Coburn J.W. Algebra and Trigonometry
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50 CHAPTER R A Review of Basic Concepts and Skills R-50
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:
Solution
The LCD for all fractions is
distribute
simplify
factor and write in lowest terms
Now try Exercises 65 through 74
E. Rewriting Formulas and Algebraic Models
In many fields of study, formulas and algebraic models involve rational expressions
and we often need to write them in an alternative form.
EXAMPLE 8
Rewriting a Formula
In an electrical circuit with two resistors in parallel, the total resistance R is related
to resistors and by the formula Rewrite the right-hand side as
a single term.
Solution
LCD for the right-hand side is
build equivalent expressions using LCD
write as a single expression
Now try Exercises 75 and 76
R
2
R
1
R
1
R
2
R
2
R
1
R
2
R
1
R
1
R
2
R
1
R
2
1
R
1
R
1
1
R
2
1
R
1
R
1
1
R
2
.R
2
R
1
2m14
1
9m2
9m 4
2m
8m 18m
2
9m 4
a
2
3m
b a
12m
2
1
b a
3
2
b a
12m
2
1
b
a
3
4m
b a
12m
2
1
b a
1
3m
2
b a
12m
2
1
b
2
3m
3
2
3
4m
1
3m
2
a
2
3m
3
2
b a
12m
2
1
b
a
3
4m
1
3m
2
b a
12m
2
1
b
12m
2
.
2
3m
3
2
3
4m
1
3m
2
multiply numerator and
denominator by 12m
2
12m
2
1
D. You’ve just reviewed
how to simplify compound
fractions
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R-51 Section R.5 Rational Expressions 51
EXAMPLE 9
Simplifying an Algebraic Model
When studying rational expressions and rates of change, we encounter the
expression Simplify the compound fraction.
Solution
Using Method I gives:
LCD for the numerator is x (x h)
write numerator as a single expression
simplify
invert and multiply
result
Now try Exercises 77 through 80
1
x1x h2
h
x1x h2
#
1
h
h
x1x h2
h
x 1x h2
x1x h2
h
1
x h
1
x
h
x
x1x h2
x h
x1x h2
h
1
x h
1
x
h
.
E. You’ve just reviewed
how to rewrite formulas and
algebraic models
R.5 EXERCISES
CONCEPTS AND VOCABULARY
DEVELOPING YOUR SKILLS
Fill in each blank with the appropriate word or phrase.
Carefully reread the section, if necessary.
1. In simplest form, is equal to ,
while is equal to .
2. A rational expression is in when the
numerator and denominator have no common
factors, other than .
3. As with numeric fractions, algebraic fractions require
a for addition and subtraction.
1a b2/1b a2
1a b2/1a b2
4. Since is prime, the expression
is already written in .
State T or F and discuss/explain your response.
5.
6.
1x 32
1x 22
1x 221x 32
0
x
x 3
x 1
x 3
1
x 3
1x 32
1x
2
92/x
2
9
Reduce to lowest terms.
7. a. b.
8. a. b.
3x 18
6x
2
12x
x 4
7x 28
2x 6
4x
2
8x
a 7
3a 21
9. a. b.
10. a. b.
m
2
3m 4
m
2
4m
r
2
3r 10
r
2
r 6
a
2
3a 28
a
2
49
x
2
5x 14
x
2
6x 7
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11. a. b.
12. a. b.
13. a. b.
c. d.
14. a. b.
c. d.
15. a. b.
c. d.
16. a. b.
c. d.
Compute as indicated. Write final results in lowest terms.
17.
18.
19.
20.
21.
22.
23.
24.
25.
26.
27.
