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APPENDIX
B
B.1 Review of Integer
Exponents
B.2 Review of nth Roots
B.3 Review of Rational
Exponents
B.4 Review of Factoring
B.5 Review of Fractional
Expressions
B.6 Properties of the Real
Numbers
B.1
REVIEW OF INTEGER EXPONENTS
I write a1, a2, a3, etc., for
1 1 1
, ,
, etc. —Isaac Newton (June 13, 1676)
a aa aaa
In basic algebra you learned the exponential notation an, where a is a real number and
n is a natural number. Because that definition is basic to all that follows in this and
the next two sections of this appendix, we repeat it here.
Definition 1 Base and Exponent
Given a real number a and a natural number n, we define an by
⎧
⎪
⎨
⎪
⎩
an a # a # a p a
n factors
n
In the expression a the number a is the base, and n is the exponent or power to
which the base is raised.
EXAMPLE
1 Using Algebraic Notation
Rewrite each expression using algebraic notation:
(a)
(b)
(c)
(d)
SOLUTION
x to the fourth power, plus five;
the fourth power of the quantity x plus five;
x plus y, to the fourth power;
x plus y to the fourth power.
(a) x 4 5
(b) (x 5)4
(c) (x y)4
(d) x y 4
Caution: Note that parts (c) and (d) of Example 1 differ only in the use of the comma.
So for spoken purposes the idea in (c) would be more clearly conveyed by saying
“the fourth power of the quantity x plus y.” In the case of (d), if you read it aloud to
another student or your instructor, chances are you’ll be asked, “Do you mean x y 4
or do you mean (x y)4?”
Moral: Algebra is a precise language; use it carefully. Learn to ask yourself whether
what you’ve written will be interpreted in the manner you intended.
B-1
B-2
Appendix B
In basic algebra the four properties in the following box are developed for working with exponents that are natural numbers. Each of these properties is a direct consequence of the definition of an. For instance, according to the first property, we have
a2a3 a5. To verify that this is indeed correct, we note that
a2a3 (aa)(aaa) a 5
PROPERTY SUMMARY
Properties of Exponents
Property
Examples
1. a a a
2. (a m)n amn
m n
a5a6 a11; (x 1)(x 1)2 (x 1)3
(23)4 212; [(x 1)2]3 (x 1)6
mn
amn
a
1
3. n d nm
a
a
1
if m n
m
1 a5
a2
a6
4
; 1
a
;
a2
a6 a4 a 5
if m n
if m n
4. (ab)m ambm;
a m am
a b m
b
b
(2x 2)3 23 # (x 2)3 8x 6; a
x2 4
x8
b
y3
y12
Of course, we assume that all of these expressions make sense; in particular, we assume that no denominator of a fraction is zero.
Now we want to extend our definition of an to allow for exponents that are integers but not necessarily natural numbers. We begin by defining a0.
Definition 2
Zero Exponent
For any nonzero real number a,
a0 1
(00 is not defined.)
EXAMPLES
(a) 20 1
(b) (p)0 1
0
3
(c) a
2
2b 1
1a b
It’s easy to see the motivation for defining a0 to be 1. Assuming that the exponent zero
is to have the same properties as exponents that are natural numbers, we can write
a0an a0n
That is,
a0an an
Now we divide both sides of this last equation by an to obtain a0 1, which agrees
with our definition.
Our next definition (see the box that follows) assigns a meaning to the expression
an when n is a natural number. Again, it’s easy to see the motivation for this definition.
We would like to have
anan an(n) a0 1
B.1 Review of Integer Exponents
That is,
B-3
anan 1
Now we divide both sides of this last equation by an to obtain an 1an, in agreement with Definition 3.
Definition 3
Negative Exponent
For any nonzero real number a
and natural number n,
1
an n
a
EXAMPLES
1
1
1 2
2
1 1
1
a b 1 1 10
10
1 10 2
1
x 2 2
x
1
1
(a2b)3 2 3 6 3
ab
(a b)
1
1
23 8
3 2
123
(a) 21 (b)
(c)
(d)
(e)
It can be shown that the four properties of exponents that we listed earlier continue to hold now for all integer exponents. We make use of this fact in the next three
examples.
EXAMPLE
2 Using Properties of Exponents
Simplify the following expression. Write the answer in such a way that only positive
exponents appear.
(a2b3)2(a 5b)1
SOLUTION
First Method
Alternative Method
1
a 5b
4 6
b61 b5
ab
(a2b3)2(a 5b)1 5 54 a
ab
a
(a2b3)2(a 5b)1 (a4b6)(a5b1)
(a2b3)2(a 5b)1 (a2b3)2 #
EXAMPLE
(a2b3)2(a 5b)1 a45b61 a1b5
b5
a
as obtained
previously
3 Using Properties of Exponents
Simplify the following expression, writing the answer so that negative exponents are
not used.
a
a5b2c0 3
b
a3b1
B-4
Appendix B
SOLUTION
We show two solutions. The first makes immediate use of the property (am)n amn.
In the second solution we begin by working within the parentheses.
First Solution
a5b2c0 3 a15b6
a 3 1 b 9 3
ab
ab
( a5b2c0 )3 EXAMPLE
Alternative Solution
a5b2c0 3
b3 3
a 3 1 b a 8 b
ab
a
b6(3)
b9
a9(15)
a24
( a5b2c0 )3 b9
a24
4 Simplifying with Variable Exponents
Simplify the following expressions, writing the answers so that negative exponents
are not used. (Assume that p and q are natural numbers.)
(a)
SOLUTION
EXAMPLE
a4pq
a pq
(b) (apbq)2(a3pbq)1
a4pq
a4pq(pq)
a pq
a4pqpq a3p2q
(b) (a pbq)2(a3pbq)1 (a2pb2q)(a3pbq)
apb3q
b3q
p
a
(a)
5 Simplifying Numerical Quantities Involving Exponents
Use the properties of exponents to compute the quantity
SOLUTION
210 # 313
.
27 # 612
210 # 313
210 # 313
3
12
27 # 6
(3 )(3 # 2)12
210 # 313
210 # 313
33 # 312 # 212
315 # 212
1
1
1
1513 1210 2 2 36
(3
)(2
)
3 #2
As an application of some of the ideas in this section, we briefly discuss scientific
notation, which is a convenient form for writing very large or very small numbers. Such
numbers occur often in the sciences. For instance, the speed of light in a vacuum is
29,979,000,000 cm/sec
As written, this number would be awkward to work with in calculating. In fact, the
number as written cannot even be displayed on some hand-held calculators, because
there are too many digits. To write the number 29,979,000,000 (or any positive number) in scientific notation, we express it as a number between 1 and 10, multiplied by
an appropriate power of 10. That is, we write it in the form
a 10n
where 1 a 10 and n is an integer.
B.1 Review of Integer Exponents
B-5
According to this convention, the number 4.03 106 is in scientific notation, but the
same quantity written as 40.3 105 is not in scientific notation. To convert a given
number into scientific notation, we’ll rely on the following two-step procedure.
To Express a Number Using Scientific Notation
1. First move the decimal point until it is to the immediate right of the first nonzero
digit.
2. Then multiply by 10 n or 10n, depending on whether the decimal point was
moved n places to the left or to the right, respectively.
For example, to express the number 29,979,000,000 in scientific notation, first
we move the decimal point 10 places to the left so that it’s located between the 2
and the 9, then we multiply by 1010. The result is
29,979,000,000 2.9979000000 1010
or, more simply,
29,979,000,000 2.9979 1010
As additional examples we list the following numbers expressed in both ordinary
and scientific notation. For practice, you should verify each conversion for yourself
using our two-step procedure.
55,708 5.5708 104
0.000099 9.9 105
0.0000002 2 107
All scientific calculators have keys for entering numbers in scientific notation.
(Indeed, many calculators will convert numbers into scientific notation for you.) If
you have questions about how your own calculator operates with respect to scientific
notation, you should consult the user’s manual. We’ll indicate only one example
here, using a Texas Instruments graphing calculator. The keystrokes on other brands
are quite similar. Consider the following number expressed in scientific notation:
3.159 1020. (This gargantuan number is the diameter, in miles, of a galaxy at the
center of a distant star cluster known as Abell 2029.) The sequence of keystrokes for
entering this number on the Texas Instruments calculator is
3.159
EXERCISE SET
In Exercises 1–4, evaluate each expression using the given
value of x.
