGEBRA
If x ≠ 0, then x 0 = 1. (_37 ) = 1
0 1
1 =_
3 -2 = _
9
32
1
1 =_
2 -4 = _
16
24 If x ≠ 0 and n is an integer,
1.
then x -n = _
xn Notice the phrase “nonzero number” in the table above. This is because 0 0 and
0 raised to a negative power are both undefined. For example, if you use the
pattern given above the table with a base of 0 instead of 5, you would get 0 0 = __00 .
1
Also, 0 -6 would be __
= __10 . Since division by 0 is undefined, neither value exists.
6
0 460 Chapter 7 Exponents and Polynomials EXAMPLE 1 Manufacturing Application
The diameter for the Model A Ford piston could vary by at most 10 -3 inch.
Simplify this expression.
1
1 =_
1
10 -3 = _
=_
10 3 10 · 10 · 10 1000
1
inch, or 0.001 inch.
10 -3 inch is equal to ____
1000 1. EXAMPLE 2 A sand fly may have a wingspan up to 5 -3 m. Simplify this
expression. Zero and Negative Exponents
Simplify. A 2 -3 B 50 1
1 =_
1
2 -3 = _
=_
23 2 · 2 · 2 8 to the zero power is 1. C (-3)-4 In (-3) , the base
is negative because
the negative sign
is inside the
parentheses.
In -3 -4 the base (3)
is positive.
-4 1 = __
1
1
(-3)-4 = _
=_
(-3) 4 (-3)(-3)(-3)(-3) 81 D -3 -4
1
1 = -_
1
-3 -4 = - _
= -_
3·3·3·3
81
34
Simplify.
2a. 10 -4 EXAMPLE 5 0 = 1 Any nonzero number raised 3 2b. (-2)-4 2c. (-2)-5 2d. -2 -5 Evaluating Expressions with Zero and Negative Exponents
Evaluate each expression for the given value(s) of the variable(s). A x -1 for x = 2
2 -1
2 -1 Substitute 2 for x. 1
1 =_
=_
21 2 1 .
Use the definition x -n = _
xn B a 0b -3 for a = 8 and b = -2
8 0 · (-2)-3
Substitute 8 for a and -2 for b.
1·
1· 1
_ (-2)3 1
__ (-2)(-2)(-2) _ 1· 1
-8
1
-_
8 Simplify expressions with exponents.
Write the power in the denominator as a product.
Simplify the power in the denominator.
Simplify. Evaluate each expression for the given value(s) of the variable(s).
3a. p -3 for p = 4
3b. 8a -2b 0 for a = -2 and b = 6
7-1 Integer Exponents 461 What if you have an expression with a negative exponent in a denominator,
1
such as ___
?
-8
x 1 , or _
1 = x -n
x -n = _
xn
xn
1 = x -(-8)
_
x -8 Definition of negative exponent
Substitute -8 for n. = x8 Simplify the exponent on the right side. So if a base with a negative exponent is in a denominator, it is equivalent to the
same base with the opposite (positive) exponent in the numerator.
An expression that contains negative or zero exponents is not considered to be
simplified. Expressions should be rewritten with only positive exponents. EXAMPLE 4 Simplifying Expressions with Zero and Negative Exponents
Simplify.
-4
B _
-4 A 3y -2
3y -2 =3·y k
-4 = -4 · 1
_
k -4
k -4 _ -2 _ = 3 · 12
y
3
= _2
y = -4 · k 4
= -4k 4 x -3
C _
0 5 a y
x -3 = _
1
_
0 5
3
a y
x · 1 · y5
1
=_
x 3y 5 1.
a 0 = 1 and x -3 = _
x3 Simplify.
r -3
4b. _
7 4a. 2r 0m -3 g4
4c. _
h -6 THINK AND DISCUSS
-3 s =_
1 , ? -2 = _
2,_
1
1. Complete each equation: 2b ? = _
b2 k ?
s3
t2
2. GET ORGANIZED Copy and complete the graphic organizer. In each
box, describe how to simplify, and give an example.
