Download SECTION 1-7 Radicals

1-7 Radicals
Problems 47–50 illustrate common errors involving rational
exponents. In each case, find numerical examples that show
that the left side is not always equal to the right side.
65.
(2x 1)1/2 (x 2)( 12 )(2x 1)1/2(2)
2x 1
47. (x y)1/2 x1/2 y1/2
66.
(x 1)1/2 x( 12 )(x 1)1/2
x1
67.
2(3x 1)1/3 (2x 1)( 13 )(3x 1)2/3(3)
(3x 1)2/3
68.
(x 2)2/3 x( 23 )(x 2)1/3
(x 2)4/3
49. (x y)1/3 1
(x y)3
48. (x3 y3)1/3 x y
50. (x y)1/2 1
(x y)2
Problems 51–56 are calculus-related. Write each problem
in the form ax p bx q, where a and b are real numbers and
p and q are rational numbers. For example,
2x1/3 4
4x
2x1/3
4
1
x1/31 x1
4x
4x 2
1
x2/3 x1
2
51.
12x1/2 3
4x1/2
52.
x2/3 2
2x1/3
53.
3x2/3 x1/2
5x
54.
2x3/4 3x1/3
3x
55.
x2 4x1/2
2x1/3
56.
2x1/3 x1/2
4x1/2
C
In Problems 57–60, m and n represent positive integers.
Simplify and express answers using positive exponents.
3/n 3/m 1/3
57. (a b )
59. (x
m/4 n/3 12
y )
n/2 n/3 1/n
58. (a b )
m/3 n/2 6
60. (a b )
61. If possible, find a real value of x such that:
(A) (x2)1/2 x
(B) (x2)1/2 x
(C) (x3)1/3 x
62. If possible, find a real value of x such that:
(A) (x2)1/2 x
(B) (x2)1/2 x
(C) (x3)1/3 x
63. If n is even and b is negative, then b1/n is not real. If m is
odd, n is even, and b is negative, is (bm)1/n real?
64. If we assume that m is odd and n is even, is it possible that
one of (b1/n)m and (bm)1/n is real and the other is not?
Problems 65–68 are calculus-related. Simplify by writing
each expression as a simple fraction reduced to lowest terms
and without negative exponents.
SECTION
1-7
APPLICATIONS
69. Economics. The number of units N of a finished product
produced from the use of x units of labor and y units of capital for a particular Third World country is approximated by
N 10x3/4y1/4
Cobb-Douglas equation
Estimate how many units of a finished product will be produced using 256 units of labor and 81 units of capital.
70. Economics. The number of units N of a finished product
produced by a particular automobile company where x
units of labor and y units of capital are used is approximated by
N 50x1/2y1/2
Cobb-Douglas equation
Estimate how many units will be produced using 256 units
of labor and 144 units of capital.
71. Braking Distance. R. A. Moyer of Iowa State College
found, in comprehensive tests carried out on 41 wet pavements, that the braking distance d (in feet) for a particular
automobile traveling at v miles per hour was given approximately by
d 0.0212v7/3
Approximate the braking distance to the nearest foot for the
car traveling on wet pavement at 70 miles per hour.
72. Braking Distance. Approximately how many feet would it
take the car in Problem 71 to stop on wet pavement if it
were traveling at 50 miles per hour? (Compute answer to
the nearest foot.)
Radicals
•
•
•
•
•
•
57
From Rational Exponents to Radicals, and Vice Versa
Properties of Radicals
Simplifying Radicals
Sums and Differences
Products
Rationalizing Operations
58
1 Basic Algebraic Operations
What do the following algebraic expressions have in common?
21/2
2x2/3
2
3 2
2
x
1/2
x
1
y1/2
1
x y
Each vertical pair represents the same quantity, one in rational exponent form and the
other in radical form. There are occasions when it is more convenient to work with
radicals than with rational exponents, or vice versa. In this section we see how the
two forms are related and investigate some basic operations on radicals.