3.
1
1 2x2
;x
2
1 2x3
20
B.1
A
1. 2x 3 x 4; x 2
EE
2. 1 x 2x 2 3x 3; x 1
4.
1 (x 1)2
1 (x 1)2
; x 1
In Exercises 5–16, use the properties of exponents to simplify
each expression.
5. (a)
(b)
(c)
7. (a)
(b)
(c)
a3a12
(a 1)3(a 1)12
(a 1)12(a 1)3
yy 2y 8
(y 1)(y 1)2(y 1)8
[(y 1)(y 1)8]2
6. (a) (32)3 (23)2
(b) (x 2)3 (x 3)2
(c) (x 2)a (x a )2
B-6
Appendix B
x12
x10
x10
(b) 12
x
210
(c) 12
2
8. (a)
9. (a)
(b)
(c)
10. (a) (y2y 3)2
11. (a)
(b) y 2(y3)2
(b)
(c) 2m(2n)2
(c)
12. (a)
x6
x
13. (a)
(b)
x
x6
(b)
(c)
14. (a)
(b)
(c)
(x2 3x 2)6
x2 3x 2
(x 2y 3z)4
2(x 2y 3z)4
(2x 2y 3z)4
(x 2 3)10
(x 2 3)9
(x 2 3)9
(x 2 3)10
1210
129
t 15
t9
t9
t 15
(t 2 3)15
(t 2 3)9
x 6y15
x 2y20
b p2q
(b2)q
38. (2pq)(2qp1)
36.
In Exercises 39–42, use the properties of exponents in computing each quantity, as in Example 5. (The point here is to do as
little arithmetic as possible.)
212 # 513
1012
144 # 125 1
42. a 3 2 b
2 #3
28 # 315
9 # 310 # 12
245
41.
32 # 124
39.
40.
For Exercises 43–50, express each number in scientific notation.
x 6y15
(c) a
43. The average distance (in miles) from Earth to the Sun:
x 2y20
x 6y15
15. (a) 4(x 3)2
(b) (4x 3)2
(4x 2)3
(c)
(4x 3)2
b
2
92,900,000
44. The average distance (in miles) from the planet Pluto to the
Sun:
3,666,000,000
45. The average orbital speed (in miles per hour) of Earth:
66,800
46. The average orbital speed (in miles per hour) of the planet
Mercury:
107,300
For Exercises 17–38, simplify each expression. Write the
answers in such a way that negative exponents do not appear.
In Exercises 35–38, assume that the letters p and q represent
natural numbers.
17. (a) 64
(b) (643)0
(c) (640)3
19. (a) 101 102
(b) (101 102)1
(c) [(101)(102)]1
21. 1 13 2 1 1 14 2 1
23. (a) 52 102
(b) (5 10)2
25. (a2bc 0)3
27. (a2b1c 3)2
x 3y2z 3
29. a 2 3 b
xy z
x4y8z2 2
31. a 2 6 b
xy z
34. (2x 2)3 2(x 2)3
x 2y20
16. (a) 2(x 1)7 (x 1)7
(b) [2(x 1)]7 (x 1)7
(c) 2(x 1)7 [2(x 1)]7
0
x2
x2
3
3
y
y
b p1
35.
bp
37. (x p)(x p)
33.
18. (a) 2 3
(b) (20 30)3
(c) (23 33)0
20. (a) 42 41
(b) [1 14 2 1 1 14 2 2]1
(c) [1 14 2 1 1 14 2 2]1
22. 1 13 14 2 1
24. (a) 1 14 2 2 1 34 2 2
(b) 1 14 34 2 2
26. (a3b)3(a2b4)1
28. (22 21 20)2
x4y8z2 2
30. a 2 6 b
xy z
0
32. a
3 9 2
0
ab c
b
a5b2c4
0
47. The average distance (in miles) from the Sun to the nearest
star:
25,000,000,000,000,000,000
48. The equatorial diameter (in miles) of
(a) Mercury: 3031
(c) Jupiter: 88,733
(b) Earth: 7927
(d) the Sun: 865,000
49. The time (in seconds) for light to travel
(a) one foot:
0.000000001
(b) across an atom:
0.000000000000000001
(c) across the nucleus of an atom:
0.000000000000000000000001
50. The mass (in grams) of
(a) a proton:
0.00000000000000000000000167
(b) an electron:
0.000000000000000000000000000911
B.2 Review of nth Roots
B.2
Hindu mathematicians first recognized
negative roots, and the two square roots
of a positive number, . . . though they
were suspicious also. —David Wells in
The Penguin Dictionary of Curious and
Interesting Numbers (Harmondsworth,
Middlesex, England: Penguin Books,
Ltd., 1986)
B-7
REVIEW OF nTH ROOTS
In this section we generalize the notion of square root that you studied in elementary
algebra. The new idea is that of an nth root; the definition and some basic examples
are given in the box that follows.
Definition 1 nth Roots
Let n be a natural number. If a
and b are real numbers and
an b
then we say that a is an nth root of b.
When n 2 and when n 3, we
refer to the roots as square roots
and cube roots, respectively.
EXAMPLES
Both 3 and 3 are square roots of 9
because 32 9 and (3)2 9.
Both 2 and 2 are fourth roots of 16
because 24 16 and (2)4 16.
2 is a cube root of 8 because 23 8.
3 is a fifth root of 243 because
(3)5 243.
As the examples in the box suggest, square roots, fourth roots, and all even roots
of positive numbers occur in pairs, one positive and one negative. In these cases we
n
use the notation 2b to denote the positive, or principal, nth root of b. As examples
of this notation, we can write
4
1
81 3
4
181 3
4
1
81 3
(The principal fourth root of 81 is 3.)
(The principal fourth root of 81 is not 3.)
(The negative of the principal fourth root of 81 is 3.)
The symbol 1 is called a radical sign, and the number within the radical sign is the
n
radicand. The natural number n used in the notation 1 is called the index of the
radical. For square roots, as you know from basic algebra, we suppress the index and
2
simply write 1 rather than 1
. So, for example, 125 5.
On the other hand, as we saw in the examples, cube roots, fifth roots, and in fact,
n
all odd roots occur singly, not in pairs. In these cases, we again use the notation 1b
for the nth root. The definition and examples in the following box summarize our
discussion up to this point.
Definition 2 Principal nth Root
1. Let n be a natural number. If a and b
are nonnegative real numbers, then
n
1 b a if and only if b an
The number a is the principal nth
root of b.
2. If a and b are negative and n is an
odd natural number, then
n
1 b a if and only if b an
EXAMPLES
125 5; 125 5; 125 5
1
1 3
; 18 2
B 16 2
4
1
1
B 32
2
5
There are five properties of nth roots that are frequently used in simplifying certain expressions. The first four are similar to the properties of square roots that are
developed in elementary algebra. For reference we list these properties side by side
B-8
Appendix B
in the following box. Property 5 is listed here only for the sake of completeness;
we’ll postpone discussing it until Appendix B.3.
PROPERTY SUMMARY
Properties of nth Roots
Suppose that x and y are real numbers and that m and n are
natural numbers. Then each of the following properties holds,
provided only that the expressions on both sides of the
equation are defined (and so represent real numbers).
1. 11 x2 n x
11x2 2 x
n
n
n
n
2. 1 xy 1 x1 y
1xy 1x1y
n
3.
x
1x
n
By
1y
x
1x
By
1y
n
n
4. n even: 2x 0 x 0
n
n odd: 2x n x
2x 2 0 x 0
n
m
CORRESPONDING
PROPERTIES FOR
SQUARE ROOTS
mn
n
5. 21
x 1x
Our immediate use for these properties will be in simplifying expressions involving nth roots. In general, we try to factor the expression under the radical so
that one factor is the largest perfect nth power that we can find. Then we apply
Property 2 or 3. For instance, the expression 172 is simplified as follows:
172 1(36)(2) 13612 612
In this procedure we began by factoring 72 as (36)(2). Note that 36 is the largest factor of 72 that is a perfect square. If we were to begin instead with a different factorization, for example, 72 (9)(8), we could still arrive at the same answer, but it would
3
take longer. (Check this for yourself.) As another example, let’s simplify 1
40. First,
what (if any) is the largest perfect-cube factor of 40? The first few perfect cubes are
13 1
23 8
33 27
43 64
so we see that 8 is a perfect-cube factor of 40, and we write
3
3
3
3
3
1
40 1(8)(5) 1815 215
EXAMPLE
1 Simplifying Square Roots
Simplify: (a) 112 175;
SOLUTION
(b)
162
.