-«vÞ}Ê
Ý«ÀiÃÃÃÊÜÌ
Ê
i}>ÌÛiÊ
Ý«iÌÃ
ÀÊ>Êi}>ÌÛiÊiÝ«iÌÊ
ÊÌ
iÊÕiÀ>ÌÀÊ°Ê°Ê° 462 Chapter 7 Exponents and Polynomials ÀÊ>Êi}>ÌÛiÊiÝ«iÌÊ
ÊÌ
iÊ`i>ÌÀÊ°Ê°Ê° 7-1 Exercises KEYWORD: MA7 7-1
KEYWORD: MA7 Parent GUIDED PRACTICE
SEE EXAMPLE 1 1. Medicine A typical virus is about 10 -7 m in size. Simplify this expression. p. 461 SEE EXAMPLE 2 p. 461 SEE EXAMPLE 3 p. 461 SEE EXAMPLE 4
p. 462 Simplify.
2. 6 -2 3. 3 0 4. -5 -2 7. -8 -3 8. 10 -2 9. (4.2)0 5. 3 -3 6. 1 -8 10. (-3)-3 11. 4-2 Evaluate each expression for the given value(s) of the variable(s).
12. b -2 for b = -3 13. (2t)-4 for t = 2 14. (m - 4)-5 for m = 6 15. 2x 0y -3 for x = 7 and y = -4 Simplify.
16. 4m 0 17. 3k -4 7
18. _
r -7 x 10
19. _
d -3 20. 2x 0y -4 f -4
21. _
g -6 c4
22. _
d -3 23. p 7q -1 PRACTICE AND PROBLEM SOLVING
Independent Practice
For
See
Exercises Example 24
25–36
37–42
43–57 1
2
3
4 Extra Practice
Skills Practice p. S16
Application Practice p. S34 24. Biology One of the smallest bats is the northern blossom bat,
which is found from Southeast Asia to Australia. This bat weighs
about 2 -1 ounce. Simplify this expression.
Simplify.
25. 8 0 26. 5 -4 27. 3 -4 29. -6 -2 30. 7 -2 31. 33. (-3)-1 34. (-4)2 (_25 )
1
35. (_
2) 0 -2 28. -9 -2
32. 13 -2
36. -7 -1 Evaluate each expression for the given value(s) of the variable(s). (_23 v) -3 37. x -4 for x = 4 38. 0
39. (10 - d) for d = 11 40. 10m -1n -5 for m = 10 and n = -2 -2
1 and b = 8
41. (3ab) for a = _
2 42. 4w vx v for w = 3, v = 0, and x = -5 for v = 9 Simplify.
43. k -4 44. 2z -8 1
45. _
2b -3 46. c -2d 47. -5x -3 48. 4x -6y -2 2f 0
49. _
7g -10 r -5
50. _
s -1 s5
51. _
-12
t 3w -5
52. _
x -6 53. b 0c 0 2 m -1n 5
54. _
3 q -2r 0
55. _
s0 a -7b 2
56. _
c 3d -4 3 -1
k
_
57. h
6m 2 7-1 Integer Exponents 463 Evaluate each expression for x = 3, y = -1, and z = 2.
58. z -5
62.
66. 59. (x + y)-4 61. (xyz)-1 60. (yz) 0 63. x -y
64. (yz) -x
65. xy -4
(xy - 3)-2
/////ERROR ANALYSIS///// Look at the two equations below. Which is incorrect?
Explain the error.
.q , *
XXXXXXX
.q , .
.q , XXXXX
q, Simplify. Biology (a 2b -7) 0 67. a 3b -2 68. c -4d 3 69. v 0w 2y -1 70. 2a -5
72. _
b -6 2a 3
73. _
b -1 m2
74. _
n -3 x -8
75. _
3y 12 77. Biology Human blood contains red blood
cells, white blood cells, and platelets. The table
shows the sizes of these components. Simplify
each expression.
Tell whether each statement is sometimes,
always, or never true.
When bleeding occurs,
platelets (which
appear green in the
image above) help to
form a clot to reduce
blood loss. Calcium
and vitamin K are
also necessary for clot
formation. 71. -5y -6
20p -1
76. - _
5q -3
Blood Components
Part Size (m)
125,000 -1 Red blood cell
White blood cell 3(500)-2 Platelet 3(1000)-2 1
.
78. If n is a positive integer, then x -n = __
xn 79. If x is positive, then x -n is negative.