• From Rational
We start this discussion by defining an nth-root radical:
Exponents to
Radicals, and
Vice Versa
DEFINITION 1
n
b , nth-Root Radical
n
For n a natural number greater than 1 and b a real number, we define b to be
the principal nth root of b (see Definition 2 in Section 1-6); that is,
n
b b1/n
2
If n 2, we write b in place of b
25
251/2
25
5
251/2
5
32
5
5
32
321/5
2
(32)1/5
2
4
0 01/4 0
25 is not real
The symbol 1 is called a radical, n is called the index, and b is called the
radicand.
As stated above, it is often an advantage to be able to shift back and forth between
rational exponent forms and radical forms. The following relationships, which are direct
consequences of Definition 1 and Theorem 2 in Section 1-6, are useful in this regard:
Rational Exponent/Radical Conversions
For m and n positive integers (n 1), and b not negative when n is even,
m/n
b
(bm)1/n
(b1/n)m
n
bm
n
(b)m
(22 )
3
22/3 3
2
2
1-7 Radicals
59
Note: Unless stated to the contrary, all variables in the rest of the discussion are
restricted so that all quantities involved are real numbers.
EXPLORE-DISCUSS 1
In each of the following, evaluate both radical forms.
163/2 163 (16)3
3
3
272/3 272 (
27)2
Which radical conversion form is easier to use if you are performing the calculations by hand?
EXAMPLE 1
Rational Exponents/Radical Conversions
Change from rational exponent form to radical form.
7
(A) x1/ 7 x
5
(B) (3u2v3)3/5 (3u2v3)3
(C) y2/3 1
1
3 2
y2/3 y
or
or
5
(
3u2v3)3
3
y2
The first is usually preferred.
3
or
1
y2
Change from radical form to rational exponent form.
5
(D) 6 61/5
Matched Problem 1
3
(E) x2 x2/3
(F) x2 y2 (x2 y2)1/ 2
Change from rational exponent form to radical form.
(A) u1/5
(B) (6x2y5)2/9
(C) (3xy)3/5
Change from radical form to rational exponent form.
4
(D) 9u
• Properties of
Radicals
7
(E) (2x)4
3
(F) x3 y3
The process of changing and simplifying radical expressions is aided by the introduction of several properties of radicals that follow directly from exponent properties
considered earlier.
60
1 Basic Algebraic Operations
Theorem 1
Properties of Radicals
For n a natural number greater than 1, and x and y positive real numbers:
n
1. x n x
n
n
3
x3 x
n
2. xy xy
n
3.
EXAMPLE 2
n
x x
n
y y
5
5
5
xy x y
4
4
x x
4
y y
Simplifying Radicals
Simplify:
5
(A) (3x2y)5 3x2y
(B) 105 50 25 2 25 2 52
3
(C)
Matched Problem 2
3
3
x
x
x
3
27 27
3
1 3
x
3
Simplify:
7
(A) (u2 v2)7
CAUTION
or
(B) 62
3
(C)
x2
8
In general, properties of radicals can be used to simplify terms raised to powers,
not sums of terms raised to powers. Thus, for x and y positive real numbers,
x2 y2 x2 y2 x y
but
x2 2xy y2 (x y)2 x y
• Simplifying
Radicals
The properties of radicals provide us with the means of changing algebraic expressions containing radicals to a variety of equivalent forms. One form that is often
useful is a simplified form. An algebraic expression that contains radicals is said to
be in simplified form if all four of the conditions listed in the following definition
are satisfied.
1-7 Radicals
DEFINITION 2
61
Simplified (Radical) Form
1. No radicand (the expression within the radical sign) contains a factor to a
power greater than or equal to the index of the radical.
For example, x5 violates this condition.
2. No power of the radicand and the index of the radical have a common factor other than 1.
6 4
For example, x violates this condition.
3. No radical appears in a denominator.
For example, y / x violates this condition.
4. No fraction appears within a radical.
For example, 35 violates this condition.
EXAMPLE 3
Finding Simplified Form
Express radicals in simplified form.
(A) 12x3y5z2 (4x2y4z2)(3xy)
Condition 1 is not met.
(2xy2z)2(3xy)
x pmy pn (x my n)p
(2xy2z)2 3xy
n
n
n
xy x y
2xy2z3xy
n
xn x
3
3
(B) 6x2y
4x5y2
3
(6x2y)(4x5y2)
n
n
n
x y xy
3
24x7y3
3
(8x6y3)(3x)
Condition 1 is not met.
