C. Addition and Subtraction of Rational Expressions
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EXAMPLE 5
ᮣ
Adding and Subtracting Rational Expressions
Compute as indicated:
7
3
a.
ϩ
10x
25x2
Solution
ᮣ
57
b.
5
10x
Ϫ
xϪ3
x Ϫ9
2
a. The LCM for 10x and 25x2 is 50x2.
3
3 2
7
7 # 5x
#
ϩ
ϩ
ϭ
2
10x
10x 5x
25x
25x2 2
35x
6
ϭ
ϩ
2
50x
50x2
35x ϩ 6
ϭ
50x2
find the LCD
write equivalent expressions
simplify
add the numerators and write the result over the LCD
The result is in simplest form.
b. The LCM for x2 Ϫ 9 and x Ϫ 3 is 1x Ϫ 32 1x ϩ 32.
5
5
10x
10x
#xϩ3
Ϫ
Ϫ
ϭ
2
xϪ3
xϪ3 xϩ3
1x Ϫ 321x ϩ 32
x Ϫ9
10x Ϫ 51x ϩ 32
ϭ
1x Ϫ 321x ϩ 32
10x Ϫ 5x Ϫ 15
ϭ
1x Ϫ 321x ϩ 32
5x Ϫ 15
ϭ
1x Ϫ 32 1x ϩ 32
find the LCD
write equivalent expressions
subtract numerators, write
the result over the LCD
distribute
combine like terms
1
ϭ
51x Ϫ 32
1x Ϫ 321x ϩ 32
ϭ
5
xϩ3
factor and reduce
Now try Exercises 43 through 48
EXAMPLE 6
ᮣ
ᮣ
Adding and Subtracting Rational Expressions
Perform the operations indicated:
nϪ3
c
5
b2
a.
b.
Ϫ 2
Ϫ
2
a
nϩ2
n Ϫ4
4a
Solution
ᮣ
a. The LCM for n ϩ 2 and n2 Ϫ 4 is 1n ϩ 22 1n Ϫ 22.
5
nϪ3
nϪ3
5
#nϪ2Ϫ
ϭ
Ϫ 2
nϩ2
1n ϩ 22 n Ϫ 2
1n ϩ 22 1n Ϫ 22
n Ϫ4
51n Ϫ 22 Ϫ 1n Ϫ 32
ϭ
1n ϩ 221n Ϫ 22
5n Ϫ 10 Ϫ n ϩ 3
ϭ
1n ϩ 221n Ϫ 22
4n Ϫ 7
ϭ
1n ϩ 221n Ϫ 22
write equivalent
expressions
subtract numerators, write
the result over the LCD
distribute
result
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b. The LCM for a and 4a2 is 4a2:
b2
c
c 4a
b2
Ϫ
Ϫ #
ϭ
a
a 4a
4a2
4a2
b2
4ac
ϭ 2Ϫ 2
4a
4a
b2 Ϫ 4ac
ϭ
4a2
write equivalent
expressions
simplify
subtract numerators, write
the result over the LCD
Now try Exercises 49 through 64
ᮣ
CAUTION
ᮣ
When the second term in a subtraction has a binomial numerator as in Example 6a, be
sure the subtraction is applied to both terms. It is a common error to write
51n Ϫ 22
nϪ3
5n Ϫ 10 Ϫ n Ϫ 3 X
in which the subtraction is applied
Ϫ
ϭ
1n ϩ 221n Ϫ 22
1n ϩ 221n Ϫ 22
1n ϩ 221n Ϫ 22
to the first term only. This is incorrect!
C. You’ve just seen how
we can add and subtract
rational expressions
D. Simplifying Compound Fractions
Rational expressions whose numerator or denominator contain a fraction are called
3
2
Ϫ
3m
2
compound fractions. The expression
is a compound fraction with a
3
1
Ϫ
4m
3m2
2
3
3
1
numerator of
Ϫ and a denominator of
Ϫ
. The two methods commonly
3m
2
4m
3m2
used to simplify compound fractions are summarized in the following boxes.