8
a
2
25
#
1a
2
2a 352
p
2
36
2p
#
4p
2
2p
2
12p
a
2
a
a
2
3a
#
3a 9
2a 2
5b 10
7b 28
2 b
5b 20
3x 9
4x 12
3 x
5x 15
a
2
3a 28
a
2
5a 14
a
3
4a
2
a
3
8
p
3
64
p
3
p
2
p
2
4p 16
p
2
5p 4
6v
2
23v 21
4v
2
4v 15
#
4v
2
25
3v 7
x
2
7x 18
x
2
6x 27
#
2x
2
7x 3
2x
2
5x 2
b
2
5b 24
b
2
6b 9
#
b
b
2
64
a
2
4a 4
a
2
9
#
a
2
2a 3
a
2
4
ax
2
5x
2
3a 15
ax 5x 5a 25
12y
2
13y 3
27y
3
1
5p
2
14p 3
5p
2
11p 2
x
3
4x
2
5x
x
3
x
mn
2
n
2
4m 4
mn n 2m 2
x
3
8
x
2
2x 4
6x
2
x 15
4x
2
9
2n
3
n
2
3n
n
3
n
2
w
4
w
4
v
w
3
v w
3
n
2
4
2 n
5v 20
25
5m
3
n
5
10mn
2
m
3
n m
3
m
4
m
4
n
y
2
9
3 y
7x 21
63
12a
3
b
5
4a
2
b
4
u
2
10u 25
25 u
2
v
2
3v 28
49 v
2
5 x
x 5
x 7
7 x
28.
29.
30.
31.
32.
33.
34.
35.
36.
37.
38.
39.
40.
41.
42.
Compute as indicated. Write answers in lowest terms
[recall that a b 1(b a)].
43. 44.
45. 46.
47. 48.
3q
q
2
49
3
2q 14
4p
p
2
36
2
p 6
3
6a
3
b
5
9ab
3
7
4x
2
y
3
1
8xy
4
15
16y
7
2y
2
3
8x
2
5
2x
a
4x
2
25
x
2
11x 30
2x
2
x 15
x
2
9x 18
b
4x
2
25x 21
12x
2
5x 3
4n
2
1
12n
2
5n 3
#
6n
2
5n 1
2n
2
n
#
12n
2
17n 6
6n
2
7n 2
p
3
p
2
49p 49
p
2
6p 7
p
2
p 1
p
3
1
3a
3
24a
2
12a 96
a
2
11a 24
6a
2
24
3a
3
81
q
2
9
25
q
2
1
10
q
3
10
q
2
17
20
q
3
20
q
2
1
16
n
2
4
9
n
2
13
15
n
2
15
n
2
4
3
n
4
9
n
2
1
25
x
2
0.25
x
2
0.1x 0.2
x
2
0.8x 0.15
x
2
0.16
x
2
0.49
x
2
0.5x 0.14
x
2
0.10x 0.21
x
2
0.09
x
2
4x 5
x
2
5x 14
x
2
1
x
2
4
#
x 1
x 5
y 3
3y
2
9y
#
y
2
7y 12
y
2
16
y
2
4y
y
2
4y
18 6x
x
2
25
2x
2
18
x
3
2x
2
25x 50
m
2
2m 8
m
2
2m
m
2
16
m
2
2a ab 7b 14
b
2
14b 49
ab 2a
ab 7a
xy 3x 2y 6
x
2
3x 10
xy 3x
xy 5y
1m
2
162
#
m
2
5m
m
2
m 20
52 CHAPTER R A Review of Basic Concepts and Skills R-52
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49. 50.
51. 52.
53.
54.
55.
56.
57.
58.
59.
60.
61.
62.
Write each term as a rational expression. Then
compute the sum or difference indicated.
63. a. b.
64. a. b.
2y
1
13y2
1
3a
1
12a2
1
x
2
2x
3
p
2
5p
1
m 4
3m
2
11m 6
m
2m
2
m 15
y 2
5y
2
11y 2
5
y
2
y 6
m 2
m
2
25
m 6
m
2
10m 25
2
m
2
9
m 5
m
2
6m 9
2
3a 12
7
a
2
4a
3y 4
y
2
2y 1
2y 5
y
2
2y 1
2x 1
x
2
3x 4
x 5
x
2
3x 4
1
a 4
a
a
2
a 20
4n
n
2
5n
3
4n 20
y 1
y
2
y 30
2
y 6
4
x 1
9
2
m 7
5
2
4 p
2
p
p 2
m
m
2
16
4
4 m
Simplify each compound rational expression. Use
either method.
