Download 7.5 Rationalizing Denominators and Numerators of Radical

440 CHAPTER 7
Rational Exponents, Radicals, and Complex Numbers
86. Find the area and perimeter of the trapezoid. (Hint: The
area of a trapezoid is the product of half the height 6 23
meters and the sum of the bases 2 263 and 7 27 meters.)
88. a. Add: 2 25 + 25
b. Multiply: 2 25 # 25
c. Describe the differences in parts (a) and (b).
2兹63 m
89. Multiply: 1 22 + 23 - 1 2 2 .
2兹27 m
90. Multiply: 1 25 - 22 + 1 2 2
6兹3 m
91. Explain how simplifying 2x + 3x is similar to simplifying
2 2x + 3 2x.
7兹7 m
92. Explain how multiplying 1x - 221x + 32 is similar to multiplying 1 2x - 22 21 2x + 3 2 .
87. a. Add: 23 + 23.
Multiply: 23 # 23.
b.
c. Describe the differences in parts (a) and (b).
7.5
Rationalizing Denominators and Numerators
of Radical Expressions
OBJECTIVE
OBJECTIVES
1 Rationalize Denominators.
2 Rationalize Denominators
Having Two Terms.
3 Rationalize Numerators.
1
Rationalizing Denominators of Radical Expressions
13
either
12
without a radical in the denominator or without a radical in the numerator. The process of writing this expression as an equivalent expression but without a radical in the
denominator is called rationalizing the denominator. To rationalize the denominator
13
of
, we use the fundamental principle of fractions and multiply the numerator and
12
12
the denominator by 22. Recall that this is the same as multiplying by
, which
12
simplifies to 1.
Often in mathematics, it is helpful to write a radical expression such as
23
=
22
23 # 22
22 # 22
=
26
24
=
26
2
In this section, we continue to assume that variables represent positive real numbers.
EXAMPLE 1
a.
2
Rationalize the denominator of each expression.
b.
25
2216
c.
29x
3 1
A2
Solution
a. To rationalize the denominator, we multiply the numerator and denominator by a
factor that makes the radicand in the denominator a perfect square.
2
25
=
2 # 25
25 # 25
225
5
=
The denominator is now rationalized.
b. First, we simplify the radicals and then rationalize the denominator.
2216
29x
=
2142
32x
=
8
32x
To rationalize the denominator, multiply the numerator and denominator by 2x.
Then
8
32x
=
8 # 2x
32x # 2x
=
82x
3x
Section 7.5
c.
Rationalizing Denominators and Numerators of Radical Expressions
441
3
1
2
1
1
3
= 3 = 3 . Now we rationalize the denominator. Since 2
2 is a cube root,
A2
22
22
we want to multiply by a value that will make the radicand 2 a perfect cube. If we
3
3 2
3 3
3
multiply 2
2 by 2
2 , we get 2
2 = 2
8 = 2.
3
3 2
1# 2
2
=
22 # 22
3
3
2
3
2
4
3
22
3
=
3
2
4
2
Multiply the numerator and denominator
3 2
by 2
2 and then simplify.
PRACTICE
1
a.
Rationalize the denominator of each expression.
5
b.
23
3225
c.
24x
2
A9
3
CONCEPT CHECK
Determine by which number both the numerator and denominator can be multiplied to rationalize the denominator of
the radical expression.
1
1
a. 3
b. 4
27
28
EXAMPLE 2
Rationalize the denominator of
7x
27x
=
A 3y
23y
27x # 23y
=
23y # 23y
Solution
221xy
3y
=
Use the quotient rule. No radical may be simplified further.
Multiply numerator and denominator by 13y so that
the radicand in the denominator is a perfect square.
Use the product rule in the numerator and denominator.
Remember that 13y # 13y = 3y .
PRACTICE
2
Rationalize the denominator of
EXAMPLE 3
7x
.
A 3y
3z
.
A 5y
Rationalize the denominator of
4
2
x
4
2
81y 5
.
Solution First, simplify each radical if possible.
4
2
x
4
2
81y 5
=
=
=
=
=
3
a. 27 or 249
2
4
4
2
81y 4 # 2
y
4
2
x
4
3y2y
4
4 3
2
x# 2
y
4
4 3
3y2
y# 2
y
4
2
xy 3
4 4
3y2
y
4
2
xy 3
3y
2
Use the product rule in the denominator.
4
Write 2
81y 4 as 3y.
4 3
Multiply numerator and denominator by 2
y so that
the radicand in the denominator is a perfect fourth power.