Simplifying Compound Fractions (Method I)
1. Add/subtract fractions in the numerator, writing them as a
single expression.
2. Add/subtract fractions in the denominator, also writing them
as a single expression.
3. Multiply the numerator by the reciprocal of the denominator
and simplify if possible.
Simplifying Compound Fractions (Method II)
1. Find the LCD of all fractions in the numerator and denominator.
2. Multiply the numerator and denominator by this LCD and simplify.
3. Simplify further if possible.
Method II is illustrated in Example 7.
EXAMPLE 7
ᮣ
Simplifying a Compound Fraction
Simplify the compound fraction:
2
3
Ϫ
3m
2
3
1
Ϫ
4m
3m2
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Section R.5 Rational Expressions and Equations
Solution
ᮣ
The LCD for all fractions is 12m2.
2
3
3 12m2
2
Ϫ
Ϫ ba
b
a
3m
2
3m
2
1
ϭ
1
1
3
3
12m2
Ϫ
Ϫ
b
a
ba
4m
4m
1
3m2
3m2
2
12m2
3 12m2
a ba
b Ϫ a ba
b
3m
1
2
1
ϭ
12m2
1
3
12m2
b Ϫ a 2 ba
b
a ba
4m
1
1
3m
8m Ϫ 18m2
ϭ
9m Ϫ 4
multiply numerator and
denominator by 12m 2 ϭ
59
12m 2
1
distribute
simplify
Ϫ1
ϭ
2m14 Ϫ 9m2
9m Ϫ 4
ϭ Ϫ2m
D. You’ve just seen how
we can simplify compound
fractions
factor and write in lowest terms
Now try Exercises 65 through 74
ᮣ
E. Solving Rational Equations
In Section R.3 we solved linear equations using basic properties of equality. If any
equation contained fractional terms, we “cleared the fractions” using the least common
denominator (LCD). We can also use this idea to solve rational equations, or equations that contain rational expressions.
Solving Rational Equations
1. Identify and exclude any values that cause
a zero denominator.
2. Multiply both sides by the LCD and simplify
(this will eliminate all denominators).
3. Solve the resulting equation.
4. Check all solutions in the original equation.
EXAMPLE 8
ᮣ
Solving a Rational Equation
Solve for m:
Solution
ᮣ
4
1
2
.
ϭ 2
Ϫ
m
mϪ1
m Ϫm
Since m2 Ϫ m ϭ m1m Ϫ 12, the LCD is m1m Ϫ 12, where m 0 and m 1.
4
1
2
d multiply by LCD
b ϭ m1m Ϫ 12 c
m1m Ϫ 12a Ϫ
m
mϪ1
m1m Ϫ 12
m1m Ϫ 12 2
m1m Ϫ 12
m1m Ϫ 12
1
4
a bϪ
a
bϭ
a
b distribute and simplify
m
1
1
mϪ1
1
m1m Ϫ 12
denominators are eliminated
21m Ϫ 12 Ϫ m ϭ 4
2m Ϫ 2 Ϫ m ϭ 4
distribute
mϭ6
solve for m
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Checking by substitution we have:
2
4
1
ϭ 2
original equation
Ϫ
m
mϪ1
m Ϫm
1
4
2
Ϫ
ϭ
substitute 6 for m
2
162
162 Ϫ 1
162 Ϫ 162
1
4
1
Ϫ ϭ
simplify
3
5
30
3
2
5
Ϫ
ϭ
common denominator
15
15
15
2
2
✓
result
ϭ
15
15
A calculator check is shown in the figure.
Now try Exercises 75 through 80 ᮣ
Multiplying both sides of an equation by a variable sometimes introduces a solution that satisfies the resulting equation, but not the original equation—the one we’re
trying to solve. Such “solutions” are called extraneous roots and illustrate the need to
check all apparent solutions in the original equation. In the case of rational equations,
we are particularly aware that any value that causes a zero denominator is outside the
domain and cannot be a solution.
