Algebra and Trigonometry (3rd Edition)

1058 Pages·2007·22.34 MB·English

by  Judith A. Beecher

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Basic Concepts
R
of Algebra
R.1 The Real-Number System
R.2 Integer Exponents,Scientific Notation,
and Order of Operations
R.3 Addition,Subtraction,and
Multiplication of Polynomials
R.4 Factoring
R.5 Rational Expressions
R.6 Radical Notation and Rational Exponents
R.7 The Basics of Equation Solving
SUMMARYANDREVIEW
TEST
A P P L I C A T I O N
G
ina wants to establish a college fund for
her newborn daughter that will have
accumulated $120,000 at the end of
18 yr.Ifshe can count on an interest rate of6%,
compounded monthly,how much should she deposit
each month to accomplish this?
This problem appears asExercise 95 in Section R.2.
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Chapter R • Basic Concepts of Algebra
2..11
R Identify various kinds ofreal numbers.
Use interval notation to write a set ofnumbers.
Identify the properties ofreal numbers.
Find the absolute value ofa real number.
The RePaol-lyNnuommbiearl
FunctioSnyss taenmd
Modeling Real Numbers
In applications of algebraic concepts, we use real numbers to represent
quantities such as distance, time, speed, area, profit, loss, and tempera-
ture. Some frequently used sets of real numbers and the relationships
among them are shown below.
Natural numbers
(positive integers):
1, 2, 3, …
Whole numbers:
0, 1, 2, 3, …
Integers:
…, −3, −2, −1, 0, Zero: 0
1, 2, 3, …
Rational Negative integers:
numbers −1, −2, −3, …
Rational numbers
that are not integers:
Real
numbers −23, − −45, −−1−95, −−8−7, 8.3,
−
Irrational numbers: 0.56, …
√2, p, −5√3, 4√27,
−4.030030003…, …
Numbers that can be expressed in the form p(cid:1)q,where pand qare in-
tegers and q (cid:2) 0, are rational numbers. Decimal notation for rational
numbers either terminates (ends) or repeats. Each of the following is a
rational number.
0
a) 0 0 (cid:2) for any nonzero integer a
a
b)(cid:1)7 (cid:1)7 (cid:2) (cid:1)7,or 7
1 (cid:1)1
1
c) (cid:1) 0.25 Terminating decimal
4
5
d)(cid:1) (cid:1) (cid:1)0.45 Repeating decimal
11
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Section R.1 • The Real-Number System
The real numbers that are not rational are irrational numbers.Decimal
notation for irrational numbers neither terminates nor repeats.Each of the
following is an irrational number.
a) (cid:7)(cid:1) 3.1415926535... There is no repeating block ofdigits.
(cid:7) (cid:3)
22and 3.14 are rational approximationsofthe irrational number (cid:7).
7
b)(cid:8)2 (cid:1) 1.414213562... There is no repeating block ofdigits.
c) (cid:1)6.12122122212222... Although there is a pattern,there is no
repeating block ofdigits.
The set of all rational numbers combined with the set of all irrational
numbers gives us the set of real numbers. The real numbers are modeled
using a number line,as shown below.
Each point on the line represents a real number,and every real number
is represented by a point on the line.
(cid:1)2.9 (cid:1)E (cid:8)3 p *
(cid:1)5 (cid:1)4 (cid:1)3 (cid:1)2 (cid:1)1 0 1 2 3 4 5
The order of the real numbers can be determined from the number
line. If a number a is to the left of a number b, then a is less than b
(cid:7)a (cid:4) b(cid:3).Similarly,a is greater than b (cid:7)a (cid:6) b(cid:3) if a is to the right of b on
the number line. For example, we see from the number line above that
(cid:1)2.9 (cid:4) (cid:1)3,because (cid:1)2.9 is to the left of (cid:1)3.Also,17 (cid:6) (cid:8)3,because 17
5 5 4 4
is to the right of (cid:8)3.