B 49
(a) 112 175 1(4)(3) 1(25)(3)
1413 12513
213 513 (2 5)13
713
(b)
162
1162
18112
912
B 49
7
7
149
B.2 Review of nth Roots
EXAMPLE
B-9
2 Simplifying nth Roots
3
3
3
Simplify: 1
16 1
250 1
128.
SOLUTION
3
3
3
3
3
3
3
3
3
1
16 1
250 1
128 1
81
21
1251
21
641
2
3
3
3
21
2 51
2 41
2
3
31
2
EXAMPLE
3 Simplifying Radicals Containing Variables
Simplify each of the following expressions by removing the largest possible perfectsquare or perfect-cube factor from within the radical:
(a) 28x2;
(b) 28x2, where x 0;
(c) 218a7, where a 0;
3
(d) 216y5.
SOLUTION
(a) 28x2 2(4)(2)(x2) 14122x2 212 0 x 0
(b) 28x2 14122x2
212x
2x2 x because x 0
(c) 218a7 2(9a6)(2a) 29a612a
3a312a
3
3
3
3
(d) 216y5 28y3 22y2 2y22y2
EXAMPLE
4 Simplifying nth Roots Containing Variables
Simplify each of the following, assuming that a, b, and c are positive:
(a) 28ab2c5;
SOLUTION
(b)
32a6b5
.
B c8
4
(a) 28ab2c5 2(4b2c4)(2ac) 24b2c412ac
2bc212ac
(b)
4
32a6b5
2
32a6b5
4 8
B c8
2
c
4
4
4
4
2
16a4b4 2
2a2b
2ab2
2a2b
2
2
c
c
B-10
Appendix B
In Examples 1 through 4 we used the definitions and properties of nth roots to simplify certain expressions. The box that follows shows some common errors to avoid
in working with roots. Use the error box to test yourself: Cover up the columns labeled
“Correction” and “Comment,” and try to decide for yourself where the errors lie.
Errors to Avoid
Error
4
1
16 2
Correction
4
1
16 2
125 5
125 5
1a b 1a 1b
The expression
1a b cannot
be simplified.
3
3
3
1
ab 1
a 1
b
The expression
3
1
a b cannot
be simplified.
Comment
Although 2 is one of the fourth
roots of 16, it is not the principal
4
fourth root. The notation 116 is
reserved for the principal fourth root.
Although 5 is one of the square
roots of 25, it is not the principal
square root.
The properties of roots differ with
respect to addition versus multiplication. For 1ab we do have the
simplification 1ab 1a1b
(assuming that a and b are
nonnegative).
3
3
3
3
For 1ab we do have 1ab 1a1b.
There are times when it is convenient to rewrite fractions involving radicals in alternative forms. Suppose, for example, that we want to rewrite the fraction 513 in
an equivalent form that does not involve a radical in the denominator. This is called
rationalizing the denominator. The procedure here is to multiply by 1 in this way:
5
5 #
5 # 13
513
1
3
13
13
13 13
513
5
, as required.
3
13
To rationalize a denominator of the form a 1b, we need to multiply the fraction
not by 1b1b, but rather by 1a 1b 21a 1b 2 ( 1). To see why this is necessary,
notice that
That is,
1a 1b 21b a1b b
which still contains a radical, whereas
1a 1b 2 1a 1b2 a2 a1b a1b b
a2 b
which is free of radicals. Note that we multiply a 1b by a 1b to obtain an
expression that is free of square roots by taking advantage of the form of a difference
B.2 Review of nth Roots
B-11
of two squares. Similarly, to rationalize a denominator of the form a 1b, we multiply the fraction by 1a 1b 21a 1b 2. (The quantities a 1b and a 1b are said
to be conjugates of each other.) The next two examples make use of these ideas.
EXAMPLE
5 Rationalizing a Square Root Denominator
Simplify:
SOLUTION
1
3150.
12
First, we rationalize the denominator in the fraction 112:
1 #
1 # 12
12
1
1
2
12
12
12 12
Next, we simplify the expression 3150:
3150 31(25)(2) 312512 (3)(5)12 1512
Now, putting things together, we have
3012
1
12
12
1512 3150 2
2
2
12
EXAMPLE
(1 30)12
12 3012
2
2
2912
2
6 Using the Conjugate to Rationalize a Denominator
Rationalize the denominator in the expression
SOLUTION
We multiply by 1, writing it as
4
.
2 13
2 13
.
2 13
4
# 1 4 # 2 13
2 13
2 13 2 13
412 132
412 132
8 413
2
43
1
4 1132
8 413
2 13
Note: Check for yourself that multiplying the original fraction by
does not
2 13
eliminate radicals in the denominator.
In the next example we are asked to rationalize the numerator rather than the
denominator. This is useful at times in calculus.
B-12
Appendix B
EXAMPLE
7 Rationalizing a Numerator Using the Conjugate
Rationalize the numerator of:
1x 13
, where x 0, x 3.
x3
1x 13 # 1x 13
1x 13 #
1
x3
x3
1x 13
SOLUTION
11x2 2 1132 2
(x 3)11x 132
1
1x 13
x3
(x 3)11x 132
The strategy for rationalizing numerators or denominators involving nth roots is
similar to that used for square roots. To rationalize a numerator or a denominator involving an nth root, we multiply the numerator or denominator by a factor that yields
a product that itself is a perfect nth power. The next example displays two instances
of this.
EXAMPLE
8 Rationalizing Denominators Containing nth Roots
(a) Rationalize the denominator of:
(b) Rationalize the denominator of:
SOLUTION
(a)
6
3
1
7
#1
(b)
ab
4
6
3
6
3
17
.
ab
4
2a2b3
, where a 0, b 0.
# 23 72
2
3
1
7 27
3 2
7
62
3 3
2
7
3
61
49
7
#1
3
2a2b
as required
4
4
# 24 a2b
3
ab
2a2b
2a b
4
ab2a2b
4
2a4b4
2
4 2
ab2
ab
4 2
2
ab
ab
Finally, just as we used the difference of squares formula to motivate the method
of rationalizing certain two-term denominators containing a square root term, we can
use the difference or sum of cubes formula (see Appendix B.4) to rationalize a
denominator involving a cube root term. In particular, in the sum of cubes formula
x3 y3 (x y)(x2 xy y2 ), the expression (x2 xy y2 ) is the conjugate of
the expression x y.
B.2 Review of nth Roots
9 Using the Conjugate to Rationalize a Denominator Involving a Cube Root
EXAMPLE
Rationalize the denominator in the expression
SOLUTION
3
2a
3
2b,
B-13
If we let x is
1
3
3
2a 2b
.
3
3
the conjugate of the denominator x y 2a 2b
and y x2 xy y2 1 2a2 2 2a 2b 1 2b2 2 2a2 2ab 2b2
3
3
3
3
3
3
3
3
3
3
3
where, for example, 1 2a 2 2 2a 2a 2a2. Then
1
3
2a 2
b
3
EXERCISE SET
In Exercises 1–8, determine whether each statement is TRUE
or FALSE.
181 9
1100 10
243 213
3
1 110 2 3 10
2.
4.
6.
8.
1256 16
149 7
110 6 110 16
2(5)2 5
For Exercises 9–18, evaluate each expression. If the expression
is undefined (i.e., does not represent a real number), say so.
9. (a)
(b)
11. (a)
(b)
13. (a)
(b)
15. (a)
(b)
17. (a)
(b)
3
164
4
1
64
3
2
8125
3
28125
116
4
1
16
4
225681
3
227125
5
1
32
5
1
32
3
2
3
3 2
2
ab 2
b
3
3
3 2
3
3 2
2
a 2
b 2
a 2
ab 2
b
3 2
3
3 2
2
a 2
ab 2
b
3
3
12
a2 3 1 2
b2 3
10. (a)
(b)
12. (a)
(b)
14. (a)
(b)
16. (a)
(b)
18. (a)
(b)
21. (a) 198
5
(b) 1
64
23. (a) 2254
4
25.
27.
28.
5
132
5
1
32
3
2
11000
6
211000
4
1
16
4
1
16
6
164
6
1
64
4
2
(10)4
3
2(10)3
In Exercises 19–44, simplify each expression. Unless otherwise
specified, assume that all letters in Exercises 35–44 represent positive numbers.