EXAMPLE 9
ᮣ
Solving a Rational Equation
Solve: x ϩ
Solution
ᮣ
4x
12
ϭ1ϩ
.
xϪ3
xϪ3
The LCD is x Ϫ 3, where x 3.
multiply both
4x
12
b ϭ 1x Ϫ 32a1 ϩ
b
1x Ϫ 32ax ϩ
sides by LCD
xϪ3
xϪ3
12
xϪ3
4x
xϪ3
distribute and
ba
ba
b ϭ 1x Ϫ 32 112 ϩ a
b simplify
1x Ϫ 32x ϩ a
1
xϪ3
1
xϪ3
x2 Ϫ 3x ϩ 12 ϭ x Ϫ 3 ϩ 4x denominators are eliminated
set equation equal to zero
x2 Ϫ 8x ϩ 15 ϭ 0
factor
1x Ϫ 321x Ϫ 52 ϭ 0
zero factor property
x ϭ 3 or x ϭ 5
Checking shows x ϭ 3 is an extraneous root, and x ϭ 5 is the only valid solution.
E. You’ve just seen how we
can solve rational equations
Now try Exercises 81 through 86 ᮣ
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Section R.5 Rational Expressions and Equations
61
R.5 EXERCISES
ᮣ
CONCEPTS AND VOCABULARY
Fill in each blank with the appropriate word or phrase. Carefully reread the section, if necessary.
1. In simplest form, 1a Ϫ b2/1a Ϫ b2 is equal to
, while 1a Ϫ b2/1b Ϫ a2 is equal to
3. As with numeric fractions, algebraic fractions
require a
for addition and
subtraction.
.
2. A rational expression is in
when the
numerator and denominator have no common
factors, other than
.
4. Since x2 ϩ 9 is prime, the expression 1x2 ϩ 92/
1x ϩ 32 is already written in
.
State T or F and discuss/explain your response.
5.
ᮣ
x
xϩ1
1
Ϫ
ϭ
xϩ3
xϩ3
xϩ3
6.
1x ϩ 321x Ϫ 22
1x Ϫ 221x ϩ 32
ϭ0
DEVELOPING YOUR SKILLS
Reduce to lowest terms.
7. a.
8. a.
aϪ7
Ϫ3a ϩ 21
b.
xϪ4
Ϫ7x ϩ 28
b.
x2 Ϫ 5x Ϫ 14
9. a. 2
x ϩ 6x Ϫ 7
2x ϩ 6
4x2 Ϫ 8x
3x Ϫ 18
6x2 Ϫ 12x
a2 ϩ 3a Ϫ 28
b.
a2 Ϫ 49
16. a.
c.
x3 ϩ 4x2 Ϫ 5x
x3 Ϫ x
12y2 Ϫ 13y ϩ 3
27y3 Ϫ 1
b.
d.
5p2 Ϫ 14p Ϫ 3
5p2 ϩ 11p ϩ 2
ax2 Ϫ 5x2 Ϫ 3a ϩ 15
ax Ϫ 5x ϩ 5a Ϫ 25
Compute as indicated. Write final results in lowest terms.
17.
a2 Ϫ 4a ϩ 4 a2 Ϫ 2a Ϫ 3
#
a2 Ϫ 9
a2 Ϫ 4
10. a.
r2 ϩ 3r Ϫ 10
r2 ϩ r Ϫ 6
b.
m2 ϩ 3m Ϫ 4
m2 Ϫ 4m
18.
11. a.
xϪ7
7Ϫx
b.
5Ϫx
xϪ5
b2 ϩ 5b Ϫ 24
# 2 b
2
b Ϫ 6b ϩ 9 b Ϫ 64
19.
12. a.
v2 Ϫ 3v Ϫ 28
49 Ϫ v2
b.
u2 Ϫ 10u ϩ 25
25 Ϫ u2
x2 Ϫ 7x Ϫ 18 # 2x2 ϩ 7x ϩ 3
x2 Ϫ 6x Ϫ 27 2x2 ϩ 5x ϩ 2
20.
13. a.
