A-7 Radicals
A-57
Section A-7 Radicals
From Rational Exponents to Radicals, and Vice Versa
Properties of Radicals
Simplifying Radicals
Sums and Differences
Products
Rationalizing Operations
What do the following algebraic expressions have in common?
21/2
2x2/3
兹2
3 2
2兹
x
1
x1/2 ⫹ y1/2
1
x
⫹
兹
兹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 Exponents to Radicals, and Vice Versa
We start this discussion by defining an nth-root radical.
n
DEFINITION
1
兹b, nTH-ROOT RADICAL
For
n a natural number greater than 1 and b a real number, we define
n
兹b to be the principal nth root of b (see Definition 2 in Section A-6);
that is,
n
兹b ⫽ b1/n
2
If n ⫽ 2, we write 兹b in place of 兹b
兹25
⫺兹25
⫽ 251/2
⫽5
⫽ ⫺251/2
兹⫺25 is not real.
5
兹32
⫽ ⫺5
5
兹⫺32
⫽ 321/5
⫽2
⫽ (⫺32)1/5
⫽ ⫺2
4
兹0 ⫽ 01/4 ⫽ 0
The symbol 兹 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 A-6, are
useful in this regard:
A-58
Appendix A
A BASIC ALGEBRA REVIEW
RATIONAL EXPONENT/RADICAL CONVERSIONS
For m and n positive integers (n ⬎ 1), and b not negative when n is even,
agbgc
bm/n ⫽
n
(bm)1/n
⫽ 兹苶
bm
(b1/n)m
⫽ (兹苵b)m
22/3 ⫽
n
冦
3
兹22
3
(兹
2)2
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 ⫽ 兹163 ⫽ (兹16)3
3
3
272/3 ⫽ 兹
272 ⫽ (兹
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 ⫽ 兹
x
5
(B) (3u2v3)3/5 ⫽ 兹
(3u2v3)3
(C) y⫺2/3 ⫽
1
1
⫽ 3 2
2/3
y
兹y
or
or
5
(兹
3u2v3)3
3
兹y⫺2
The first is usually
preferred.
兹
3
or
1
y2
Change from radical form to rational exponent form.
5
(D) 兹
6 ⫽ 61/5
MATCHED PROBLEM
1
3 2
(E) ⫺ 兹
x ⫽ ⫺x 2/3
(F) 兹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) 兹
9u
7
(E) ⫺ 兹
(2x)4
3 3
(F) 兹
x ⫹ y3
A-7 Radicals
A-59
Properties of Radicals
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.
THEOREM
1
PROPERTIES OF RADICALS
For n a natural number greater than 1, and x and y positive real numbers,
n
1. 兹x n ⫽ x
n
3.
2
n
2. 兹xy ⫽ 兹x 兹y
n
兹
n
EXAMPLE
n
3
兹x3 ⫽ x
x 兹x
⫽ n
y 兹y
5
5
5
兹xy ⫽ 兹x兹y
兹
4
4
x 兹
x
⫽ 4
y 兹y
Simplifying Radicals
Simplify.
5
(A) 兹
(3x2y)5 ⫽ 3x2y
(B) 兹10 兹5 ⫽ 兹50 ⫽ 兹25 ⴢ 2 ⫽ 兹25 兹2 ⫽ 5兹2
兹
3
(C)
MATCHED PROBLEM
2
CAUTION
3
3
x
兹x
兹x
⫽ 3
⫽
27 兹27
3
or
13
兹x
3
Simplify.
7
(A) 兹
(u2 ⫹ v2)7
(B) 兹6 兹2
兹
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,
兹x2 ⫹ y2 ⫽ 兹x2 ⫹ 兹y2 ⫽ x ⫹ y
but
兹x2 ⫹ 2xy ⫹ y2 ⫽ 兹(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
A-60
Appendix A
A BASIC ALGEBRA REVIEW
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.
SIMPLIFIED (RADICAL) FORM
DEFINITION
2
EXAMPLE
3
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, 兹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, 兹
x violates this condition.