3
(2x2y)3(3x)
x pmy pn (x my n) p
3
3
(2x2y)3 3x
n
n
n
xy x y
3
2x2y
3x
n
x n x
6
(C) 16x 4y2 [(4x2y)2]1/6
(4x2y)2/6
Condition 2 is not met.
Note the convenience of using rational exponents.
(4x2y)1/3
3
4x2y
3
(D) 27 [(33)1/2]1/3
(33)1/6 33/6
31/2 3
62
1 Basic Algebraic Operations
Matched Problem 3
Express radicals in simplified form.
(A) 18x5y2z3
• Sums and
Differences
EXAMPLE 4
4
4
(B) 27a3b3 3a5b3
9
(C) 8x6y3
3
(D) 4
Algebraic expressions involving radicals often can be simplified by adding and subtracting terms that contain exactly the same radical expressions. We proceed in essentially the same way as we do when we combine like terms in polynomials. The distributive property of real numbers plays a central role in this process.
Combining Like Terms
Combine as many terms as possible:
(A) 53 43
(5 4)3
3
3
(B) 2
xy2 7
xy2
93
3
(2 7)
xy2
3
3
(C) 3xy 2
xy 4xy 7
xy
3
5
xy2
3
3
3xy 4xy 2
xy 7
xy
3
7xy 9
xy
Matched Problem 4
Combine as many terms as possible:
(A) 62 22
5
5
(B) 3
2x2y3 8
2x2y3
3
3
(C) 5
mn2 3mn 2
mn2 7mn
• Products
EXAMPLE 5
We will now consider several types of special products that involve radicals. The
distributive property of real numbers plays a central role in our approach to these
problems.
Multiplication with Radical Forms
Multiply and simplify:
(A) 2(10 3) 210 2 3 20 32 25 32
(B) (2 3)(2 5) 22 32 52 15
2 22 15
22 13
(C) (x 3)(x 5) xx 3x 5x 15
x 2x 15
1-7 Radicals
63
3
3 2
3
3
3
3
3
3 3
(D) (
m
n )(
m2 n) m3 m2n2 mn n
3
3
m
mn m2n2 n
Matched Problem 5
• Rationalizing
Operations
Multiply and simplify:
(A) 3(6 4)
(B) (3 2)(3 4)
(C) (y 2)(y 4)
3 2
3 2
3
3
(D) (
x y )(
x
y)
We now turn to algebraic fractions involving radicals in the denominator. Eliminating a radical from a denominator is referred to as rationalizing the denominator. To
rationalize the denominator, we multiply the numerator and denominator by a suitable factor that will rationalize the denominator—that is, will leave the denominator
free of radicals. This factor is called a rationalizing factor. The following special
products are of use in finding some rationalizing factors (see Examples 6C, D):
(a b)(a b) a2 b2
EXPLORE-DISCUSS 2
(a b)(a ab b ) a b
(2)
(a b)(a2 ab b2) a3 b3
(3)
2
3
Use special products (1) to (3) above to find a rationalizing factor for each of the
following:
(A) a b
EXAMPLE 6
(1)
3
2
(B) a b
3
3
(C) a
b
3
3
(D) a
b
Rationalizing Denominators
Rationalize denominators.
(A)
Solutions
3
5
3
(B)
2a2
3b2
(C)
x y
3x 2y
(D)
1
3
m 2
(A) 5 is a rationalizing factor for 5, since 55 52 5. Thus, we multiply the numerator and denominator by 5 to rationalize the denominator:
3
35
35
5
5 55
3
(B)
3
3
3 2
2a2 2a2 2a2
3b
3
3
2
2
2 3 2
3b
3b
3b 3 b
3
2 32a2b
3
33b3
3
18a2b
3b
64
1 Basic Algebraic Operations
(C) Special product (1) suggests that if we multiply the denominator 3x 2y
by 3x 2y, we will obtain the difference of two squares and the denominator will be rationalized.