19. (a) 118
3
(b) 1
54
# 2a
3 2
3
3 2
2
a 2
ab 2
b
ab
B.2
A
1.
3.
5.
7.
1
20. (a) 1150
3
(b) 1
375
29.
30.
31.
33.
35.
37.
39.
41.
43.
22. (a) 127
3
(b) 1
108
24. (a) 222549
(b) 216625
(a) 12 18
26.
3
3
(b) 1
21
16
(a) 4150 31128
4
4
(b) 1
32 1
162
(a) 13 112 148
5
5
5
(b) 1
2 1
64 1
486
(a) 10.09
3
(b) 1
0.008
3
3
(a) 1
2 1
2
3
(b) 281121 2
81331
32.
4124 8154 216
34.
2164
(a) 236x2
36.
(b) 236y2, where y 0
(a) 2ab2 2a2b
38.
(b) 2ab3 2a3b
40.
272a3b4c5
4
42.
2
16a4b5
3
12 2 9
44.
216a b c
5
(b) 2
256243
(a) 413 2127
3
3
(b) 21
81 31
24
3
3
3
1
192 1
81 21
9
3
2
14096
(a) 2225x4y3
4
(b) 2
16a4, where a 0
3
(a) 2125x6
4
(b) 2
64y4, where y 0
2(a b)5(16a2b2)
3
2
8a4b6
4
4 3
2ab3 2
ab
B-14
Appendix B
In Exercises 45–70, rationalize the denominators and simplify
where possible. (Assume that all letters represent positive
quantities.)
45.
47.
49.
51.
53.
55.
57.
59.
61.
63.
65.
66.
67.
69.
417
118
1 1 1 152
11 132 11 132
1 1152 4145
46.
48.
50.
52.
54.
56.
58.
60.
62.
313
1213
12 11 122
1 1a 1b2 1 1a 1b2
1 3182 1450
3
3
11
25
41
16
4
3
3
313
151
6
5
4
4 9
3 216a b
3 227a5b11
(a) 21 13 1 2
(a) 1 1215 1 2
(b) 21 13 1 2
(b) 1 1215 1 2
(a) x 11x 2 2
64. (a) 1x 1 5 1x2
(b) x 1 1x y2
(b) 1x 1 1a 1x2
1
1x
122
1
1x
122
(a)
(b) 1 1x 1a2 11x 1a2
(a) 1 13 21y2 1 13 21y2
(b) 1 1x 21y2 11x 21y2
68. 1 1 1x h 1x h2
2 11x h 1x2
3
3
3
70. 1 1 1
1 1 1
a 12
a1
b2
In Exercises 71–76, rationalize the numerator.
71. 1 1x 152 (x 5)
73. 1 12 h 122h
75. 1 1x h 1x2 h
72. 1 21y 32(4y 9)
74. 111 2h 1 2 (2h)
76. 1 1x 2h 1x 2h2 h
B.3
If Descartes . . . had discarded the
radical sign altogether and had
introduced the notation for fractional as
well as integral exponents, . . . it is
conceivable that generations upon
generations of pupils would have been
saved the necessity of mastering the
operations with two . . . notations when
one alone (the exponential) would have
answered all purposes. —Florian Cajori,
A History of Mathematical Notations,
Vol. 1 (La Salle, Ill.: The Open Court
Publishing Co., 1928)
In Exercises 77–80, refer to the following table. The left-hand
column of the table lists four errors to avoid in working with
expressions containing radicals. In each case, give a numerical
example showing that the expressions on each side of the equation are, in general, not equal. Use a calculator as necessary.
Numerical Example
Showing that the
Formula Is Not,
in General, Valid
Errors to Avoid
77.
78.
79.
80.
1a b 1a 1b
2x2 y2 x y
3
3
3
1
uv1
u1
v
3 3
3
2p q p q
B
81. (a) Use a calculator to evaluate 28 217 and 17 1.
What do you observe?
(b) Prove that 28 217 17 1. Hint: In view of
the definition of a principal square root, you need to
check that 117 12 2 8 217.
82. Use a calculator to provide empirical evidence indicating
that both of the following equations may be correct:
2 13
12 22 13
2 13
12 22 13
12
3297.5 (111) p
The point of this exercise is to remind you that as useful as
calculators may be, there is still the need for proofs in mathematics. In fact, it can be shown that the first equation is indeed correct, but the second is not.
REVIEW OF RATIONAL EXPONENTS
. . . Nicole Oresme, a bishop in Normandy (about 1323–1382), first conceived the notion of fractional powers, afterwards rediscovered by [Simon] Stevin [1548–1620] . . .
—Florian Cajori in A History of Elementary Mathematics (New York: Macmillan Co., 1917)
We can use the concept of an nth root to give a meaning to fractional exponents that
is useful and, at the same time, consistent with our earlier work. First, by way of
motivation, suppose that we want to assign a value to 513. Assuming that the usual
properties of exponents continue to apply here, we can write
(513)3 51 5
That is,
(513)3 5
or
3
513 15
B.3 Review of Rational Exponents
B-15
By replacing 5 and 3 with b and n, respectively, we can see that we want to define
n
b1n to mean 1b. Also, by thinking of bmn as (b1n)m, we see that the definition for
n
bmn ought to be 11b2 m. These definitions are formalized in the box that follows.
Definition Rational Exponents
1. Let b denote a real number and n a
natural number. We define b1n by
EXAMPLES
412 14 2
3
(8)13 18 2
n
b1n 1
b
(If n is even, we require that b 0.)
2. Let mn be a rational number reduced
to lowest terms. Assume that n is
n
positive and that 1
b exists. Then
n
bmn 11
b2 m
3
823 1182 2 22 4
or, equivalently,
823 282 164 4
3
3
or, equivalently,
n
bmn 2bm
It can be shown that the four properties of exponents that we listed in Appendix B.1
(on page B-2) continue to hold for rational exponents in general. In fact, we’ll take
this for granted rather than follow the lengthy argument needed for its verification.
We will also assume that these properties apply to irrational exponents. For instance,
we have
1215 2 15 25 32
(The definition of irrational exponents is discussed in Section 5.1.) In the next three
examples we display the basic techniques for working with rational exponents.
EXAMPLE
1 Evaluating Rational Exponents
Simplify each of the following quantities. Express the answers using positive exponents. If an expression does not represent a real number, say so.
(a) 4912
SOLUTION
(b) 4912
(c) (49)12
(d) 4912
(a) 4912 149 7
(b) 4912 (4912) 149 7
(c) The quantity (49)12 does not represent a real number because there is no
real number x such that x 2 49.
1
11
(d) 4912 2491 2149 7
149
Alternatively, we have
4912 (4912)1 71 1
7
B-16
Appendix B
EXAMPLE
2 Simplifying Expressions Containing Rational Exponents
Simplify each of the following. Write the answers using positive exponents. (Assume
that a 0.)
13
5 16a
(b)
(c) (x 2 1)15(x 2 1)45
(a) (5a23)(4a34)
B a14
SOLUTION
(a) (5a23)(4a34) 20a(23)(34)
20a1712 because 32 34 17
12
16a13
16a13 15
a 14 b
14
B a
a
(16a112)15
because 13 14 121
1615a160
2
15 2
(c) (x 1) (x 1)45 (x 2 1)1 x 2 1
(b)
EXAMPLE
5
3 Evaluating Rational Exponents
Simplify: (a) 3225;
SOLUTION
(b) (8)43.
5
(a) 3225 11322 2 22 1
1
22 4
Alternatively, we have
1
1
22 4
3
(b) (8)43 1182 4 (2)4 16
3225 (25)25 22 Alternatively, we can write
(8)43 [(2)3]43 (2)4 16
Rational exponents can be used to simplify certain expressions containing radicals. For example, one of the properties of nth roots that we listed but did not discuss
mn
m n
in Appendix B.2 is 21x 1x. Using exponents, it is easy to verify this property.
We have
m n
21
x (x1n)1m
mn
x1mm 1x
EXAMPLE
as we wished to show
4 Using Rational Exponents to Combine Radicals
3
Consider the expression 1x2y2, where x and y are positive.
(a) Rewrite the expression using rational exponents.
(b) Rewrite the expression using only one radical sign.