The statement a (cid:5) b,read “ais less than or equal to b,”is true if either
a (cid:4) bis true or a (cid:1) bis true.
The symbol (cid:3)is used to indicate that a member,or element,belongs to
a set.Thus ifwe let (cid:2)represent the set ofrational numbers,we can see from
the diagram on page 2 that 0.56 (cid:3) (cid:2).We can also write (cid:8)2 (cid:5)(cid:2) to indi-
cate that (cid:8)2is notan element ofthe set ofrational numbers.
When allthe elements ofone set are elements ofa second set,we say that
(cid:4)
the first set is a subsetofthe second set.The symbol is used to denote this.
For instance,if we let (cid:1)represent the set of real numbers,we can see from
the diagram that (cid:2) (cid:4) (cid:1)(read “(cid:2)is a subset of(cid:1)”).
Interval Notation
Sets ofreal numbers can be expressed using interval notation.For example,
for real numbers aand bsuch that a (cid:4) b,the open interval(cid:7)a,b(cid:3)is the set
ofreal numbers between,but not including,aand b.That is,
( ) (cid:7)a,b(cid:3) (cid:1) (cid:4)x(cid:5)a (cid:4) x (cid:4) b(cid:6).
a (a, b) b
The points a and b are endpoints of the interval.The parentheses indicate
that the endpoints are not included in the interval.
Some intervals extend without bound in one or both directions. The
interval (cid:2)a,(cid:2)(cid:3), for example, begins at a and extends to the right without
bound.That is,
[ (cid:2)a,(cid:2)(cid:3) (cid:1) (cid:4)x(cid:5)x (cid:3) a(cid:6).
a [a, ∞)
The bracket indicates that ais included in the interval.
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Chapter R • Basic Concepts of Algebra
The various types ofintervals are listed below.
Intervals:Types, Notation, and Graphs
INTERVAL SET
TYPE NOTATION NOTATION GRAPH
Open (cid:7)a,b(cid:3) (cid:4)x(cid:5)a(cid:4)x(cid:4)b(cid:6) ( )
a b
Closed (cid:2)a,b(cid:9) (cid:4)x(cid:5)a(cid:5)x(cid:5)b(cid:6) [ ]
a b
Half-open (cid:2)a,b(cid:3) (cid:4)x(cid:5)a(cid:5)x(cid:4)b(cid:6) [ )
a b
Half-open (cid:7)a,b(cid:9) (cid:4)x(cid:5)a(cid:4)x(cid:5)b(cid:6) ( ]
a b
Open (cid:7)a,(cid:2)(cid:3) (cid:4)x(cid:5)x(cid:6)a(cid:6) (
a
Half-open (cid:2)a,(cid:2)(cid:3) (cid:4)x(cid:5)x(cid:3)a(cid:6) [
a
Open (cid:7)(cid:1)(cid:2),b(cid:3) (cid:4)x(cid:5)x(cid:4)b(cid:6) )
b
Half-open (cid:7)(cid:1)(cid:2),b(cid:9) (cid:4)x(cid:5)x(cid:5)b(cid:6) ]
b
The interval (cid:7)(cid:1)(cid:2),(cid:2)(cid:3),graphed below,names the set of all real num-
bers,(cid:1).
EXAMPLE 1 Write interval notation for each set and graph the set.