Use the product rule in the numerator and denominator.
4 4
In the denominator, 2
y = y and 3y # y = 3y 2 .
PRACTICE
Answer to Concept Check:
3
4
2
x
3
4
b. 22
Rationalize the denominator of
3 2
2
z
3
2
27x4
.
442 CHAPTER 7
Rational Exponents, Radicals, and Complex Numbers
OBJECTIVE
Rationalizing Denominators Having Two Terms
2
Remember the product of the sum and difference of two terms?
1a + b21a - b2 = a 2 - b2
a
Q
These two expressions are called conjugates of each other.
To rationalize a numerator or denominator that is a sum or difference of two terms,
we use conjugates. To see how and why this works, let’s rationalize the denominator of
5
. To do so, we multiply both the numerator and the denominator
the expression
13 - 2
by 23 + 2, the conjugate of the denominator 23 - 2, and see what happens.
5
23 - 2
=
=
51 23 + 22
1 23 - 221 23 + 22
51 23 + 22
1 232 - 2
2
2
Multiply the sum and difference
of two terms: 1a + b21a - b2 = a 2 - b2.
51 23 + 22
3 - 4
51 23 + 22
=
-1
= -51 23 + 22
=
or
-523 - 10
Notice in the denominator that the product of 1 23 - 22 and its conjugate,
1 23 + 22 , is -1. In general, the product of an expression and its conjugate will contain no radical terms. This is why, when rationalizing a denominator or a numerator
containing two terms, we multiply by its conjugate. Examples of conjugates are
1a - 2b
x + 1y
EXAMPLE 4
a.
2
322 + 4
1a + 2b
x - 1y
and
and
Rationalize each denominator.
b.
26 + 2
25 - 23
c.
22m
32x + 2m
Solution
a. Multiply the numerator and denominator by the conjugate of the denominator,
312 + 4.
2
322 + 4
=
=
21322 - 42
1322 + 421322 - 42
21322 - 42
13222 2 - 42
21322 - 42
18 - 16
21322 - 42
=
, or 322 - 4
2
=
It is often useful to leave a numerator in factored form to help determine whether
the expression can be simplified.
Section 7.5
Rationalizing Denominators and Numerators of Radical Expressions
443
b. Multiply the numerator and denominator by the conjugate of 25 - 23.
26 + 2
25 - 23
1 26
=
125
+ 2 2125 + 23 2
- 23 2125 + 23 2
2625 + 2623 + 225 + 223
=
1 25 2 2
-
1 23 2 2
=
230 + 218 + 225 + 223
5 - 3
=
230 + 322 + 225 + 223
2
c. Multiply by the conjugate of 32x + 2m to eliminate the radicals from the
denominator.
22m
32x + 2m
=
=
22m 1 32x - 2m 2
1 32x
=
+ 2m 21 32x - 2m 2
62mx - 2m
1 32x 2 2
- 12m 2 2
62mx - 2m
9x - m
PRACTICE
4
a.
Rationalize the denominator.
5
b.
325 + 2
22 + 5
c.
23 - 25
32x
22x + 2y
OBJECTIVE
3
Rationalizing Numerators
As mentioned earlier, it is also often helpful to write an expression such as
23
as
22
an equivalent expression without a radical in the numerator. This process is called
23
rationalizing the numerator. To rationalize the numerator of
, we multiply the
22
numerator and the denominator by 23.
23
22
EXAMPLE 5
=
23 # 23
=
22 # 23
29
26
Rationalize the numerator of
3
=
26
27
245
.
Solution First we simplify 245.
27
245
=
27
29 # 5
=
27
325
Next we rationalize the numerator by multiplying the numerator and the denominator
by 27.
27
325
=
27 # 27
325 # 27
PRACTICE
5
Rationalize the numerator of
=
232
280
.
7
325 # 7
=
7
3235
444 CHAPTER 7
Rational Exponents, Radicals, and Complex Numbers
EXAMPLE 6
Rationalize the numerator of
3
2
2x2
.
3
2
5y
Solution The numerator and the denominator of this expression are already simplified.
3
To rationalize the numerator, 2
2x2 , we multiply the numerator and denominator by a
3
3
factor that will make the radicand a perfect cube. If we multiply 2
2x2 by 2
4x, we get
3
2
8x3 = 2x.
3
3
3
3
2
2x2
2
2x2 # 2
4x
2
8x3
2x
=
=
= 3
3
3
3
3
#
25y
25y 24x
220xy
220xy
PRACTICE
6
Rationalize the numerator of
3
2
5b
3
2
2a
.