Ϫ12a3b5
4a2bϪ4
b.
7x ϩ 21
63
6v2 ϩ 23v ϩ 21 4v2 Ϫ 25
#
3v ϩ 7
4v2 Ϫ 4v Ϫ 15
21.
m3n Ϫ m3
d. 4
m Ϫ m4n
22.
a3 Ϫ 4a2
a2 ϩ 3a Ϫ 28
Ϭ 3
2
a ϩ 5a Ϫ 14
a Ϫ8
y2 Ϫ 9
c.
3Ϫy
14. a.
c.
15. a.
c.
p3 Ϫ 64
p3 Ϫ p2
Ϭ
p2 ϩ 4p ϩ 16
p2 Ϫ 5p ϩ 4
5mϪ3n5
Ϫ10mn2
b.
Ϫ5v ϩ 20
25
23.
n2 Ϫ 4
2Ϫn
d.
w4 Ϫ w4v
w3v Ϫ w3
3Ϫx
3x Ϫ 9
Ϭ
4x ϩ 12
5x ϩ 15
24.
2n3 ϩ n2 Ϫ 3n
n3 Ϫ n2
b.
6x2 ϩ x Ϫ 15
4x2 Ϫ 9
2Ϫb
5b Ϫ 10
Ϭ
7b Ϫ 28
5b Ϫ 20
25.
x3 ϩ 8
x2 Ϫ 2x ϩ 4
d.
mn2 ϩ n2 Ϫ 4m Ϫ 4
mn ϩ n ϩ 2m ϩ 2
a2 ϩ a 3a Ϫ 9
#
a2 Ϫ 3a 2a ϩ 2
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26.
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CHAPTER R A Review of Basic Concepts and Skills
2
p2 Ϫ 36
# 2 4p
2p
2p ϩ 12p
Compute as indicated. Write answers in lowest terms
[recall that a ؊ b ؍؊1(b ؊ a)].
8
# 1a2 Ϫ 2a Ϫ 352
27. 2
a Ϫ 25
43.
3
5
ϩ
2
2x
8x
44.
15
7
Ϫ 2
16y
2y
m2 Ϫ 5m
m2 Ϫ m Ϫ 20
45.
7
1
Ϫ
4x2y3
8xy4
46.
5
3
ϩ
6a3b
9ab3
28. 1m2 Ϫ 162
29.
#
xy Ϫ 3x ϩ 2y Ϫ 6
x2 Ϫ 3x Ϫ 10
xy Ϫ 3x
xy Ϫ 5y
Ϭ
47.
4p
p Ϫ 36
2
Ϫ
2
pϪ6
48.
3q
q Ϫ 49
2
Ϫ
3
2q Ϫ 14
30.
ab Ϫ 2a
2a Ϫ ab ϩ 7b Ϫ 14
Ϭ
2
ab Ϫ 7a
b Ϫ 14b ϩ 49
49.
4
m
ϩ
4Ϫm
m Ϫ 16
50.
p
2
ϩ
2
pϪ2
4Ϫp
31.
m2 Ϫ 16
m2 ϩ 2m Ϫ 8
Ϭ
m2 Ϫ 2m
m2
51.
2
Ϫ5
mϪ7
52.
4
Ϫ9
xϪ1
32.
2x2 Ϫ 18
18 Ϫ 6x
Ϭ 3
2
x Ϫ 25
x Ϫ 2x2 Ϫ 25x ϩ 50
53.
33.
yϩ3
2
#y
3y2 ϩ 9y
ϩ 7y ϩ 12
y2 Ϫ 16
Ϭ
y2 ϩ 4y
2
yϩ1
y ϩ y Ϫ 30
2
Ϫ
2
yϩ6
54.
3
4n
Ϫ
4n Ϫ 20
n Ϫ 5n
y2 Ϫ 4y
2
34.
x2 Ϫ 1 x ϩ 1
x2 ϩ 4x Ϫ 5
#
Ϭ
x2 Ϫ 5x Ϫ 14
x2 Ϫ 4 x ϩ 5
55.