a) (cid:4)x(cid:5)(cid:1)4 (cid:4) x (cid:4) 5(cid:6) b) (cid:4)x(cid:5)x (cid:3) 1.7(cid:6)
(cid:4) (cid:6)
c) (cid:4)x(cid:5)(cid:1)5 (cid:4) x (cid:5) (cid:1)2(cid:6) d) x(cid:5)x (cid:4) (cid:8)5
Solution
a) (cid:4)x(cid:5)(cid:1)4 (cid:4) x (cid:4) 5(cid:6) (cid:1) (cid:7)(cid:1)4,5(cid:3);
(cid:1)5(cid:1)4(cid:1)3(cid:1)2(cid:1)1 0 1 2 3 4 5
b) (cid:4)x(cid:5)x (cid:3) 1.7(cid:6) (cid:1) (cid:2)1.7,(cid:2)(cid:3);
(cid:1)5(cid:1)4(cid:1)3(cid:1)2(cid:1)1 0 1 2 3 4 5
c) (cid:4)x(cid:5)(cid:1)5 (cid:4) x (cid:5) (cid:1)2(cid:6) (cid:1) (cid:7)(cid:1)5,(cid:1)2(cid:9);
(cid:1)5(cid:1)4(cid:1)3(cid:1)2(cid:1)1 0 1 2 3 4 5
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Section R.1 • The Real-Number System
d) (cid:4)x(cid:5)x (cid:4) (cid:8)5(cid:6) (cid:1) (cid:7)(cid:1)(cid:2),(cid:8)5(cid:3);
(cid:1)5(cid:1)4(cid:1)3(cid:1)2(cid:1)1 0 1 2 3 4 5
Properties of the Real Numbers
The following properties can be used to manipulate algebraic expressions as
well as real numbers.
Properties of the Real Numbers
For any real numbers a,b,and c:
a (cid:8) b (cid:1) b (cid:8) aand Commutative properties of
ab (cid:1) ba addition and multiplication
a (cid:8) (cid:7)b (cid:8) c(cid:3) (cid:1) (cid:7)a (cid:8) b(cid:3) (cid:8) c and Associative properties of
a(cid:7)bc(cid:3) (cid:1) (cid:7)ab(cid:3)c addition and multiplication
a (cid:8) 0 (cid:1) 0 (cid:8) a (cid:1) a Additive identity property
(cid:1)a (cid:8) a (cid:1) a (cid:8) (cid:7)(cid:1)a(cid:3) (cid:1) 0 Additive inverse property
a (cid:9) 1 (cid:1) 1 (cid:9) a (cid:1) a Multiplicative identity property
1 1
a (cid:9) (cid:1) (cid:9) a (cid:1) 1 (cid:7)a (cid:2) 0(cid:3) Multiplicative inverse property
a a
a(cid:7)b (cid:8) c(cid:3) (cid:1) ab (cid:8) ac Distributive property
Note that the distributive property is also true for subtraction since
a(cid:7)b (cid:1) c(cid:3) (cid:1) a(cid:2)b (cid:8) (cid:7)(cid:1)c(cid:3)(cid:9) (cid:1) ab (cid:8) a(cid:7)(cid:1)c(cid:3) (cid:1) ab (cid:1) ac.
EXAMPLE 2 State the property being illustrated in each sentence.
a) 8 (cid:9) 5 (cid:1) 5 (cid:9) 8 b) 5 (cid:8) (cid:7)m (cid:8) n(cid:3) (cid:1) (cid:7)5 (cid:8) m(cid:3) (cid:8) n
c) 14 (cid:8) (cid:7)(cid:1)14(cid:3) (cid:1) 0 d) 6 (cid:9) 1 (cid:1) 1 (cid:9) 6 (cid:1) 6
e) 2(cid:7)a (cid:1) b(cid:3) (cid:1) 2a (cid:1) 2b
Solution
SENTENCE PROPERTY
a) 8 (cid:9) 5 (cid:1) 5 (cid:9) 8 Commutative property ofmultiplication:
ab (cid:1) ba
b) 5 (cid:8) (cid:7)m (cid:8) n(cid:3) (cid:1) (cid:7)5 (cid:8) m(cid:3) (cid:8) n Associative property ofaddition:
a (cid:8) (cid:7)b (cid:8) c(cid:3) (cid:1) (cid:7)a (cid:8) b(cid:3) (cid:8) c
c) 14 (cid:8) (cid:7)(cid:1)14(cid:3) (cid:1) 0 Additive inverse property: a(cid:8)(cid:7)(cid:1)a(cid:3)(cid:1)0
d) 6 (cid:9) 1 (cid:1) 1 (cid:9) 6 (cid:1) 6 Multiplicative identity property:
a (cid:9) 1 (cid:1) 1 (cid:9) a (cid:1) a
e) 2(cid:7)a (cid:1) b(cid:3) (cid:1) 2a (cid:1) 2b Distributive property:
a(cid:7)b (cid:8) c(cid:3) (cid:1) ab (cid:8) ac
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Chapter R • Basic Concepts of Algebra
Absolute Value
The number line can be used to provide a geometric interpretation of
absolute value. The absolute value of a number a, denoted (cid:5)a(cid:5), is its dis-
tance from 0 on the number line. For(cid:11) (cid:11)example, (cid:5)(cid:1)5(cid:5) (cid:1) 5, because the
distance of (cid:1)5 from 0 is 5. Similarly, 3 (cid:1) 3, because the distance of 3
4 4 4
from 0 is 3.