SOLUTION
3
(a) 2x2y2 x12y23
3
(b) 2x2y2 x12y23
x36y46
6 3 6 4
6 3 4
2
x 2y 2
xy
rewriting the fractions using
a common denominator
B.3 Review of Rational Exponents
EXAMPLE
B-17
5 Using Rational Exponents to Simplify a Radical Expression
Rewrite the following expression using rational exponents (assume that x, y, and z
are positive):
3
4
2
x2y3
C
SOLUTION
3
4 3
2
x2
y
C
5
2z4
a
5 4
2
z
3
4 3
2
x2
y
5
b
12
a
x13y34
45
z
2z4
x y
25 x16y38z25
z
b
12
16 38
EXERCISE SET
B.3
A
Exercises 1–14 are warm-up exercises to help you become
n
n
familiar with the definition bmn 11
b 2 m 2bm. In
Exercises 1–8, write the expression in the two equivalent forms
n
n
11
b 2 m and 2bm. In Exercises 9–14, write the expression in
mn
the form b .
1. a35
4. 1045
7. 2xy3
5 3
10. 2
R
p
2
13. 2(a
b2)3
2. x 37
5. (x 2 1)34
8. 2(a1)b
7
11. 2
(1 u)4
3p
14. 2(a2 b2)2p
3. 523
6. (a b)710
3 2
9. 2
p
6
12. 2
(1 u)5
For Exercises 51–58, follow Example 4 in the text to rewrite each
expression in two ways: (a) using rational exponents and (b) using
only one radical sign. (Assume that x, y, and z are positive.)
3
51. 131
6
3
4
53. 1612
3 2 5 4
55. 2
x 2y
3
52. 151
7
3
5
54. 1212
3
4
56. 1x1
y1
z
4 a 3 b
57. 2
x 2x 2x a6
3
58. 2
27164x
In Exercises 59–66, rewrite each expression using rational exponents rather than radicals. Assume that x, y, and z are positive.
3
59. 2
(x 1)2
5
61. 11x y2 2
3
3
63. 2
1x 21
x
In Exercises 15–50, evaluate or simplify each expression.
Express the answers using positive exponents. If an expression
is undefined (that is, does not represent a real number), say so.
Assume that all letters represent positive numbers.
15.
18.
21.
24.
27.
30.
33.
36.
39.
41.
43.
45.
1612
16. 10012
17. (136)12
12
12
0.09
19. (16)
20. (1)12
62514
22. (181)14
23. 813
13
23
0.001
25. 8
26. 6423
15
13
(32)
28. (1125)
29. (1000)13
15
12
(243)
31. 49
32. 12112
(49)12
34. (64)13
35. 3632
23
23
(0.001)
37. 125
38. 12523
35
43
(1)
40. 27 2743 270
45
45
32 32
42. 6412 6412 6443
(916)52 (100027)43 44. (25634)43
5
(2a13)(3a14)
46. 23a4
4
47. 264a23a13
48. (x 2 1)23(x 2 1)43
2
34
(x 1)
49. 2
(x 1)14
(2x2 1)65(2x2 1)65(x2 1)15
50.
(x2 1)95
60. 111x2 1x
62. 21x
3
64. 2
12
3
4
5
3
4 2
65. 21
66. 3
x1
y
1x1
y 2
z
67. Use a calculator to determine which is larger, 9109 or 10910.
68. Use a calculator to determine which number is closer to 3:
1337 or 656018.
In Exercises 69–74, refer to the following table. The left-hand
column of the table lists six errors to avoid in working with expressions containing fractional exponents. In each case, give a
numerical example showing that the expressions on each side of
the equation are not equal. Use a calculator as necessary.
Errors to Avoid
(a b)12 a12 b12
(x 2 y 2)12 x y
(u v)13 u13 v13
(p3 q3)13 p q
1
73. x 1m m
x
1
74. x 12 2
x
69.
70.
71.
72.
Numerical Example
Showing That the
Formula Is Not,
in General, Valid
B-18
Appendix B
B.4
#
fac tor (fakt 0 r), n. . . . 2. Math.
one of two or more numbers, algebraic
expressions, or the like, that when
multiplied together produce a given
product; . . . v.t. 10. Math. to express
(a mathematical quantity) as a product
of two or more quantities of like kind,
as 30 2 3 5, or x 2 y 2 (x y) (x y). —The Random House
Dictionary of the English Language,
2nd ed. (New York: Random House,
1987)
# #
REVIEW OF FACTORING
There are many cases in algebra in which the process of factoring simplifies the work
at hand. To factor a polynomial means to write it as a product of two or more nonconstant polynomials. For instance, a factorization of x 2 9 is given by
x2 9 (x 3)(x 3)
In this case, x 3 and x 3 are the factors of x 2 9.
There is one convention that we need to agree on at the outset. If the polynomial
or expression that we wish to factor contains only integer coefficients, then the factors (if any) should involve only integer coefficients. For example, according to this
convention, we will not consider the following type of factorization in this section:
x 2 2 1x 12 2 1x 122
because it involves coefficients that are irrational numbers. (We should point out,
however, that factorizations such as this are useful at times, particularly in calculus.)
As it happens, x 2 2 is an example of a polynomial that cannot be factored using integer coefficients. In such a case we say that the polynomial is irreducible over the
integers.
If the coefficients of a polynomial are rational numbers, then we do allow factors
with rational coefficients. For instance, the factorization of y 2 14 over the rational
numbers is given by
y2 1
1
1
ay b ay b
4
2
2
We’ll consider five techniques for factoring in this section. These techniques will
be applied in the next two sections and throughout the text. In Table 1 we summarize
these techniques. Notice that three of the formulas in the table are just restatements
TABLE 1
Basic Factoring Techniques
Technique
Example or Formula
Remark
Common factor
3x 4 6x 3 12x 2 3x 2(x 2 2x 4)
4(x 2 1) x(x 2 1) (x 2 1)(4 x)
x 2 a2 (x a)(x a)
In any factoring problem, the first step always is
to look for the common factor of highest degree.
Difference of
squares
Trial and error
x 2 2x 3 (x 3)(x 1)
Difference of cubes x 3 a3 (x a)(x 2 ax a2)
Sum of cubes
x 3 a3 (x a) (x 2 ax a2)
x3 x2 x 1 (x3 x2) (x 1)
Grouping
x2(x 1) (x 1) # 1
(x 1)(x2 1)
There is no corresponding formula for a sum of
squares; x 2 a2 is irreducible over the integers.
In this example the only possibilities, or trials, are
(a) (x 3)(x 1)
(c) (x 3)(x 1)
(b) (x 3)(x 1)
(d) (x 3)(x 1)
By inspection or by carrying out the indicated
multiplications, we find that only case (c) checks.
Verify these formulas for yourself by carrying out
the multiplications. Then memorize the formulas.
This is actually an application of the common
factor technique.
B.4 Review of Factoring
B-19
of Special Products formulas. Remember that it is the form or pattern in the formula
that is important, not the specific choice of letters.
The idea in factoring is to use one or more of these techniques until each of the factors obtained is irreducible. The examples that follow show how this works in practice.
EXAMPLE
1 Factoring Using the Difference of Squares Technique
Factor:
SOLUTION
(a) x 2 49;
(b) (2a 3b)2 49.
(a) x2 49 x2 72
(x 7)(x 7)
difference of squares
(b) Notice that the pattern or form is the same as in part (a): It’s a difference of
squares. We have
(2a 3b)2 49 (2a 3b)2 72
[(2a 3b) 7][(2a 3b) 7]
(2a 3b 7)(2a 3b 7)
EXAMPLE
2 Factoring Using Common Factor and Difference of Squares
Factor:
SOLUTION
EXAMPLE
(a) 2x 3 50x;
(b) 3x 5 3x.
(a) 2x3 50x 2x(x2 25)
2x(x 5)(x 5)
(b) 3x5 3x 3x(x4 1)
3x[(x2)2 12]
3x(x2 1)(x2 1)
3x(x 1)(x 1)(x2 1)
common factor
difference of squares
common factor
difference of squares
difference of squares, again
3 Factoring by Trial and Error
Factor:
SOLUTION
difference of squares
(a) x 2 4x 5;
(b) (a b)2 4(a b) 5.