1
a
ϩ 2
aϩ4
a Ϫ a Ϫ 20
35.
x2 Ϫ x ϩ 0.21
x2 Ϫ 0.49
Ϭ
x2 ϩ 0.5x Ϫ 0.14
x2 Ϫ 0.09
56.
xϪ5
2x Ϫ 1
Ϫ 2
x ϩ 3x Ϫ 4
x ϩ 3x Ϫ 4
36.
x2 Ϫ 0.8x ϩ 0.15
x2 Ϫ 0.25
Ϭ
x2 ϩ 0.1x Ϫ 0.2
x2 Ϫ 0.16
57.
4
4
n2 ϩ n ϩ
3
9
37.
Ϭ
13
1
2
n2 Ϫ n ϩ
n2 Ϫ
15
15
25
n2 Ϫ
q2 Ϫ
38.
q2 Ϫ
40.
41.
9
25
1
3
qϪ
10
10
3
17
qϩ
20
20
1
q2 Ϫ
16
q2 ϩ
Ϭ
2
p3 ϩ p2 Ϫ 49p Ϫ 49
p2 ϩ 6p Ϫ 7
42. a
4x Ϫ 25
2x Ϫ x Ϫ 15 # 4x ϩ 25x Ϫ 21
Ϭ 2
b
x Ϫ 11x ϩ 30
x Ϫ 9x ϩ 18
12x2 Ϫ 5x Ϫ 3
2
2
y ϩ 2y ϩ 1
mϪ5
2
ϩ 2
m Ϫ9
m ϩ 6m ϩ 9
60.
mϩ6
mϩ2
Ϫ 2
2
m Ϫ 25
m Ϫ 10m ϩ 25
2
yϩ2
5y ϩ 11y ϩ 2
2
ϩ
5
y ϩyϪ6
2
m
mϪ4
ϩ
2
3m Ϫ 11m ϩ 6
2m Ϫ m Ϫ 15
2
Write each term as a rational expression. Then compute
the sum or difference indicated.
p3 Ϫ 1
4n2 Ϫ 1
6n2 ϩ 5n ϩ 1 # 12n2 Ϫ 17n ϩ 6
#
12n2 Ϫ 5n Ϫ 3
2n2 ϩ n
6n2 Ϫ 7n ϩ 2
2
2y Ϫ 5
2
59.
62.
p2 ϩ p ϩ 1
y ϩ 2y ϩ 1
Ϫ
Ϫ2
7
Ϫ 2
3a ϩ 12
a ϩ 4a
2
Ϭ
3y Ϫ 4
2
58.
61.
6a Ϫ 24
3a Ϫ 24a Ϫ 12a ϩ 96
Ϭ 3
2
a Ϫ 11a ϩ 24
3a Ϫ 81
3
39.
4
9
2
2
63. a. pϪ2 Ϫ 5pϪ1
64. a. 3aϪ1 ϩ 12a2 Ϫ1
b. xϪ2 ϩ 2xϪ3
b. 2yϪ1 Ϫ 13y2 Ϫ1
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Simplify each compound fraction. Use either method.
5
1
Ϫ
a
4
65.
25
1
Ϫ
2
16
a
1
8
Ϫ
3
27
x
66.
2
1
Ϫ
x
3
1
pϪ2
67.
1
1ϩ
pϪ2
3
yϪ6
68.
9
yϩ
yϪ6
3
2
ϩ
3Ϫx
xϪ3
69.
4
5
ϩ
x
xϪ3
2
1
Ϫ
yϪ5
5Ϫy
70.
3
2
Ϫ
y
yϪ5
2
y Ϫ y Ϫ 20
71.
3
4
Ϫ
yϩ4
yϪ5
2
x Ϫ 3x Ϫ 10
72.
6
4
Ϫ
xϩ2
xϪ5
pϩ
2
1ϩ
73. a.
74. a.
1 ϩ 3mϪ1
1 Ϫ 3mϪ1
b.