4
Absolute Value
For any real number a,
(cid:10)
a, if a (cid:3) 0,
(cid:5)a(cid:5) (cid:1)
(cid:1)a, if a (cid:4) 0.
When ais nonnegative,the absolute value ofais a.When ais negative,
the absolute value ofais the opposite,or additive inverse,ofa.Thus,
(cid:5)a(cid:5)is never negative;that is,for any real number a,(cid:5)a(cid:5) (cid:3) 0.
Absolute value can be used to find the distance between two points on
the number line.
Distance Between Two Points on the Number Line
a b For any real numbers aand b,the distance between aand bis
(cid:5)a (cid:1) b(cid:5) (cid:10) (cid:5)b (cid:1) a(cid:5) (cid:5)a (cid:1) b(cid:5),or equivalently,(cid:5)b (cid:1) a(cid:5).
GCM EXAMPLE 3 Find the distance between (cid:1)2 and 3.
Solution The distance is
(cid:5)(cid:1)2 (cid:1) 3(cid:5) (cid:1) (cid:5)(cid:1)5(cid:5) (cid:1) 5, or equivalently,
(cid:5)3 (cid:1) (cid:7)(cid:1)2(cid:3)(cid:5) (cid:1) (cid:5)3 (cid:8) 2(cid:5) (cid:1) (cid:5)5(cid:5) (cid:1) 5.
We can also use the absolute-value operation on a graphing calculator to
find the distance between two points. On many graphing calculators, ab-
solute value is denoted “abs”and is found in the MATH NUMmenu and also
in the CATALOG.
abs ((cid:1)2(cid:1)3)
5
abs (3(cid:1)((cid:1)2))
5
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Section R.1 • The Real-Number System
.1
R
Exercise Set
In Exercises 1–10,consider the numbers (cid:1)12,(cid:8)7,5.3, 25.(cid:2)x,x (cid:8)h(cid:9)
(cid:1)7,(cid:8)3 8,0,5.242242224...,(cid:1)(cid:8)14,(cid:8)5 5,(cid:1)1.96,9, [ ]
3 x x (cid:8) h
42,(cid:8)25,(cid:8)3 4,5.