2x + 2
.
5
Solution We multiply the numerator and the denominator by the conjugate of the
numerator, 1x + 2.
EXAMPLE 7
2x + 2
=
5
=
=
Rationalize the numerator of
1 2x
+ 2 21 2x - 2 2
5 1 2x - 2 2
1 2x 2 2 - 22
5 1 2x - 2 2
1a + b21a - b2 = a 2 - b2
x - 4
5 1 2x - 2 2
PRACTICE
7
Multiply by 2x - 2 , the conjugate of 2x + 2 .
Rationalize the numerator of
2x - 3
.
4
Vocabulary, Readiness & Video Check
Use the choices below to fill in each blank. Not all choices will be used.
rationalizing the numerator
conjugate
rationalizing the denominator
5
5
23
23
1. The
of a + b is a - b.
2. The process of writing an equivalent expression, but without a radical in the denominator is, called
.
3. The process of writing an equivalent expression, but without a radical in the numerator, is called
5
4. To rationalize the denominator of
, we multiply by
.
23
Martin-Gay Interactive Videos
Watch the section lecture video and answer the following questions.
OBJECTIVE
1
5. From
Examples 1–3, what is the goal of rationalizing a denominator?
OBJECTIVE
2
6. From Example 4, why will multiplying a denominator by its conjugate always rationalize the denominator?
OBJECTIVE
3
See Video 7.5
7. From Example 5, is the process of rationalizing a numerator any different from rationalizing a denominator?
.
Section 7.5
7.5
Rationalizing Denominators and Numerators of Radical Expressions
Exercise Set
Rationalize each denominator. See Examples 1 through 3.
1.
22
2.
27
45.
23
4.
1
A2
4
Ax
6.
25
Ay
7.
9.
11.
13.
15.
17.
4
8.
3
2
3
3
10.
28x
3
3
24x
2
9
12.
14.
23a
3
16.
3
2
2
2 23
18.
27
47.
49.
6
3
2
9
51.
5
227a
5
5
- 5 22
7
22.
A 10
25.
27.
1
212z
3
2
2y 2
3
29x
2
26.
28.
29.
81
A8
30.
31.
16
A 9x 7
32.
33.
4
4
5a
5
28a b
9 11
34.
11y
B 45
1
232x
3
23x
3
24y
4
1
A9
4
32
5
Am n
6 13
24y 9
37. 5 - 2a
38. 6 - 2b
39. - 7 25 + 8 2x
40. - 9 22 - 6 2y
-7
2x - 3
44.
54.
2 2x - 2y
4 25 + 22
2 25 - 22
3
A2
57.
18
A5
58.
12
A7
59.
24x
7
60.
23x 5
6
3
2
5y 2
3
2
4x
62.
3
2
4x
3 4
2
z
63.
2
A5
64.
3
A7
65.
22x
11
66.
2y
7
67.
3
7
A8
68.
25
A2
3
69.
23x 5
10
3 9y
70. A
7
71.
18x 4y 6
B 3z
72.
3
8x 5y
B 2z
Rationalize each numerator. See Example 7.
73.
2 - 211
6
74.
75.
2 - 27
-5
76.
Rationalize each denominator. See Example 4.
43.
4 23 - 26
2 2a
56.
77.
42.
2 23 + 26
26 - 2
9y
36. 23 + y
6
2x + 2y
52.
-3
4
35. 22 + x
2 - 27
2x
2 2a + 2b
5
A3
61.
Write the conjugate of each expression.
41.
1 + 210
50.
2 2a - 3
55.
3
2
9
3
24.
8
22 - 23
Rationalize each numerator. See Examples 5 and 6.
25
3
21.
A5
3x
A 50
2 2a - 2b
48.
23 + 24
23y
x
2x
19.
A 5y
23.
53.
2a + 1
46.
3
211
13a
20.
A 2b
3
22 + 23
22
1
3.
A5
5.
22 - 23
3
27 - 4
-8
2y + 4
79.
81.
2x + 3
2x
22 - 1
22 + 1
2x + 1
2x - 1
78.
80.
82.
215 + 1
2
25 + 2
22
5 + 22
22x
28 - 23
22 + 23
2x + 2y
2x - 2y
445
446 CHAPTER 7
Rational Exponents, Radicals, and Complex Numbers
REVIEW AND PREVIEW
Solve each equation. See Sections 2.1 and 5.8.