4 Ϫ 9aϪ2
3aϪ2
b.
Solve each equation. Identify any extraneous roots.
75.
2
5
1
ϭ 2
ϩ
x
xϩ1
x ϩx
76.
1
5
3
ϭ
Ϫ 2
m
mϩ3
m ϩ 3m
77.
3
21
ϭ
aϩ2
aϪ1
78.
4
7
ϭ
2y Ϫ 3
3y Ϫ 5
79.
1
1
1
Ϫ
ϭ 2
3y
4y
y
81. x ϩ
80.
1
1
3
Ϫ
ϭ 2
5x
2x
x
2x
14
ϭ1ϩ
xϪ7
xϪ7
82.
2x
10
ϩxϭ1ϩ
xϪ5
xϪ5
83.
5
20
6
ϭ
ϩ 2
nϩ3
nϪ2
n ϩnϪ6
84.
1 ϩ 2xϪ2
1 Ϫ 2xϪ2
2
1
7
ϭϪ
Ϫ 2
pϩ2
p
ϩ
3
p ϩ 5p ϩ 6
85.
3 ϩ 2nϪ1
5nϪ2
3
2a2 ϩ 5
a
ϭ
Ϫ 2
2a ϩ 1
aϪ3
2a Ϫ 5a Ϫ 3
86.
Ϫ18
3n
4n
ϩ
ϭ
2n Ϫ 1
3n ϩ 1
6n Ϫ n Ϫ 1
2
Rewrite each expression as a compound fraction. Then
simplify using either method.
ᮣ
63
Section R.5 Rational Expressions and Equations
2
WORKING WITH FORMULAS
87. Cost to seize illegal drugs: C ؍
450P
100 ؊ P
The cost C, in millions of
450P
P
dollars, for a government to find
100 ؊ P
and seize P% 10 Յ P 6 1002 of
40
a certain illegal drug is modeled
60
by the rational equation shown.
80
Complete the table (round to the
nearest dollar) and answer the
90
following questions.
93
a. What is the cost of seizing
95
40% of the drugs? Estimate
98
the cost at 85%.
100
b. Why does cost increase
dramatically the closer you get to 100%?
c. Will 100% of the drugs ever be seized?
88. Chemicals in the bloodstream: C ؍
200H2
H3 ؉ 40
Rational equations are often used
200H2
to model chemical concentrations H
H3 ؉ 40
in the bloodstream. The percent
0
concentration C of a certain drug
1
H hours after injection into
muscle tissue can be modeled by
2
the equation shown (H Ն 0).
3
Complete the table (round to the
4
nearest tenth of a percent) and
5
answer the following questions.
6
a. What is the percent
7
concentration of the drug
3 hr after injection?
b. Why is the concentration virtually equal at
H ϭ 4 and H ϭ 5?
c. Why does the concentration begin to decrease?
d. How long will it take for the concentration to
become less than 10%?
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ᮣ
APPLICATIONS
89. Stock prices: When a hot new stock hits the market,
its price will often rise dramatically and then taper
5017d 2 ϩ 102
off over time. The equation P ϭ
d3 ϩ 50
models the price of stock XYZ d days after it has
“hit the market.” (a) Create a table of values showing
the price of the stock for the first 10 days (rounded to
the nearest dollar) and comment on what you notice.
(b) Find the opening price of the stock. (c) Does the
stock ever return to its original price?
90. Population growth: The Department of Wildlife
introduces 60 elk into a new game reserve. It is
projected that the size of the herd will grow
1016 ϩ 3t2
, where
according to the equation N ϭ
1 ϩ 0.05t
N is the number of elk and t is the time in years.
(a) Approximate the population of elk after 14 yr.
(b) If recent counts find 225 elk, approximately
how many years have passed?
91. Typing speed: The number of words per minute
that a beginner can type is approximated by the
60t Ϫ 120
, where N is the number
equation N ϭ
t
ᮣ
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CHAPTER R A Review of Basic Concepts and Skills
of words per minute after t weeks, 3 6 t 6 12.