65. 66.
67. 68.
69. 70.
71. 72.
Rewrite each expression as a compound fraction.
Then simplify using either method.
73. a. b.
74. a. b.
Rewrite each expression as a single term.
75. 76.
77. 78.
79. 80.
a
1x h2
2
a
x
2
h
1
21x h2
2
1
2x
2
h
a
h x
a
x
h
a
x h
a
x
h
1
w
1
x
1
y
1
f
1
1
f
2
3 2n
1
5n
2
4 9a
2
3a
2
1 2x
2
1 2x
2
1 3m
1
1 3m
1
2
x
2
3x 10
6
x 2
4
x 5
2
y
2
y 20
3
y 4
4
y 5
1
y 5
2
5 y
3
y 5
2
y
2
3 x
3
x 3
4
x
5
x 3
1
3
y 6
y
9
y 6
p
1
p 2
1
1
p 2
8
x
3
1
27
2
x
1
3
5
a
1
4
25
a
2
1
16
R-53 Section R.5 Rational Expressions 53
WORKING WITH FORMULAS
81. Cost to seize illegal drugs:
The cost C, in millions of dollars, for a government
to find and seize P% of a certain
illegal drug is modeled by the rational equation
shown. Complete the table (round to the nearest
dollar) and answer the following questions.
10 P 6 1002
C
450P
100 P
a. What is the cost of seizing
40% of the drugs? Estimate
the cost at 85%.
b. Why does cost increase
dramatically the closer you
get to 100%?
c. Will 100% of the drugs ever
be seized?
P
40
60
80
90
93
95
98
100
450P
100 P
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b. Why is the concentration
virtually equal at and
c. Why does the concentration
begin to decrease?
d. How long will it take for the
concentration to become less
than 10%?
H 5?
H 4
82. Chemicals in the bloodstream:
Rational equations are often used to model
chemical concentrations in the bloodstream. The
percent concentration C of a certain drug H hours
after injection into muscle tissue can be modeled
by the equation shown ( ). Complete the table
(round to the nearest tenth of a percent) and answer
the following questions.
a. What is the percent concentration of the drug
3 hr after injection?
H 0
C
200H
2
H
3
40
EXTENDING THE CONCEPT
87. One of these expressions is not equal to the others.
Identify which and explain why.
a. b.
c. d.
88. 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
a. b.
c. d.
31x y2
5
21x y2
5
2x 3y
5
3x 2y
5
20
10
#
n
n
20n
#
1
10n
20
#
n 10
#
n
20n
10n
89. Given the rational numbers and what is the
reciprocal of the sum of their reciprocals? Given
that and are any two numbers—what is the
reciprocal of the sum of their reciprocals?
c
d
a
b
3
4
,
2
5
APPLICATIONS
83. Stock prices: When a hot new stock hits the
market, its price will often rise dramatically and
then taper off over time. The equation
models the price
of stock XYZ d days after it has “hit the market.”
Create a table of values showing the price of the
stock for the first 10 days and comment on what
you notice. Find the opening price of the stock—
does the stock ever return to its original price?
84. 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
according to the equation where
N is the number of elk and t is the time in years.
Approximate the population of elk after 14 yr.
N
1016 3t2
1 0.05t
,
P
5017d
2
102
d
3
50
85. Typing speed: The number of words per minute
that a beginner can type is approximated by the
equation where N is the number
of words per minute after t weeks, Use
a table to determine how many weeks it takes for a
student to be typing an average of forty-five words
per minute.
86. 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 by the
equation How many 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?
N
5D 35
D
1D 12.
2 6 t 6 12.
N
60t 120
t
,
H
0
1
2
3
4
5
6
7
200H
2
H
3
40
54 CHAPTER R A Review of Basic Concepts and Skills R-54
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R.6 Radicals and Rational Exponents
Square roots and cube roots come from a much larger family called radical expres-
sions. Expressions containing radicals can be found in virtually every field of mathe-
matical study, and are an invaluable tool for modeling many real-world phenomena.