83. 2x - 7 = 31x - 42
84. 9x - 4 = 71x - 22
85. 1x - 6212x + 12 = 0
86. 1y + 2215y + 42 = 0
87. x 2 - 8x = -12
88. x 3 = x
25y 3
, rationalize the denominator by following
212x 3
parts (a) and (b).
91. Given
a. Multiply the numerator and denominator by 212x 3 .
b. Multiply the numerator and denominator by 23x.
c. What can you conclude from parts (a) and (b)?
CONCEPT EXTENSIONS
89. The formula of the radius r of a sphere with surface area A is
92. Given
3
2
5y
3
2
4
(a) and (b).
A
r =
A 4p
Rationalize the denominator of the radical expression in
this formula.
, rationalize the denominator by following parts
3
a. Multiply the numerator and denominator by 216.
3
b. Multiply the numerator and denominator by 2
2.
c. What can you conclude from parts (a) and (b)?
r
Determine the smallest number both the numerator and denominator should be multiplied by to rationalize the denominator of the
radical expression. See the Concept Check in this section.
93.
90. The formula for the radius r of a cone with height 7 centimeters and volume V is
9
3
25
94.
5
227
95. When rationalizing the denominator of
25
, explain why
27
both the numerator and the denominator must be multiplied by 27.
25
96. When rationalizing the numerator of
, explain why both
27
the numerator and the denominator must be multiplied by
25.
3V
r =
A 7p
Rationalize the numerator of the radical expression in this
formula.
7 cm
97. Explain why rationalizing the denominator does not change
the value of the original expression.
r
98. Explain why rationalizing the numerator does not change
the value of the original expression.
Integrated Review RADICALS AND RATIONAL EXPONENTS
Sections 7.1–7.5
Throughout this review, assume that all variables represent positive real numbers.
Find each root.
3
4 1
1. 281
2. 2 -8
3.
A 16
3 9
5. 2
y
6. 24y 10
5
7. 2
-32y 5
4. 2x6
4
8. 2
81b12
Use radical notation to write each expression. Simplify if possible.
9. 361/2
10. 13y2 1/4
11. 64-2/3
12. 1x + 12 3/5
Use the properties of exponents to simplify each expression. Write with positive exponents.
13. y -1/6 # y 7/6
14.
12x1/3 2 4
x5/6
Use rational exponents to simplify each radical.
3
17. 2
8x6
12
18. 2
a 9b 6
15.
x1/4x3/4
x -1/4
16. 41/3 # 42/5
Section 7.6
Radical Equations and Problem Solving 447
Use rational exponents to write each as a single radical expression.
4
19. 2x # 2x
3
20. 25 # 2
2
Simplify.
4
22. 2
16x7y 10
21. 240
3
23. 2
54x4
5
24. 2
-64b10
Multiply or divide. Then simplify if possible.
25. 25 # 2x
3
3
26. 2
8x # 2
8x2
27.
Perform each indicated operation.
1 27
+ 23 2
28.
22y
2
33.
1 2x
4
2
48a 9b3
4
2
ab3
31. 23 1 25 - 22 2
3
3
30. 2
54y 4 - y2
16y
29. 220 - 275 + 527
32.
298y 6
- 25 21 2x + 25 2
34.
1 2x
+ 1 - 12
2
Rationalize each denominator.
35.
7
A3
36.
5
3
22x
37.
2
23 - 27
223 + 27
Rationalize each numerator.
38.
7
A3
7.6
39.
9y
A 11
3
40.
2x - 2
2x
Radical Equations and Problem Solving
OBJECTIVE
OBJECTIVES
1 Solve Equations That Contain
Radical Expressions.
2 Use the Pythagorean Theorem
to Model Problems.
1
Solving Equations That Contain Radical Expressions
In this section, we present techniques to solve equations containing radical expressions
such as
22x - 3 = 9
We use the power rule to help us solve these radical equations.
Power Rule
If both sides of an equation are raised to the same power, all solutions of the original
equation are among the solutions of the new equation.
This property does not say that raising both sides of an equation to a power yields an
equivalent equation. A solution of the new equation may or may not be a solution of
the original equation. For example, 1 -22 2 = 22 , but -2 ⬆ 2. Thus, each solution of
the new equation must be checked to make sure it is a solution of the original equation.
Recall that a proposed solution that is not a solution of the original equation is called
an extraneous solution.
EXAMPLE 1
Solve: 22x - 3 = 9.
Solution We use the power rule to square both sides of the equation to eliminate the
radical.
22x - 3
1 22x - 32 2
2x - 3
2x
x
=
=
=
=
=
9
92
81
84
42
Now we check the solution in the original equation.
(Continued on next page)