Use a table to determine how many weeks it takes
for a student to be typing an average of forty-five
words per minute.
92. Memory retention: A group of students is asked
to memorize 50 Russian words that are unfamiliar
to them. The number N of these words that the
average student remembers D days later is modeled
5D ϩ 35
1D Ն 12. How many
by the equation N ϭ
D
words are remembered after (a) 1 day? (b) 5 days?
(c) 12 days? (d) 35 days? (e) 100 days? According
to this model, is there a certain number of words
that the average student never forgets?
How many?
93. Pollution removal: For a steel mill, the cost C (in
millions of dollars) to remove toxins from the
22P
resulting sludge is given by C ϭ
, where
100 Ϫ P
P is the percent of the toxins removed. What percent
can be removed if the mill spends $88,000,000 on
the cleanup? Round to tenths of a percent.
EXTENDING THE CONCEPT
94. One of these expressions is not equal to the others.
Identify which and explain why.
20n
a.
b. 20 # n Ϭ 10 # n
10n
20 n
1
#
c. 20n #
d.
10n
10 n
95. The average of A and B is x. The average of C, D,
and E is y. The average of A, B, C, D, and E is:
3x ϩ 2y
2x ϩ 3y
a.
b.
5
5
21x ϩ y2
31x ϩ y2
c.
d.
5
5
3
2
and , what is the
5
4
reciprocal of the sum of their reciprocals? Given
c
a
that and are any two numbers—what is the
b
d
reciprocal of the sum of their reciprocals?
96. Given the rational numbers
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College Algebra Graphs & Models—
R.6
Radicals, Rational Exponents, and Radical Equations
LEARNING OBJECTIVES
In Section R.6 you will review how to:
A. Simplify radical
B.
C.
D.
E.
F.
expressions of the form
n
1a n
Rewrite and simplify
radical expressions using
rational exponents
Use properties of radicals
to simplify radical
expressions
Add and subtract radical
expressions
Multiply and divide
radical expressions; write
a radical expression in
simplest form
Solve equations and use
formulas involving
radicals
Square roots and cube roots come from a much larger family called radical expressions. Expressions containing radicals can be found in virtually every field of mathematical study, and are an invaluable tool for modeling many real-world phenomena.
n
A. Simplifying Radical Expressions of the Form 1an
In previous coursework, you likely noted that 1a ϭ b only if b2 ϭ a. This definition
cannot be applied to expressions like 1Ϫ16, since there is no number b such that
b2 ϭ Ϫ16. In other words, the expression 1a represents a real number only if a Ն 0
(for a full review of the real numbers and other sets of numbers, see Appendix I at
www.mhhe.com/coburn). Of particular interest to us now is an inverse operation for a2.
In other words, what operation can be applied to a2 to return a? Consider the following.
EXAMPLE 1
ᮣ
Evaluating a Radical Expression
Evaluate 2a2 for the values given:
a. a ϭ 3
b. a ϭ 5
c. a ϭ Ϫ6
Solution
ᮣ
a. 232 ϭ 19
ϭ3
b. 252 ϭ 125
ϭ5
c. 21Ϫ62 2 ϭ 136
ϭ6
Now try Exercises 7 and 8
ᮣ
The pattern seemed to indicate that 2a2 ϭ a and that our search for an inverse
operation was complete—until Example 1(c), where we found that 21Ϫ62 2 Ϫ6.
Using the absolute value concept, we can “repair” this apparent discrepancy and state a
general rule for simplifying these expressions: 2a2 ϭ ͿaͿ. For expressions like 249x2
and 2y6, the radicands can be rewritten as perfect squares and simplified in the same
manner: 249x2 ϭ 217x2 2 ϭ 7ͿxͿ and 2y6 ϭ 21y3 2 2 ϭ Ϳy3Ϳ.
The Square Root of a2: 2a2
For any real number a,
2a2 ͦ ؍aͦ.