80. If n is zero, then x -n is 1.
81. If n is a negative integer, then x -n = 1.
82. If x is zero, then x -n is 1.
83. If n is an integer, then x -n > 1.
84. Critical Thinking Find the value of 2 3 · 2 -3. Then find the value of 3 2 · 3 -2. Make a
conjecture about the value of a n · a -n.
1
85. Write About It Explain in your own words why 2 -3 is the same as __
.
3
2 Find the missing value.
1 =2
86. _
4 1
87. 9 -2 = _ 1
90. 7 -2 = _ 91. 10 1
=_
1000 1 =
88. _
64 -2 3
92. 3 · 4 -2 = _ 89. _ = 3 -1
3
1 =2·5
93. 2 · _
5 94. This problem will prepare you for the Multi-Step Test Prep on page 494.
a. The product of the frequency f and the wavelength w of light in air is a constant v.
Write an equation for this relationship.
b. Solve this equation for wavelength. Then write this equation as an equation with
f raised to a negative exponent.
c. The units for frequency are hertz (Hz). One hertz is one cycle per second, which
is often written as __1s . Rewrite this expression using a negative exponent. 464 Chapter 7 Exponents and Polynomials 95. Which is NOT equivalent to the other three?
1
_
25 5 -2 0.04 -25 (-6)(-6) 1
-_
6·6 1
_
6·6 a 3b 2
_
-c a3
_
-b 2c c
_
3 2
ab 96. Which is equal to 6 -2?
6 (-2)
a 3b -2 .
97. Simplify _
c -1
3
ac
_
2
b 98. Gridded Response Simplify ⎡⎣2 -2 + (6 + 2)0⎤⎦.
99. Short Response If a and b are real numbers and n is a positive integer, write a
simplified expression for the product a -n · b 0 that contains only positive exponents.
Explain your answer. CHALLENGE AND EXTEND
100. Multi-Step Copy and complete the table of values below. Then graph the ordered
pairs and describe the shape of the graph.
-4 x
y=2 -3 -2 -1 0 1 2 3 4 x 101. Multi-Step Copy and complete the table. Then write a rule for the values of 1 n and
(-1)n when n is any negative integer.
n -1 -2 -3 -4 -5 1n (-1)n SPIRAL REVIEW
Solve each equation. (Lesson 2-3)
102. 6x - 4 = 8 103. -9 = 3 (p - 1) y
104. _ - 8 = -12
5 105. 1.5h - 5 = 1 106. 2w + 6 - 3w = -10 1n+2-n
107. -12 = _
2 Identify the independent and dependent variables. Write a rule in function notation
for each situation. (Lesson 4-3)
108. Pink roses cost $1.50 per stem.
109. For dog-sitting, Beth charges a $30 flat fee plus $10 a day.
Write the equation that describes each line in slope-intercept form. (Lesson 5-7)
110. slope = 3, y-intercept = -4 111. slope = __13 , y-intercept = 5 112. slope = 0, y-intercept = __23 113. slope = -4, the point (1, 5) is on the line.
7-1 Integer Exponents 465 7-2 Powers of 10 and
Scientific Notation Objectives
Evaluate and multiply by
powers of 10. Why learn this?
Powers of 10 can be used to read and
write very large and very small numbers,
such as the masses of atomic particles.
(See Exercise 44.) Convert between
standard notation and
scientific notation.
Vocabulary
scientific notation The table shows relationships between
several powers of 10.
÷ 10 ÷ 10
2 Power 10 3 10 Value 1000 100 ÷ 10 Nucleus of a silicon atom ÷ 10 ÷ 10 ÷ 10 10 1
10 0 10 -1 10 -2 10 -3 10 1 1 = 0.1
_
10 1 = 0.01
_
100 1 = 0.001
_
1000 × 10
× 10
× 10
× 10
× 10
× 10
• Each time you divide by 10, the exponent decreases by 1 and the decimal
point moves one place to the left.
• Each time you multiply by 10, the exponent increases by 1 and the decimal
point moves one place to the right.
Powers of 10
WORDS NUMBERS Positive Integer Exponent
10 4 = 1 0, 0 0 0 ⎧
⎨
⎩ If n is a positive integer, find the value
of 10 n by starting with 1 and moving the
decimal point n places to the right. 4 places Negative Integer Exponent EXAMPLE 1 1 = 0.0 0 0 0 0 1
10 -6 = _
10 6 ⎧
⎨
⎩ If n is a positive integer, find the value of
10 -n by starting with 1 and moving the
decimal point n places to the left. 6 places Evaluating Powers of 10
Find the value of each power of 10. A 10 -3
You may need to add
zeros to the right or
left of a number in
order to move the
decimal point in that
direction.