(x y)(3x 2y)
x y
3x 2y (3x 2y)(3x 2y)
3x2 2xy 3xy 2y2
(3x)2 (2y)2
3x 5xy 2y
9x 4y
3
(D) Special product (3) above suggests that if we multiply the denominator m2
3
3
2
2
by (m) 2m 2 , we will obtain the sum of two cubes and the denominator will be rationalized.
3
3
1
1[(
m)2 2
m 22]
3
3
3
3
2
m 2 (m 2)[(m) 2m 22]
Matched Problem 6
3
3
m2 2m 4
3
(m)3 23
3
3
m2 2m 4
m8
Rationalize denominators.
(A)
6
2x
(B)
10x3
3
4x
(C)
x 2
2x 3
(D)
1
3
1
y
Answers to Matched Problems
5
9
9
5
1. (A) (B) (C) 1/ (D) (9u)1/4
u
(6x2y5)2 or (
6x2y5)2
(3xy)3
(E) (2x)4/7
(F) (x3 y3)1/3
3 2
3 2
2. (A) u2 v2
(B) 23
(C) (
x )/ 2 or 12 x
4 2
3
3
2
2
2
3. (A) 3x yz2xz
(B) 3a bb 3a bb
(C) (D) 2x2y
2
5
3
2 3
2
4. (A) 82
(B) 52x y
(C) 3mn 4mn
3 2
3
5. (A) 32 43
(B) 23 5
(C) y 2y 8
(D) x xy
xy2 y
3
3 2
1
32x
2x
6
y
y
x
3
6. (A)
(B) 5x2
(C)
(D)
2x2
x
4x 9
1y
1-7 Radicals
EXERCISE
1-7
Unless stated to the contrary, all variables are restricted so
that all quantities involved are real numbers.
3 3
51. a b3
52. x2 y2
In Problems 53–64, rationalize denominators and write in
simplified form.
A
In Problems 1–8, change to radical form. Do not simplify.
1. m2/3
2. n4/5
2/5
3. 6x3/5
3 2/5
4. 7y
5. (4xy )
7. (x y)1/2
8. x1/2 y1/2
2
5
9. b
4 3
11. 5
x
5 2
12. 7m
n
5
13. (2x2y)3
9
14. (3m4n)2
3
3
15. x
y
3
16. xy
3
17. 8
3
18. 27
19. 9x8y4
20. 16m4y8
4
21. 16m4n8
5
22. 32a15b10
23. 8a3b5
24. 27m2n7
3 4 4 7
25. 2xy
26. 2 x y
27. m
28. n
29. xy
30. 5x
3
3
31. 9x2 9x
5
3
4
2
4
54.
6 8c
18c
55.
4a3b2
3
2ab2
56.
8x3y5
3
4x2y
57.
58.
In Problems 17–32, write in simplified form.
4 5 8
2m 5
20m
6. (7x y)
10. c
4
53.
5/7
In Problems 9–16, change to rational exponent form. Do not
simplify.
10
6
32. 2x 8xy
In Problems 33–40, rationalize denominators, and write in
simplified form.
4
3y3
4x3
5
4x2
16y3
59.
3y
2y 3
60.
5x
3 2x
61.
25 32
55 22
62.
32 23
33 22
33.
1
5
34.
1
7
35.
6x
3x
63.
x2
x 9 3
36.
12y2
6y
37.
2
2 1
38.
4
6 2
64.
y2
2 y2 4
39.
2
6 2
40.
2
10 2
In Problems 41–52, write in simplified form.
5 6 7 11
41. x
3xy
5
32u12v8
uv
2
Problems 65–68 are calculus-related. Rationalize the numerators; that is, perform operations on the fractions that eliminate radicals from the numerators. (This is a particularly
useful operation in some problems in calculus.)
B
44.
65
4
32m7n9
2mn
3
42. 2a
8a8b13
43.
6 4
45. a (b a)2
8 6
46. 3 (u v)6
3 4 9 3
47. a b
6 8 6
48. xy
3
3
49. 2x2y4 3x5y
4
4
50. 4m5n 6m3n4
65.
t x
tx
66.
x y
x y
67.
x h x
h
68.