EXAMPLE 2
ᮣ
Simplifying Square Root Expressions
Simplify each expression.
a. 2169x2
Solution
ᮣ
b. 2x2 Ϫ 10x ϩ 25
a. 2169x2 ϭ Ϳ13xͿ
ϭ 13ͿxͿ
b. 2x Ϫ 10x ϩ 25 ϭ 21x Ϫ 52
ϭ Ϳx Ϫ 5Ϳ
2
since x could be negative
2
since x ؊ 5 could be negative
Now try Exercises 9 and 10
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ᮣ
65
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CHAPTER R A Review of Basic Concepts and Skills
ᮣ
CAUTION
In Section R.2, we noted that 1A ϩ B2 2 A2 ϩ B2, indicating that you cannot square
the individual terms in a sum (the square of a binomial results in a perfect square
trinomial). In a similar way, 2A2 ϩ B2 A ϩ B, and you cannot take the square root
of individual terms. There is a big difference between the expressions 2A2 ϩ B2 and
21A ϩ B2 2 ϭ ͿA ϩ BͿ. Try evaluating each when A ϭ 3 and B ϭ 4.
3 3
3
3
To investigate expressions like 2
x , note the radicand in both 1
8 and 1
Ϫ64 can
be written as a perfect cube. From our earlier definition of cube roots we know
3
3
3
3
1
8ϭ 2
122 3 ϭ 2, 1
Ϫ64 ϭ 2
1Ϫ42 3 ϭ Ϫ4, and that every real number has only
one real cube root. For this reason, absolute value notation is not used or needed when
taking cube roots.
3 3
The Cube Root of a3: 2
a
For any real number a,
3 3
2
a ؍a.
EXAMPLE 3
ᮣ
Simplifying Cube Root Expressions
Simplify each expression.
3
3
a. 2
b. 2
Ϫ27x3
Ϫ64n6
Solution
ᮣ
3
3
a. 2
Ϫ27x3 ϭ 2
1Ϫ3x2 3
ϭ Ϫ3x
3
3
b. 2
Ϫ64n6 ϭ 2
1Ϫ4n2 2 3
ϭ Ϫ4n2
Now try Exercises 11 and 12
ᮣ
We can extend these ideas to fourth roots, fifth roots, and so on. For example, the
5
fifth root of a is b only if b5 ϭ a. In symbols, 1a ϭ b implies b5 ϭ a. Since an odd
number of negative factors is always negative: 1Ϫ22 5 ϭ Ϫ32, and an even number of
negative factors is always positive: 1Ϫ22 4 ϭ 16, we must take the index into account
n
when evaluating expressions like 1an. If n is even and the radicand is unknown,
absolute value notation must be used.
n
WORTHY OF NOTE
2
Just as 1
Ϫ16 is not a real number,
4
6
1 Ϫ16 and 1
Ϫ16 do not represent
real numbers. An even number of
repeated factors is always positive!
EXAMPLE 4
ᮣ
The nth Root of an: 2an
For any real number a,
n
1. 1an ϭ ͿaͿ when n is even.
Simplifying Radical Expressions
Simplify each expression.
4
4
a. 1
b. 1
81
Ϫ81
4
5
4
e. 216m
f. 232p5
Solution
A. You’ve just seen how
we can simplify radical
n
expressions of the form 1a n
ᮣ
n
2. 1an ϭ a when n is odd.
4
1
81 ϭ 3
5
132 ϭ 2
4
4
2
16m4 ϭ 2
12m2 4
ϭ Ϳ2mͿ or 2ͿmͿ
6
g. 2
1m ϩ 52 6 ϭ Ϳm ϩ 5Ϳ
a.
c.
e.
5
c. 1
32
6
g. 2 1m ϩ 52 6
5
d. 1
Ϫ32
7
h. 2 1x Ϫ 22 7
4
b. 1
Ϫ81 is not a real number
5
d.
1
Ϫ32 ϭ Ϫ2
5
5
f.
232p5 ϭ 2 12p2 5
ϭ 2p
7
h. 2
1x Ϫ 22 7 ϭ x Ϫ 2
Now try Exercises 13 and 14
ᮣ