466 B 10 2 C 10 0 Start with 1 and
move the decimal
point three places
to the left. Start with 1 and
move the decimal
point two places
to the right. Start with 1 and
move the decimal
point zero places. 0. 0 0 1 1 0 0 1 0.001 100 Chapter 7 Exponents and Polynomials Find the value of each power of 10.
1a. 10 -2
1b. 10 5 EXAMPLE 2 1c. 10 10 Writing Powers of 10
Write each number as a power of 10. A 10,000,000
If you do not see
a decimal point
in a number, it is
understood to be
at the end of the
number. B 0.001 C 10 The decimal point is
seven places to the
right of 1, so the
exponent is 7. The decimal point
is three places to
the left of 1, so the
exponent is -3. The decimal point
is one place to the
right of 1, so the
exponent is 1. 10 7 10 -3 10 1 Write each number as a power of 10.
2a. 100,000,000
2b. 0.0001 2c. 0.1 You can also move the decimal point to find the product of any number and a
power of 10. You start with the number instead of starting with 1.
Multiplying by Powers of 10
125 × 10 5 = 12,5 0 0, 0 0 0 ⎧
⎨
⎩ If the exponent is a positive integer,
move the decimal point to the right. 5 places 36.2 × 10 = 0.0 3 6 2 ⎧
⎨
⎩ If the exponent is a negative integer,
move the decimal point to the left. -3 3 places EXAMPLE 3 Multiplying by Powers of 10
Find the value of each expression. A 97.86 × 10 6
97.8 6 0 0 0 0 Move the decimal point 6 places to the right. 97,860,000 B 19.5 × 10 -4
0 0 1 9.5 Move the decimal point 4 places to the left. 0.00195
Find the value of each expression.
3a. 853.4 × 10 5
3b. 0.163 × 10 -2
Scientific notation is a method of writing numbers that are very large or very
small. A number written in scientific notation has two parts that are multiplied.
The first part is a number that is greater than or equal to 1 and less than 10. The second part is a power of 10.
7- 2 Powers of 10 and Scientific Notation 467 EXAMPLE 4 Astronomy Application
Jupiter has a diameter of about
143,000 km. Its shortest distance
from Earth is about 5.91 × 10 8 km,
and its average distance from the
Sun is about 778,400,000 km.
Jupiter’s orbital speed is
approximately 1.3 × 10 4 m/s. £{Î]äääÊ A Write Jupiter’s shortest distance
from Earth in standard form.
5.91 × 10 8
5.9 1 0 0 0 0 0 0
Move the decimal point 8 places to the right. Standard form
refers to the usual
way that numbers
are written. 591,000,000 km B Write Jupiter’s average distance from the Sun in scientific notation. ⎩
⎨
⎧ 778,400,000
7 7 8, 4 0 0, 0 0 0
8 places 7.784 × 10 8 km Count the number of places you need to move
the decimal point to get a number between 1
and 10.
Use that number as the exponent of 10. 4a. Use the information above to write Jupiter’s diameter in
scientific notation.
4b. Use the information above to write Jupiter’s orbital speed
in standard form. EXAMPLE 5 Comparing and Ordering Numbers in Scientific Notation
Order the list of numbers from least to greatest.
1.2 × 10 -1, 8.2 × 10 4, 6.2 × 10 5, 2.4 × 10 5, 1 × 10 -1, 9.9 × 10 -4
Step 1 List the numbers in order by powers of 10.
9.9 × 10 -4, 1.2 × 10 -1, 1 × 10 -1, 8.2 × 10 4, 6.2 × 10 5, 2.4 × 10 5
Step 2 Order the numbers that have the same power of 10.
9.9 × 10 -4, 1 × 10 -1, 1.2 × 10 -1, 8.2 × 10 4, 2.4 × 10 5, 6.2 × 10 5
5. Order the list of numbers from least to greatest.
5.2 × 10 -3, 3 × 10 14, 4 × 10 -3, 2 × 10 -12, 4.5 × 10 30, 4...

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- Fall '14
- Decimal, Orders of magnitude, Exponentiation