(a) x 2 4x 5 (x 5)(x 1) trial and error
(b) Note that the form of this expression is
( )2 4( ) 5
This is the same form as the expression in part (a), so we need only replace x with
the quantity a b in the solution for part (a). This yields
(a b)2 4(a b) 5 [(a b) 5][(a b) 1]
(a b 5)(a b 1)
EXAMPLE
4 Factoring by Trial and Error
Factor: 2z4 9z2 4.
SOLUTION
We use trial and error to look for a factorization of the form (2z2 ?)(z2 ?). There
are three possibilities:
(i) (2z2 2)(z2 2) (ii) (2z2 1)(z2 4) (iii) (2z2 4)(z2 1)
B-20
Appendix B
Each of these yields the appropriate first term and last term, but (after checking) only
possibility (ii) yields 9z2 for the middle term. The required factorization is then
2z4 9z2 4 (2z2 1)(z2 4).
Question: Why didn’t we consider any possibilities with subtraction signs in place
of addition signs?
EXAMPLE
5 Two Irreducible Expressions
Factor:
SOLUTION
EXAMPLE
(a) x 2 9;
(b) x 2 2x 3.
(a) The expression x 2 9 is irreducible over the integers. (This can be discovered
by trial and error.) If the given expression had instead been x 2 9, then it could
have been factored as a difference of squares. Sums of squares, however, cannot
in general be factored over the integers.
(b) The expression x 2 2x 3 is irreducible over the integers. (Check this for
yourself by trial and error.)
6 Factoring Using Grouping and a Special Product
Factor: x 2 y 2 10x 25.
SOLUTION
EXAMPLE
Familiarity with the special product (a b)2 a2 2ab b2 suggests that we try
grouping the terms this way:
(x2 10x 25) y2
Then we have
x2 y2 10x 25 (x2 10x 25) y2
(x 5)2 y2
[(x 5) y][(x 5) y] difference of squares
(x y 5)(x y 5)
7 Factoring Using Grouping
Factor: ax ay 2 bx by 2.
SOLUTION
We factor a from the first two terms and b from the second two to obtain
ax ay2 bx by2 a(x y2) b(x y2)
We now recognize the quantity x y 2 as a common expression that can be factored
out. We then have
a(x y2) b(x y2) (x y2)(a b)
The required factorization is therefore
ax ay2 bx by2 (x y2)(a b)
Alternatively,
ax ay2 bx by2
(ax bx) (ay2 by2)
x(a b) y2(a b)
(a b)(x y2)
B.4 Review of Factoring
EXAMPLE
8 Factoring Sum and Difference of Cubes
Factor:
SOLUTION
B-21
(a) t 3 125;
(b) 8 (a 2)3.
(a) t3 125 t3 53
(t 5)(t2 5t 25)
difference of cubes
3
3
3
(b) 8 (a 2) 2 (a 2)
[2 (a 2)][22 2(a 2) (a 2)2]
a(4 2a 4 a2 4a 4)
a(a2 6a 12)
sum of cubes
As you can check now, the expression a2 6a 12 is irreducible over the integers.
Therefore the required factorization is a(a2 6a 12).
For some calculations (particularly in calculus) it’s helpful to be able to factor an
expression involving fractional exponents. For instance, suppose that we want to factor the expression
x(2x 1)12 (2x 1)32
The common expression to factor out here is (2x 1)12; the technique is to choose
the expression with the smaller exponent.
EXAMPLE
9 Factoring with Rational Exponents
Factor: x(2x 1)12 (2x 1)32.
SOLUTION
x(2x 1)12 (2x 1)32 (2x 1)12[x (2x 1)3212]
(2x 1)12[x (2x 1)2]
(2x 1)12(4x2 3x 1)
We conclude this section with examples of four common errors to avoid in factoring. The first two errors in the box that follows are easy to detect; simply multiplying out
the supposed factorizations shows that they do not check. The third error may result
from a lack of familiarity with the basic factoring techniques listed on page B-18. The
fourth error indicates a misunderstanding of what is required in factoring; the final quantity in a factorization must be expressed as a product, not a sum, of terms or expressions.
Errors to Avoid
Error
x 2 6x 9 (x 3)2
x 2 64 (x 8)(x 8)
Correction
x 2 6x 9 (x 3)2
x 2 64 is irreducible
over the integers.
x 3 64 is irreducible
over the integers
x 2 2x 3 factors as
x(x 2) 3
x 3 64 (x 4)(x 2 4x 16)
x 2 2x 3 is irreducible over
the integers.
Comment
Check the middle term.
A sum of squares is, in general, irreducible over
the integers. (A difference of squares, however, can
always be factored.)
Although a sum of squares is irreducible, a sum
of cubes can be factored.
The polynomial x 2 2x 3 is the sum of the expression x(x 2) and the constant 3. By definition, however, the factored form of a polynomial must be a
product (of two or more nonconstant polynomials).
B-22
Appendix B
EXERCISE SET
B.4
A
In Exercises 1–66, factor each polynomial or expression. If a
polynomial is irreducible, state this. (In Exercises 1–6 the factoring techniques are specified.)
1. (Common factor and difference of squares)
(a) x 2 64
(c) 121z z3
(b) 7x 4 14x 2
(d) a2b2 c 2
2. (Common factor and difference of squares)
(a) 1 t 4
(c) u2v2 225
6
5
4
(b) x x x
(d) 81x 4 x 2
3. (Trial and error)
(a) x 2 2x 3
(c) x 2 2x 3
2
(b) x 2x 3
(d) x 2 2x 3
4. (Trial and error)
(a) 2x 2 7x 4
(c) 2x 2 7x 4
(b) 2x 2 7x 4
(d) 2x 2 7x 4
5. (Sum and difference of cubes)
(a) x 3 1
(c) 1000 8x 6
(b) x 3 216
(d) 64a3x 3 125
6. (Grouping)
(a) x 4 2x 3 3x 6
(b) a2x bx a2z bz
7. (a) 144 x 2
8. (a) 4a2b2 9c 2
(b) 144 x 2
(b) 4a2b2 9c 2
2
(c) 144 (y 3)
(c) 4a2b2 9(ab c)2
3
5
9. (a) h h
(b) 100h3 h5
(c) 100(h 1)3 (h 1)5
10. (a) x 4 x 2
(b) 3x 4 48x 2
(c) 3(x h)4 48(x h)2
11. (a) x 2 13x 40
12. (a) x 2 10x 16
2
(b) x 13x 40
(b) x 2 10x 16
2
13. (a) x 5x 36
14. (a) x 2 x 6
(b) x 2 13x 36
(b) x 2 x 6
15. (a) 3x 2 22x 16
16. (a) x 2 4x 1
(b) 3x 2 x 16
(b) x 2 4x 3
2
17. (a) 6x 13x 5
18. (a) 16x 2 18x 9
(b) 6x 2 x 5
(b) 16x 2 143x 9
19. (a) t 4 2t 2 1
20. (a) t 4 9t 2 20
(b) t 4 2t 2 1
(b) t 4 19t 2 20
4
2
(c) t 2t 1
(c) t 4 19t 2 20
21. (a) 4x 3 20x 2 25x
22. (a) x 3 x 2 x
(b) 4x 3 20x 2 25x
(b) x 3 2x 2 x
2
23. (a) ab bc a ac
(b) (u v)x xy (u v)2 (u v)y
24. (a) 3(x 5)3 2(x 5)2
(b) a(x 5)3 b(x 5)2
25. x 2z2 xzt xyz yt
26. a2t 2 b2t 2 cb2 ca2
4
2 2 2
4 4
27. a 4a b c 4b c
28. A2 B2 16A 64
29. A2 B2
30. x 2 64
31.
33.
35.
37.
39.
41.
43.
45.
47.
49.
51.
53.
55.
56.
57.
58.
59.
60.
61.
62.
63.
64.
65.
66.
x 3 64
32. 27 (a b)3
(x y)3 y 3
34. (a b)3 8c 3
3
3
x y xy
36. 8a3 27b3 2a 3b
4
(a) p 1
38. (a) p4 4
8
(b) p 1
(b) p4 4
x 3 3x 2 3x 1
40. 1 6x 12x 2 8x 3
x 2 16y 2
42. 4u2 25v2
25
81
c2
44.
y2
16
4
(a b)2
81
a2b2
z4 46.
16
4
9
125
x3
512
1
3
48.