3 7
1.Which are whole numbers? (cid:8)3 8,0,9,(cid:8)25 26.(cid:7)x,x (cid:8)h(cid:9)
2.Which are integers? (cid:1)12,(cid:8)3 8,0,9,(cid:8)25 ( ]
(cid:8)7,5.242242224..., x x (cid:8) h
3.Which are irrational numbers? (cid:1)(cid:8)14,(cid:8)5 5,(cid:8)3 4
4.Which are natural numbers? (cid:8)3 8,9,(cid:8)25 27.(cid:7)p,(cid:2)(cid:3)
(cid:1)12,5.3,(cid:1)7,(cid:8)3 8,0, (
3
5.Which are rational numbers? (cid:1)1.96,9,42,(cid:8)25,5 p
3 7
6.Which are real numbers? All of them 5.3,(cid:1)7,(cid:1)1.96, 28.(cid:7)(cid:1)(cid:2),q(cid:9)
3
7.Which are rational numbers but not integers? 423,57 ]
8.Which are integers but not whole numbers? (cid:1)12 q
9.Which are integers but not natural numbers? (cid:1)12,0 In Exercises 29–46,the following notation is used:
(cid:6) (cid:1) the set ofnatural numbers,(cid:7) (cid:1) the set ofwhole
10.Which are real numbers but not integers? (cid:3) numbers,(cid:5) (cid:1) the set ofintegers,(cid:2) (cid:1) the set of
Write interval notation.Then graph the interval. rational numbers,(cid:4) (cid:1) the set ofirrational numbers,and
11.(cid:4)x(cid:5)(cid:1)3(cid:5)x(cid:5)3(cid:6) (cid:3) 12.(cid:4)x(cid:5)(cid:1)4(cid:4)x(cid:4)4(cid:6) (cid:3) (cid:1) (cid:1) the set ofreal numbers.Classify the statement as
true or false.
13.(cid:4)x(cid:5)(cid:1)4(cid:5)x(cid:4)(cid:1)1(cid:6) (cid:3) 14.(cid:4)x(cid:5)1(cid:4)x(cid:5)6(cid:6) (cid:3) 29.6(cid:3)(cid:6) True 30.0(cid:5)(cid:6) True
15.(cid:4)x(cid:5)x(cid:5)(cid:1)2(cid:6) (cid:3) 16.(cid:4)x(cid:5)x(cid:6)(cid:1)5(cid:6) (cid:3) 31.3.2(cid:3)(cid:5) False 32.(cid:1)10.1(cid:3)(cid:1) True
(cid:4) (cid:6)
17.(cid:4)x(cid:5)x(cid:6)3.8(cid:6) (cid:3) 18. x(cid:5)x(cid:3)(cid:8)3 (cid:3)
11
33.(cid:1) (cid:3)(cid:2) True 34.(cid:1)(cid:8)6(cid:3)(cid:2) False
19.(cid:4)x(cid:5)7(cid:4)x(cid:6) (cid:3) 20.(cid:4)x(cid:5)(cid:1)3(cid:6)x(cid:6) (cid:3) 5
Write interval notation for the graph. 35.(cid:8)11(cid:5)(cid:1) False 36.(cid:1)1(cid:3)(cid:7) False
21.(cid:7)0,5(cid:3) 37.24(cid:5)(cid:7) False 38.1(cid:3)(cid:5) True
( )
(cid:1)6 (cid:1)5 (cid:1)4 (cid:1)3 (cid:1)2 (cid:1)1 0 1 2 3 4 5 6 39.1.089(cid:5)(cid:4) True 40.(cid:6)(cid:4)(cid:7) True
41.(cid:7) (cid:4)(cid:5) True 42.(cid:5)(cid:4)(cid:6) False
22.(cid:2)(cid:1)1,2(cid:9)
[ ] 43.(cid:2)(cid:4)(cid:1) True 44.(cid:5)(cid:4)(cid:2) True
(cid:1)6 (cid:1)5 (cid:1)4 (cid:1)3 (cid:1)2 (cid:1)1 0 1 2 3 4 5 6 45.(cid:1)(cid:4)(cid:5) False 46.(cid:2)(cid:4)(cid:4) False
23.(cid:2)(cid:1)9,(cid:1)4(cid:3) Name the property illustrated by the sentence.