A. Simplifying Radical Expressions of the Form
In Section R.1 we noted only if The expression does not rep-
resent a real number because there is no number b such that showing
is a real number only if Of particular interest to us now is an inverse operation
for In other words, what operation can be applied to to return a? Consider the
following.
EXAMPLE 1
Evaluating a Radical Expression
Evaluate for the values given:
a. b. c.
Solution
a. b. c.
Now try Exercises 7 and 8
The pattern seemed to indicate that and that our search for an inverse
operation was complete—until Example 1(c), where we found that
Using the absolute value concept, we can repair this apparent discrepancy and state a
general rule for simplifying these expressions: For expressions like
and the radicands can be rewritten as perfect squares and simplified in the same
manner: and
The Square Root of
For any real number a,
EXAMPLE 2
Simplifying Square Root Expressions
Simplify each expression.
a. b.
Solution
a.
since x could be negative
b.
since could be negative
Now try Exercises 9 and 10
CAUTION
In Section R.3, we noted that indicating that you cannot square the
individual terms in a sum (the square of a binomial results in a perfect square trino-
mial). In a similar way, and you cannot take the square root of
individual terms. There is a big difference between the expressions and
Try evaluating each when and B 4.A 321A B2
2
A B
.
2A
2
B
2
2A
2
B
2
A B,
1A B2
2
A
2
B
2
,
x 5
x 5
2x
2
10x 25 21x 52
2
13
x
2169x
2
13x
2x
2
10x 252169x
2
2a
2
a
.
a
2
: 2a
2
2y
6
21y
3
2
2
y
3
.249x
2
217x2
2
7
x
2y
6
,
249x
2
2a
2
a
.
2162
2
6.
2a
2
a
6 5 3
2162
2
13625
2
12523
2
19
a 6a 5a 3
2a
2
a
2
a
2
.
a 0.
1a
b
2
16,
116
b
2
a.1a b
1
n
a
n
Learning Objectives
In Section R.6 you will learn how to:
A. Simplify radical expres-
sions of the form
B. Rewrite and simplify
radical expressions
using rational exponents
C. Use properties of
radicals to simplify
radical expressions
D. Add and subtract
radical expressions
E. Multiply and divide
radical expressions;
write a radical expres-
sion in simplest form
F. Evaluate formulas
involving radicals
1
n
a
n
R-55 55
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56 CHAPTER R A Review of Basic Concepts and Skills R-56
To investigate expressions like note the radicand in both and can
be written as a perfect cube. From our earlier definition of cube roots we know
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.
The Cube Root of
For any real number a,
EXAMPLE 3
Simplifying Cube Root Expressions
Simplify each expression.
a. b.
Solution
a. b.
Now try Exercises 11 and 12
We can extend these ideas to fourth roots, fifth roots, and so on. For example, the
fifth root of a is b only if In symbols, implies Since an odd
number of negative factors is always negative: and an even number of
negative factors is always positive: we must take the index into account
when evaluating expressions like If n is even and the radicand is unknown,
absolute value notation must be used.
The nth Root of a
n
:
For any real number a,
1. when n is even. 2. when n is odd.
EXAMPLE 4
Simplifying Radical Expressions
Simplify each expression.
a. b. c. d.
e. f. g. h.
Solution
a. b. is not a real number
c. d.
e. f.
g. h.
Now try Exercises 13 and 14
B. Radical Expressions and Rational Exponents
As an alternative to radical notation, a rational (fractional) exponent can be used, along
with the power property of exponents. For notice that an exponent of one-
third can replace the cube root notation and produce the same result:
In the same way, an exponent of one-half can replace the
square root notation: In general, we have the following:2a
2
1a
2
2
1
2
a
2
2
a
.
2
3
a
3
1a
3
2
1
3
a
3
3
a.