8
m3n3
x
1 2
2
2
x
xy
y
50.
x
x
1
4
64(x a)3 x a
52. 64(x a)4 x a
x 2 a2 y 2 2xy
54. a4 (b c)4
3
2
21x 82x 39x
x 3a2 8y 3a2 4x 3b2 32y 3b2
12xy 25 4x 2 9y 2
ax 2 (1 ab)xy by 2
ax 2 (a b)x b
(5a2 11a 10)2 (4a2 15a 6)2
(x 1)12 (x 1)32
(x 2 1)32 (x 2 1)72
(x 1)12 (x 1)32
(x 2 1)23 (x 2 1)53
(2x 3)12 13 (2x 3)32
(ax b)12 1ax bb
In Exercises 67 and 68, evaluate the given expressions using
factoring techniques. (The point here is to do as little actual
arithmetic as possible.)
67. (a) 1002 992
68. (a) 103 93
(b) 83 63
(b) 502 492
2
2
153 10 3
(c) 1000 999
(c)
152 10 2
B
In Exercises 69–73, factor each expression.
69.
70.
71.
72.
73.
74.
A3 B3 3AB(A B)
p3 q3 p(p2 q2) q(p q)2
2x(a2 x 2)12 x 3(a2 x 2)32
1
12
(x a)12 12 (x a)12(x a)32
2 (x a)
4
y (p q)y 3 (p2q pq2)y p2q2
(a) Factor x 4 2x 2y 2 y 4.
(b) Factor x 4 x 2y 2 y 4. Hint: Add and subtract a
term. [Keep part (a) in mind.]
(c) Factor x 6 y 6 as a difference of squares.
(d) Factor x 6 y 6 as a difference of cubes. [Use the result
in part (b) to obtain the same answer as in part (c).]
B.5 Review of Fractional Expressions
REVIEW OF FRACTIONAL EXPRESSIONS
B.5
But if you do use a rule involving
mechanical calculation, be patient,
accurate, and systematically neat in
the working. It is well known to
mathematical teachers that quite half the
failures in algebraic exercises arise from
arithmetical inaccuracy and slovenly
arrangement. —George Chrystal (1893)
B-23
The rules of basic arithmetic that you learned for working with simple fractions are
also used for fractions involving algebraic expressions. In algebra, as in arithmetic,
we say that a fraction is reduced to lowest terms or simplified when the numerator
and denominator contain no common factors (other than 1 and 1). The factoring
techniques that we reviewed in Appendix B.4 are used to reduce fractions. For exx2 9
, we write
ample, to reduce the fraction 2
x 3x
(x 3)(x 3)
x2 9
x3
x
x(x 3)
x2 3x
In the box that follows, we review two properties of negatives and fractions that
will be useful in this section. Notice that Property 2 follows from Property 1 because
ab
ab
1
ba
(a b)
PROPERTY SUMMARY
EXAMPLE
Negatives and Fractions
Property
Example
1. b a (a b)
ab
2.
1
ba
2 x (x 2)
2x 5
1
5 2x
1 Using Property 2 to Simplify a Fractional Expression
Simplify:
4 3x
.
15x 20
4 3x
4 3x
15x 20
5(3x 4)
1
1 4 3x
a
b 5 3x 4
5
SOLUTION
using Property 2
In Example 2 we display two more instances in which factoring is used to reduce
a fraction. After that, Example 3 indicates how these skills are used to multiply and
divide fractional expressions.
EXAMPLE
2 Using Factoring to Simplify a Fractional Expression
Simplify: (a)
SOLUTION
x3 8
;
x2 2x
(b)
x2 6x 8
.
a(x 2) b(x 2)
(x 2)(x2 2x 4)
x2 2x 4
x3 8
x
x(x 2)
x2 2x
2
(x 2)(x 4)
x 6x 8
x4
(b)
a(x 2) b(x 2)
(x 2)(a b)
ab
(a)
B-24
Appendix B
EXAMPLE
3 Using Factoring to Simplify Products and Quotients
Carry out the indicated operations and simplify:
x3
# 12x2 18 ;
2
2x 3x 4x 6x
x 12
x2 144
.
(c) 2
x2
x 4
(a)
SOLUTION
(a)
2x2 x 6 x3 1
#
;
x2 x 1 4 x2
(b)
x3
12x 18
x3
#
# 6(2x 3)
2
2
x(2x
3)
2x(2x 3)
2x 3x 4x 6x
3
3x
6x
2
2x 3
2x (2x 3)
(2x 3)(x 2) (x 1)(x2 x 1)
2x2 x 6 # x3 1
#
(2 x)(2 x)
x2 x 1 4 x2
(x2 x 1)
(2x 3)(x 1)
using the fact
2x
x2
2x2 x 3
that
1
2x
2x
x 12
x2 144
x2 144 # x 2
(c) 2
2
x2
x 4
x 4 x 12
(x 12)(x 12) x 2
x 12
#
(x 2)(x 2) x 12
x2
(b)
As in arithmetic, to combine two fractions by addition or subtraction, the fractions
must have a common denominator, that is, the denominators must be the same. The
rules in this case are
c
ac
a
b
b
b
and
a
c
ac
b
b
b
For example, we have
4x 1 (2x 1)
4x 1
2x 1
4x 1 2x 1
2x
x1
x1
x1
x1
x1
Fractions with unlike denominators are added or subtracted by first converting to
a common denominator. For instance, to add 9a and 10a2, we write
10
10
9
9 a
2 # 2
a
a a
a
a
9a
10
9a 10
2 a2
a
a2
Notice that the common denominator used was a2. This is the least common
denominator. In fact, other common denominators (such as a3 or a4) could be used
here, but that would be less efficient. In general, the least common denominator for a
given group of fractions is chosen as follows. Write down a product involving the
irreducible factors from each denominator. The power of each factor should be equal to
(but not greater than) the highest power of that factor appearing in any of the individual denominators. For example, the least common denominator for the two fractions
B.5 Review of Fractional Expressions
B-25
1
1
and
is (x 1)2(x 2). In the example that follows, notice
(x 1)(x 2)
(x 1)2
that the denominators must be in factored form before the least common denominator
can be determined.
EXAMPLE
4 Using the Least Common Denominator
Combine into a single fraction and simplify:
(a)
SOLUTION
3
7x
2;
4x
10y2
(b)
1
x
;
x3
x2 9
(c)
x1
15
.
2x
x2 x 6
(a) Denominators: 22 # x; 2 # 5 # y2; 1
Least common denominator; 22 # 5xy 2 20xy 2
3
7x
2
3 # 5y2
7x # 2x
2 20xy2
#
2 2 2
4x
1
4x 5y
1 20xy2
10y
10y 2x
15y2 14x2 40xy2
20xy2
Note: The final numerator is irreducible over the integers.
(b) Denominators: x2 9 (x 3)(x 3); x 3
Least common denominator: (x 3)(x 3)
1
x
1
x
x2 9 x 3 (x 3)(x 3) x 3
x
1 #x3
(x 3)(x 3) x 3 x 3
x (x 3)
3
(x 3)(x 3) (x 3)(x 3)
15
x1
x1
15
using the fact that
(c) 2
2 x (x 2)
2x
(x 2)(x 3)
x2
x x6
x1#x3
15
(x 2)(x 3) x 2 x 3
15 (x2 4x 3)
x2 4x 12
(x 2)(x 3)
(x 2)(x 3)
(x 2)(x 6)
x 6
x6
(x 2)(x 3)
x3
x3
Notice in the last line that we were able to simplify the answer by factoring the
numerator and reducing the fraction. (This is why we prefer to leave the least
common denominator in factored form, rather than multiplying it out, in this type
of problem.)
The next four examples illustrate using the least common denominator to
simplify compound fractions, or complex fractions, which are fractions “within”
fractions.
B-26
Appendix B
EXAMPLE
5 Simplifying a Compound Fraction
1
1
3a
4b
.
Simplify:
5
1
b
6a2
SOLUTION
The least common denominator for the four individual denominators 3a, 4b, 6a2, and
b is 12a2b. Multiplying the given expression by 12a2b12a2b, which equals 1, yields
1
1
12a2b 3a
4b
4ab 3a2
#
1
12a2b 5
10b 12a2
2 b
6a
EXAMPLE
6 Simplifying a Fraction Containing Negative Exponents
Simplify: (x 1 y 1)1. (The answer is not x y.)
SOLUTION
After applying the definition of negative exponent to rewrite the given expression,
we’ll use the method shown in Example 5.