[ ) 47.6(cid:9)x(cid:1)x(cid:9)6 Commutative property of
(cid:1)10 (cid:1)9 (cid:1)8 (cid:1)7 (cid:1)6 (cid:1)5 (cid:1)4 (cid:1)3 (cid:1)2 (cid:1)1 0 1 2 multiplication
48.3(cid:8)(cid:7)x(cid:8)y(cid:3)(cid:1)(cid:7)3(cid:8)x(cid:3)(cid:8)y Associative property
24.(cid:7)(cid:1)9,(cid:1)5(cid:9) ofaddition
49.(cid:1)3(cid:9)1(cid:1)(cid:1)3 50.x(cid:8)4(cid:1)4(cid:8)x (cid:3)
( ] Multiplicative identity property
(cid:1)10 (cid:1)9 (cid:1)8 (cid:1)7 (cid:1)6 (cid:1)5 (cid:1)4 (cid:1)3 (cid:1)2 (cid:1)1 0 1 2 51.5(cid:7)ab(cid:3)(cid:1)(cid:7)5a(cid:3)b (cid:3) 52.4(cid:7)y(cid:1)z(cid:3)(cid:1)4y(cid:1)4z
Distributive property
(cid:3)
Answers to Exercises 10–20,50,and 51 can be found on p.IA-1.
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Chapter R • Basic Concepts of Algebra
Additive inverse property
53.2(cid:7)a(cid:8)b(cid:3)(cid:1)(cid:7)a(cid:8)b(cid:3)2 54.(cid:1)7(cid:8)7(cid:1)0 not appear at the back ofthe book.They are denoted
Commutative property ofmultiplication
by the words “Discussion and Writing.”
55.(cid:1)6(cid:7)m(cid:8)n(cid:3)(cid:1)(cid:1)6(cid:7)n(cid:8)m(cid:3) Commutative property
ofaddition 79.How would you convince a classmate that division is
56.t(cid:8)0(cid:1)t Additive identity property not associative?
57.8(cid:9) 1 (cid:1)1 Multiplicative inverse property 80.Under what circumstances is (cid:8)aa rational number?
8
58.9x(cid:8)9y(cid:1)9(cid:7)x(cid:8)y(cid:3) Distributive property Synthesis
To the student and the instructor: The Synthesis
Simplify.
exercises found at the end ofevery exercise set challenge
59.(cid:5)(cid:1)7.1(cid:5) 7.1 60.(cid:5)(cid:1)86.2(cid:5) 86.2
students to combine concepts or skills studied in that
61.(cid:5)347(cid:5) 347 62.(cid:5)(cid:1)54(cid:5) 54 section or in preceding parts ofthe text.
(cid:5) (cid:5)
(cid:11) (cid:11) 12 12 Between any two (different) real numbers there are
63. (cid:1)(cid:8)97 (cid:8)97 64.
19 19 many other real numbers.Find each ofthe following.
Answers may vary.
65.(cid:5)0(cid:5) 0 66.(cid:5)15(cid:5) 15
81.An irrational number between 0.124 and 0.125
(cid:5) (cid:5)
5 5 (cid:11) (cid:11) Answers may vary;0.124124412444...
67. 68. (cid:1)(cid:8)3 (cid:8)3 82.A rational number between (cid:1)(cid:8)2.01and (cid:1)(cid:8)2
4 4
Answers may vary;(cid:1)1.415
1 1
Find the distance between the given pair ofpoints on 83.A rational number between (cid:1) and (cid:1)
the number line. Answers may vary;(cid:1)0.00999 101 100
69.(cid:1)5, 6 11 70.(cid:1)2.5, 0 2.5 84.An irrational number between (cid:8)5.99and (cid:8)6
Answers may vary;(cid:8)5.995
71.(cid:1)8, (cid:1)2 6 72.15, 23 1 85.The hypotenuse ofan isosceles right triangle with
8 12 24 legs oflength 1 unit can be used to “measure”a
73.6.7, 12.1 5.4 74.(cid:1)14, (cid:1)3 11 value for (cid:8)2by using the Pythagorean theorem,
as shown.