3. No radical appears in a denominator.
For example, y/兹x violates this condition.
4. No fraction appears within a radical.
For example, 兹35 violates this condition.
Finding Simplified Form
Express radicals in simplified form.
(A) 兹12x3y5z2 ⫽ 兹(4x2y4z2)(3xy)
Condition 1 is not met.
⫽ 兹(2xy2z)2(3xy)
x pmy pn ⫽ (x my n) p
⫽ 兹(2xy2z)2 兹3xy
n
n
n
兹xy ⫽ 兹x兹y
⫽ 2xy2z 兹3xy
n
兹x n ⫽ x
3
3
(B) 兹
6x2y 兹
4x5y2
3
⫽兹
(6x2y)(4x5y2)
n
n
n
兹x兹y ⫽ 兹xy
3
⫽兹
24x7y3
3
⫽兹
(8x6y3)(3x)
Condition 1 is not met.
3
⫽兹
(2x2y)3(3x)
x pmy pn ⫽ (x my n) p
3
3
⫽兹
(2x2y)3 兹
3x
n
n
n
兹xy ⫽ 兹x兹y
3
⫽ 2x2y 兹
3x
n
兹x n ⫽ x
6
(C) 兹
16x4y2 ⫽ [(4x2y)2]1/6
⫽ (4x2y)2/6
⫽ (4x2y)1/3
3
⫽兹
4x2y
Condition 2 is not met.
Note the convenience of using rational
exponents.
A-7 Radicals
A-61
3
(D) 兹
兹27 ⫽ [(33)1/2]1/3
⫽ (33)1/6 ⫽ 33/6
MATCHED PROBLEM
3
⫽ 31/2 ⫽ 兹3
Express radicals in simplified form.
(A) 兹18x5y2z3
4
4
(B) 兹
27a3b3 兹
3a5b3
9
(C) 兹
8x6y3
3
(D) 兹兹
4
Sums and Differences
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.
EXAMPLE
4
Combining Like Terms
Combine as many terms as possible.
⫽ (5 ⫹ 4)兹3
(A) 5兹3 ⫹ 4兹3
3
3
(B) 2兹
xy2 ⫺ 7兹
xy2
⫽ 9兹3
3
⫽ (2 ⫺ 7)兹
xy2
3
⫽ ⫺5兹
xy2
3
3
3
3
(C) 3兹xy ⫺ 2兹
xy ⫹ 4兹xy ⫺ 7兹
xy ⫽ 3兹xy ⫹ 4兹xy ⫺ 2兹xy ⫺ 7兹xy
3
⫽ 7兹xy ⫺ 9兹
xy
MATCHED PROBLEM
4
Combine as many terms as possible.
(A) 6兹2 ⫹ 2兹2
5
5
(B) 3兹
2x2y3 ⫺ 8兹
2x2y3
(C) 5兹mn2 ⫺ 3兹mn ⫺ 2兹mn2 ⫹ 7兹mn
3
3
Products
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.
EXAMPLE
5
Multiplication with Radical Forms
Multiply and simplify.
(A) 兹2(兹10 ⫺ 3) ⫽ 兹2兹10 ⫺ 兹2 ⴢ 3 ⫽ 兹20 ⫺ 3兹2 ⫽ 2兹5 ⫺ 3兹2
(B) (兹2 ⫺ 3)(兹2 ⫹ 5) ⫽ 兹2兹2 ⫺ 3兹2 ⫹ 5兹2 ⫺ 15
⫽ 2 ⫹ 2兹2 ⫺ 15
⫽ 2兹2 ⫺ 13
A-62
Appendix A
A BASIC ALGEBRA REVIEW
(C) (兹x ⫺ 3)(兹x ⫹ 5) ⫽ 兹x兹x ⫺ 3兹x ⫹ 5兹x ⫺ 15
⫽ x ⫹ 2兹x ⫺ 15
3
3 2
3
3
3
3
3
3 3
(D) (兹
m⫹兹
n )(兹
m2 ⫺ 兹
n) ⫽ 兹
m3 ⫹ 兹
m2n2 ⫺ 兹
mn ⫺ 兹
n
3
3
⫽m⫺兹
mn ⫹ 兹
m2n2 ⫺ n
MATCHED PROBLEM
5
Multiply and simplify.