1
1 1
1
b x
y
1
1
x
y
xy
xy
1
#
xy 1
yx
1
x
y
(x1 y1)1 a
EXAMPLE
multiplying by 1 xy
xy
7 Simplifying a Compound Fraction
1
1
x
xh
. (This type of expression occurs in calculus.)
Simplify:
h
SOLUTION
1
1
1
1
(x h)x x h x
xh x
#
h
(x h)x
h
EXAMPLE
x (x h)
h
1
(x h)xh
(x h)xh
x(x h)
8 Simplifying a Compound Fraction
x
Simplify:
1
x2
1
1
x2
.
multiplying by 1 (x h)x
(x h)x
B.5 Review of Fractional Expressions
SOLUTION
1
x 2
x2
x2
x
2#
multiplying by 1 2
1
x 1
x
1
1
x2
x2
x
1
x2
x3 1
x3 1
1 x2
x2 1
EXERCISE SET
25 x2
x5
x2 x 20
4.
2x2 7x 4
ab
6. 2
ax bx2
x2 9
x3
x2
3. 4
x 16
x2 2x 4
5.
x3 8
9.
11.
12.
2.
9ab 12b2
6a2 8ab
8.
a3 a2 a 1
a2 1
3
x y3
10.
a3b2 27b5
(ab 3b2)2
(x y)2(a b)
(x y2)(a2 2ab b2)
2
(x y)3
x4 y4
(x4y x2y3 x3y2 xy4)(x y)2
In Exercises 13–55, carry out the indicated operations and
simplify where possible.
13.
15.
16.
17.
1
3
x3
x2
3x
6
25.
2
x2
x 4
4
4
x4
x1
1
1
26. 1 2
x
x
23.
In Exercises 1–12, reduce the fractions to lowest terms.
7.
(x 1)(x2 x 1)
x2 x 1
(x 1)(x 1)
x1
B.5
A
1.
B-27
ax 3
a2x2 3ax
2 # x2 4
14.
x2 x2
2a 1
4a2 1
2
2
x x 2 # x 3x
x2 x 12 x2 4x 4
3t 2 2tx x 2
(3t 2 4tx x 2) t2 x2
3
3
x y
(x y)3
x2 4xy 3y2
x2 2xy 3y2
a2 a 42
a2 49
18.
a4 216a
a3 6a2 36a
2
1
4
1
1
19. 2
20.
2
x
3x
x
5x
30x3
1
6
a
1
1
21. 22. a
a
c
6
b
24.
27.
a
2ax
3ax2
2
x1
(x 1)
(x 1)3
28.
2a
a2 5a 4
2
2
a 16
2a 8a
29.
4
4
x5
5x
30.
x
a
ax
xa
b
a
a2 b2
2
2
ab
ba
a b
5
1
3
32.
2
2x 2
x1
x 1
1
1
33. 2
2
x x 20
x 8x 16
1
1
4
34. 2
2
2x 1
6x 5x 4
3x 4x
31.
35.
36.
37.
40.
43.
2q p
2p 9pq 5q
2
2
pq
p 5pq
2
1
1
1
2
3
x1
x 1
x 1
1
4
1
a
x
a
38.
1
2
1
1
x
a
1
1
1
a
a
a
b
41.
1
1
1
1
a
a
b
3
1
1
3
2
2
2h
2
x h
x
44.
h
h
1
1
x
a
39.
xa
1
1
2
x2
y
42.
1
1
x
y
a
x
2
2
x
a
45. 2
a ax x2
B-28
Appendix B
x
46.
xy
yx
2
y
x2 y2
47. (x 1 2)1
1
48.
C
(x2 2x)1
x2
aa 1 a
1 b
b aa b
b
b
55.
1 a
1 b
ab b ab b
a
a
56. Consider the three fractions
1
1 1
50. (a1 a2)1
b
a1
a2
51. x(x y)1 y(x y)1
52. x(2x 2y)1 y(2y 2x)1
49. a
B
a
a
53.
a
a
bc
,
1 bc
ab
1 ab
PROPERTIES OF THE REAL NUMBERS
B.6
PROPERTY SUMMARY
and
(a) If a 1, b 2, and c 3, find the sum of the three
fractions. Also compute their product. What do you
observe?
(b) Show that the sum and the product of the three given
fractions are, in fact, always equal.
b
ab
1x a
1x a
b
a b # ab3 a3b
54.
b
a b a2 b2
1x a
1x a
b
ab
Today’s familiar plus and minus signs
were first used in 15th-century Germany
as warehouse marks. They indicated
when a container held something that
weighed over or under a certain
standard weight. —Martin Gardner in
“Mathematical Games” (Scientific
American, June 1977)
ca
,
1 ca
I do not like as a symbol for multiplication, as it is easily confounded with x; . . .
often I simply relate two quantities by an interposed dot. . . . —G. W. Leibniz in a letter
dated July 29, 1698
In this section we will first list the basic properties for the real number system. Then
we will use those properties to prove the familiar rules of algebra for working with
signed numbers.
The set of real numbers is closed with respect to the operations of addition
and multiplication. This means that when we add or multiply two real numbers, the
result (that is, the sum or the product) is again a real number. Some of the other most
basic properties and definitions for the real-number system are listed in the following
box. In the box, the lowercase letters a, b, and c denote arbitrary real numbers.
Some Fundamental Properties of the Real Numbers
Commutative properties
abba
ab ba
Associative properties
a (b c) (a b) c
Identity properties
1. There is a unique real number 0 (called zero or the additive identity) such that
a(bc) (ab)c
a0a
and
0aa
2. There is a unique real number 1 (called one or the multiplicative identity) such that
a#1a
and
1#aa
Inverse properties
1. For each real number a there is a real number a (called the additive inverse of a
or the opposite of a) such that
a (a) 0
and
(a) a 0
1
(or 1a or a1)
a
and called the multiplicative inverse or reciprocal of a, such that
2. For each real number a 0 there is a real number denoted by
a#
Distributive properties
a(b c) ab ac
1
1
a
and
(b c)a ba ca
1
#a1
a
B.6 Properties of the Real Numbers
B-29
On reading this list of properties for the first time, many students ask the natural
question, “Why do we even bother to list such obvious properties?” One reason is
that all the other laws of arithmetic and algebra (including the not-so-obvious ones)
can be derived from our rather short list. For example, the rule 0 # a 0 can be
proved by using the distributive property, as can the rule that the product of two negative numbers is a positive number. We now state and prove those properties and several others.
Theorem
Let a and b be real numbers. Then
(a) a # 0 0
(c) (a) a
(b) a (1)a
(d) a(b) ab
(e) (a)(b) ab
Proof of Part (a)
a # 0 a # (0 0)
additive identity property
a#0a#0a#0
distributive property
Now since a # 0 is a real number, it has an additive inverse, (a # 0). Adding this to
both sides of the last equation, we obtain
a # 0 [(a # 0)] (a # 0 a # 0) [(a # 0)]
a # 0 [(a # 0)] a # 0 5a # 0 [(a # 0)]6
associative property of
addition
0a#00
additive inverse property
0a#0
additive identity property
Thus a # 0 0, as we wished to show.
Proof of Part (b)
00#a
using part (a) and the commutative
of multiplication
[1 (1)]a
additive inverse property
1 # a (1)a
distributive property
a (1)a
multiplicative identity property
Now, by adding a to both sides of this last equation, we obtain
a 0 a [a (1)a]
a (a a) (1)a
a 0 (1)a
a (1)a
additive identity property and
associative property of addition
additive inverse property
additive identity property
This last equation asserts that a (1)a, as we wished to show.
B-30
Appendix B
Proof of Part (c)
(a) (a) 0
additive inverse property
By adding a to both sides of this last equation, we obtain
[(a) (a)] a 0 a
(a) (a a) a
associative property of addition
and additive identity property
(a) 0 a
additive inverse property
(a) a
additive identity property
This last equation states that (a) a, as we wished to show.
Proof of Part (d)
a(b) a[(1)b]
using part (b)
[a(1)]b
associative property of multiplication
[(1)a]b
commutative property of multiplication
(1)(ab)
associative property of multiplication
(ab)
using part (b)
Thus a(b) ab, as we wished to show.
Proof of Part (e)
(a)(b) [(a)b]
using part (d)
[b(a)]
commutative property of multiplication
[(ba)]
using part (d)
ba
using part (c)
ab
commutative property of multiplication
We’ve now shown that (a)(b) ab, as required.