2 h 2
h
66
1 Basic Algebraic Operations
In Problems 69–80, evaluate to four significant digits using
a calculator. (Read the instruction booklet accompanying
n
your calculator for the process required to evaluate x.)
69. 0.032 965
70. 419.763
71. 45.0218
4
72. 0.098 553
8
73. 5.477 10 9
7
74. 4.892 1016
5
5
75. 9
9
3
3
76. 2
2
3
4 5
77. 100 and 100
21
1
3
and
3
9
3
79.
1
4
and
2
2
3
80.
3
mn
APPLICATIONS
3 7
78. 500 and 500
20
m n
94. Show that x x for m and n natural numbers
greater than 1.
—Relativistic Mass. The mass M of an object
95. Physics—
moving at a velocity v is given by
3
M
For what real numbers are Problems 81–84 true?
81. x2 x
82. x2 x
3 3
83. x x
3 3
84. x x
In Problems 85 and 86, evaluate each expression on a calculator and determine which pairs have the same value.
Verify these results algebraically.
85. (A) 3 5
(C) 1 3
(E) 8 60
(B) 2 3 2 3
3
(D) 10 63
(F) 6
3
86. (A) 2
2 5
(C) 3 7
(E) 10 84
(B) 8
(D) 3 8 3 8
(F) 1 5
In Problems 87–90, rationalize denominators.
1
3
3
a b
88.
1
3
3
m n
1
90.
x y z
1
89.
x y z
[Hint for Problem 89: Start by multiplying numerator and
denominator by (x y) z.]
Problems 91 and 92 are calculus-related. Rationalize
numerators.
91.
1
C
87.
M0
3
3
x h x
h
kn
92.
n
3
3
t x
tx
93. Show that x km x m for k, m, and n natural numbers
greater than 1.
v2
c2
where M0 mass at rest and c velocity of light. The mass
of an object increases with velocity and tends to infinity as
the velocity approaches the speed of light. Show that M can
be written in the form
M
M0cc2 v2
c2 v2
—Pendulum. A simple pendulum is formed by
96. Physics—
hanging a bob of mass M on a string of length L from a
fixed support (see the figure). The time it takes the bob to
swing from right to left and back again is called the period
T and is given by
T 2
L
g
where g is the gravitational constant. Show that T can be
written in the form
T
2gL
g
Chapter 1 Review
CHAPTER 1 GROUP ACTIVITY
67
Rational Number Representations
The set of real numbers can be partitioned into two disjoint subsets, the set of rational numbers and the set of
irrational numbers. Rational numbers can be represented two ways: as a/b, where a and b are integers and
b 0, and as terminating or repeating decimal expansions. The irrational numbers can be represented as nonrepeating and nonterminating decimal expansions. In this activity, we want to explore the relationship between
the two different methods for representing rational numbers.
Consider the rational number r a/b, where a and b are integers with no common factors and b 0.
1. If b 10n for a positive integer n, what kind of decimal expansion will r have?
2. If b 2m5n for positive integers m and n, what kind of decimal expansion will r have?
3. If r has a terminating decimal expansion, show that r can be expressed in the form a/b, where b 2m5n.
4. Find a and b for the following repeating expansions (the overbar indicates the repeating block):
(A) 0.63
(B) 0.486
(C) 0.846153
5. Find a and b for r 0.19 and then find a terminating decimal expansion for r.
Chapter 1 Review
1-1
ALGEBRA AND REAL NUMBERS
null set and is denoted . A variable is a symbol that represents
unspecified elements from a replacement set. A constant is a
symbol for a single object. If each element of set A is also in set
B, we say A is a subset of B and write A B.
A set is a collection of objects called elements or members of
the set. Sets are usually described by listing the elements or by
stating a rule that determines the elements. A set may be finite
or infinite. A set with no elements is called the empty set or the
Real numbers:
Real numbers (R )
NZQR
Rational numbers (Q ) Irrational numbers (I )
Integers (Z )
Natural numbers (N)
Zero
Noninteger ratios of integers
Negatives of natural numbers
Real number line:
Origin
Negative real numbers
10
5
Coordinate
Positive real numbers
0
10
5
Positive direction