3 15 21
75.(cid:1) , 76.(cid:1)3.4, 10.2 13.6
4 8 8
77.(cid:1)7, 0 7 78.3, 19 16 c2(cid:1)12(cid:8)12
c 1 c2(cid:1)2
c(cid:1)(cid:8)2
Collaborative Discussion and Writing
To the student and the instructor: The Collaborative 1
Discussion and Writing exercises are meant to be
answered with one or more sentences.These exercises Draw a right triangle that could be used to
can also be discussed and answered collaboratively by “measure”(cid:8)10units. (cid:3)
the entire class or by small groups.Because oftheir
open-ended nature,the answers to these exercises do
(cid:3)
Answer to Exercise 85 can be found on p.IA-1.
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9
Section R.2 • Integer Exponents,Scientific Notation,and Order of Operations
.2
R Simplify expressions with integer exponents.
Solve problems using scientific notation.
Use the rules for order ofoperations.
Integer
Integers as Exponents
Exponents,
When a positive integer is used as an exponent, it indicates the number of
Scientific
times a factor appears in a product.For example,73means 7 (cid:9) 7 (cid:9) 7and 51
Notation, and means 5.
Order of
Operations For any positive integer n,
an (cid:1) a (cid:9) a (cid:9) a (cid:9) (cid:9) (cid:9) a,
nfactors
where ais the baseand nis the exponent.
Zero and negative-integer exponents are defined as follows.
For any nonzero real number aand any integer m,
1
a0 (cid:1) 1 and a(cid:1)m (cid:1) .
am
EXAMPLE 1 Simplify each ofthe following.
a) 60 b) (cid:7)(cid:1)3.4(cid:3)0
Solution
a) 60 (cid:1) 1 b) (cid:7)(cid:1)3.4(cid:3)0 (cid:1) 1
EXAMPLE 2 Write each ofthe following with positive exponents.
1 x(cid:1)3
a) 4(cid:1)5 b) c)
(cid:7)0.82(cid:3)(cid:1)7 y(cid:1)8
Solution
1
a) 4(cid:1)5 (cid:1)
45
1
b) (cid:1) (cid:7)0.82(cid:3)(cid:1)(cid:7)(cid:1)7(cid:3) (cid:1) (cid:7)0.82(cid:3)7
(cid:7)0.82(cid:3)(cid:1)7
x(cid:1)3 1 1 y8
c) (cid:1) x(cid:1)3 (cid:9) (cid:1) (cid:9) y8 (cid:1)
y(cid:1)8 y(cid:1)8 x3 x3
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Chapter R • Basic Concepts of Algebra
The results in Example 2 can be generalized as follows.
For any nonzero numbers aand band any integers mand n,
a(cid:1)m bn
(cid:1) .
b(cid:1)n am
(A factor can be moved to the other side ofthe fraction bar ifthe
sign ofthe exponent is changed.)
EXAMPLE 3 Write an equivalent expression without negative exponents:
x(cid:1)3y(cid:1)8
.
z(cid:1)10
Solution Since each exponent is negative,we move each factor to the other
side ofthe fraction bar and change the sign ofeach exponent:
x(cid:1)3y(cid:1)8 z10
(cid:1) .
z(cid:1)10 x3y8
The following properties of exponents can be used to simplify
expressions.
Properties of Exponents
For any real numbers aand band any integers mand n,assuming 0 is
not raised to a nonpositive power:
am (cid:9) an (cid:1) am(cid:8)n Product rule
am
(cid:1) am(cid:1)n (cid:7)a (cid:2) 0(cid:3) Quotient rule
an
(cid:7)am(cid:3)n (cid:1) amn Power rule
(cid:7)ab(cid:3)m (cid:1) ambm Raising a product to a power
(cid:12) (cid:13)
a m am
(cid:1) (cid:7)b (cid:2) 0(cid:3) Raising a quotient to a power
b bm
EXAMPLE 4 Simplify each ofthe following.
48x12
a) y(cid:1)5 (cid:9) y3 b)
16x4
c) (cid:7)t(cid:1)3(cid:3)5 d) (cid:7)2s(cid:1)2(cid:3)5
(cid:12) (cid:13)
45x(cid:1)4y2 (cid:1)3
e)
9z(cid:1)8