2
3
a
3
a,
2
7
1x 22
7
x 22
6
1m 52
6
m 5
2p
2m
or 2
m
2
5
32p
5
2
5
12p2
5
2
4
16m
4
2
4
12m2
4
1
5
32 21
5
32 2
1
4
811
4
81 3
2
7
1x 22
7
2
6
1m 52
6
2
5
32p
5
2
4
16m
4
1
5
321
5
321
4
811
4
81
1
n
a
n
a1
n
a
n
a
2
n
a
n
1
n
a
n
.
122
4
16,
122
5
32,
b
5
a.1
5
a bb
5
a.
4n
2
3x
2
3
64n
6
2
3
14n
2
2
3
2
3
27x
3
2
3
13x2
3
2
3
64n
6
2
3
27x
3
2
3
a
3
a.
a
3
: 2
3
a
3
1
3
8 2
3
122
3
2, 1
3
64 2
3
142
3
4,
1
3
641
3
82
3
x
3
,
WORTHY OF NOTE
Just as is not a real
number, or do
not represent real numbers.
An even number of repeated
factors is always positive!
1
6
161
4
16
1
2
16
A. You’ve just reviewed
how to simplify radical
expressions of the form 1
n
a
n
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R-57 Section R.6 Radicals and Rational Exponents 57
Rational Exponents
If a is a real number and n is an integer greater than 1,
then
provided represents a real number.
EXAMPLE 5
Simplifying Radical Expressions Using Rational Exponents
Simplify by rewriting each radicand as a perfect nth power and converting to
rational exponent notation.
a. b. c. d.
Solution
a. b.
c. d.
is not a real number
Now try Exercises 15 and 16
When a rational exponent is used, as in the
denominator of the exponent represents the index number, while
the numerator of the exponent represents the original power on
a. This is true even when the exponent on a is something other
than one! In other words, the radical expression can be
rewritten as or This is further illustrated in Figure R.6 where
we see the rational exponent has the form, “power over root.” To evaluate this expres-
sion without the aid of a calculator, we use the commutative property to rewrite
as and begin with the fourth root of 16:
In general, if m and n have no common factors (other than 1) the expression
can be interpreted in the following two ways.
Rational Exponents
If is a rational number expressed in lowest terms with then
(1) or (2)
(compute then take the mth power) (compute then take the nth root)
provided represents a real number.1
n
a
a
m
,1
n
a,
a
m
n
2
n
a
m
a
m
n
A
2
n
a
B
m
n 2,
m
n
a
m
n
A
16
1
4
B
3
1
2
3
8.
A
16
1
4
B
3
1
A
16
3
1
B
1
4
16
3
4
.
A
16
3
B
1
4
A
16
3
1
B
1
4
2
4
16
3
1
n
a 2
n
a
1
a
1
n
,
2w
3
a
2w
3
b
3
3
ca
2w
3
b
3
d
1
3
B
3
8w
3
27
B
3
a
2w
3
b
3
2
4
81 1812
1
4
2
x
5
5
12x
5
2
4
4
152
3
3
12x
5
2
4
1
4
152
3
1
3
2
4
16x
20
2
4
12x
5
2
4
2
3
125 2
3
152
3
B
3
8w
3
27
2
4
812
4
16x
20
2
3
125
1
n
a
1
n
a 2
n
a
1
a
1
n
(a
m
)
n
a
m
n
Figure R.6
WORTHY OF NOTE
Any rational number can be
decomposed into the
product of a unit fraction and
an integer:
m
n
1
n
#
m.
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58 CHAPTER R A Review of Basic Concepts and Skills R-58
EXAMPLE 6
Simplifying Expressions with Rational Exponents
Find the value of each expression without a calculator, by rewriting the exponent
as the product of a unit fraction and an integer.
a. b. c.
Solution
a. b.
or 9
c.
Now try Exercises 17 and 18
Expressions with rational exponents are generally easier to evaluate if we compute
the root first, then apply the exponent. Computing the root first also helps us deter-
mine whether or not an expression represents a real number.
EXAMPLE 7
Simplifying Expressions with Rational Exponents
Simplify each expression, if possible.
a. b. c. d.
Solution
a. b.
or not a real number
c. d.
or 4
or
Now try Exercises 19 through 22
C. Using Properties of Radicals to Simplify
Radical Expressions
The properties used to simplify radical expressions are closely connected to the prop-
erties of exponents. For instance, the product to a power property holds even when n
is a rational number. This means and When the second
statement is expressed in radical form, we have with both
forms having a value of 10. This suggests the product property of radicals, which
can be extended to include cube roots, fourth roots, and so on.
Product Property of Radicals
If and represent real-valued expressions, then
and 1
n
A
#
1
n
B 1
n
AB.1
n
AB 1
n
A
#
1
n
B
1
n
B 1
n
A
14
#
25 14
#
225,
14
#
252
1
2
4
1
2
#
25
1
2
.1xy2
1
2
x
1
2
y
1
2
1
4
2
2
122
2
11
3
82
2
11
3
82
2
8
2
3
A
8
1
3
B
2
182
2
3
3
182
1
3
4
2
343 172
3
11492
3
11492
3
1492
3
2
1492
1
2
3
, 49
3
2
A
49
1
2
B
3
8
2
3
182
2
3
1492
3
2
49
3
2
c
2
x
3
3
d
5
32
x
15
243
ca
4x
6
9
b
1
2
d
5
a
4x
6
9
b
5
2
a
4x
6
9
b
1
2
#
5
2
4
16 3
2
182
1
3
4
A
27
1
3
B
2
182
4
3
182
1
3
#
4
27
2
3
27
1
3
#
2
a
4x
6
9
b
5
2
182
4
3
27
2
3
WORTHY OF NOTE
While the expression
represents the
real number the expres-
sion is not a
real number, even though
Note that the second
exponent is not in lowest
terms.
1
3
2
6
.
182
2
6
11
6
82
2
2,
182
1
3
1
3
8
B. You’ve just reviewed
how to rewrite and simplify
radical expressions using
rational exponents
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R-59 Section R.6 Radicals and Rational Exponents 59
CAUTION Note that this property applies only to a product of two terms, not to a sum or difference.
In other words, while
One application of the product property is to simplify radical expressions. In
general, the expression is in simplified form if a has no factors (other than 1) that
are perfect nth roots.
EXAMPLE 8
Simplifying Radical Expressions
Write each expression in simplest form using the product property.
a. b. c.
Solution
a. b.
c.
look for perfect square factors of 20
product property
factor and reduce
result
Now try Exercises 23 and 24
When radicals are combined using the product property, the result may contain a
perfect nth root, which should be simplified. Note that the index numbers must be the
same in order to use this property.
EXAMPLE 9
Simplifying Radical Expressions
Combine factors using the product property and simplify:
Solution
product property
Since the index is 3, we look for perfect cube factors in the radicand.
product property
rewrite as a perfect cube
simplify
result
Now try Exercises 25 and 26
The quotient property of radicals can also be established using exponential prop-
erties. The fact that suggests the following:
1100
125
A
100
25
2
4.8n
3
1.2142n
3
n
9
1.22
3
64 2
3
1n
3
2
3
1.22
3
64 2
3
n
9
1.22
3
16n
4
2
3
4n
5
1.22
3
64
#
n
9
2
3
4n
5
.1.22
3
16n
4
2 15
2
12 152
2
4 215
2
4 120
2
4 14
#
5
2
25x1
3
x
5
#
5
#
x
#
1
3
x 312
5
#
2
3
125
#
2
3
x
3
#
2
3
x
1
1912
5 2
3
125x
4
5
#
2
3
125
#
x
4
118 19
#
2
4 220
2
52
3
125x
4
118
1
n
a
29 x
2
|
3 x
|
!29x
2
3x
,
WORTHY OF NOTE
For expressions like those in
Example 8(c), students must
resist the “temptation” to
reduce individual terms as in
Remember, only factors can
be reduced.
2 120
.
4 120
2
WORTHY OF NOTE
Rational exponents also
could have been used to
simplify the expression from
Example 9, since
Also see Example 11.
4.8n
3
.
1.21
3
64 2
3
n
9
1.2142n
9
3
These steps can
be done mentally.
e
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