(A) 兹3(兹6 ⫺ 4)
(B) (兹3 ⫺ 2)(兹3 ⫹ 4)
(C) (兹y ⫺ 2)(兹y ⫹ 4)
3 2
3 2
3
3
(D) (兹
x ⫺兹
y )(兹
x⫹兹
y)
Rationalizing Operations
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
Example 6, parts C and D):
(a ⫺ b)(a ⫹ b) ⫽ a2 ⫺ b2
(a ⫺ b)(a ⫹ ab ⫹ b ) ⫽ a ⫺ b
(2)
(a ⫹ b)(a ⫺ ab ⫹ b ) ⫽ a ⫹ b
3
(3)
2
2
Explore/Discuss
2
EXAMPLE
6
2
3
3
Use special products in equations (1) to (3) to find a rationalizing factor
for each of the following:
(A) 兹a ⫺ 兹b
(B) 兹a ⫹ 兹b
(C) 兹a ⫺ 兹b
3
3
(D) 兹a ⫹ 兹b
3
3
Rationalizing Denominators
Rationalize denominators.
(A)
Solutions
(1)
3
2
3
兹5
兹
3
(B)
2a2
3b2
(C)
兹x ⫹ 兹y
3兹x ⫺ 2兹y
(D)
1
3
兹m ⫹ 2
(A) 兹5 is a rationalizing factor for 兹5, since 兹5兹5 ⫽ 兹52 ⫽ 5. Thus,
we multiply the numerator and denominator by 兹5 to rationalize the
denominator:
3兹5
3兹5
3
⫽
⫽
5
兹5 兹5兹5
A-7 Radicals
兹
3
(B)
3
3
3 2
2a2 兹
2a2 兹
2a2 兹
3b
⫽
⫽
3
3
2
2
2 3 2
3b
兹3b
兹3b 兹3 b
⫽
3
兹2 ⴢ 32a2b
3
兹33b3
⫽
A-63
3
兹18a2b
3b
(C) The special product in equation (1) suggests that if we multiply the denominator 3兹x ⫺ 2兹y by 3兹x ⫹ 2兹y, we will obtain the difference of two
squares and the denominator will be rationalized.
(兹x ⫹ 兹y)(3兹x ⴙ 2兹y)
兹x ⫹ 兹y
⫽
3兹x ⫺ 2兹y (3兹x ⫺ 2兹y)(3兹x ⴙ 2兹y)
⫽
3兹x2 ⫹ 2兹xy ⫹ 3兹xy ⫹ 2兹y2
(3兹x)2 ⫺ (2兹y)2
⫽
3x ⫹ 5兹xy ⫹ 2y
9x ⫺ 4y
(D) The special product in equation (3) suggests that if we multiply the denom3
3
3
inator 兹m ⫹ 2 by (兹m)2 ⫺ 2兹m ⫹ 22, we will obtain the sum of two cubes
and the denominator will be rationalized.
1
1[(兹m)2 ⴚ 2兹m ⴙ 22]
⫽
3
3
3
3
兹m ⫹ 2 (兹m ⫹ 2)[(兹m)2 ⴚ 2兹m ⴙ 22]
3
兹m2 ⫺ 2兹m ⫹ 4
3
(兹m)3 ⫹ 23
3
⫽
3
3
兹m2 ⫺ 2兹m ⫹ 4
⫽
m⫹8
3
MATCHED PROBLEM
3
Rationalize denominators.
6
(A)
6
兹2x
(B)
10x3
3
兹4x
(C)
兹x ⫹ 2
2兹x ⫹ 3
(D)
1
3
1⫺兹
y
Answers to Matched Problems
5
9
9
5
(B) 兹
(6x2y5)2 or (兹
6x2y5)2
(C) 1/兹
(3xy)3
(D) (9u)1/4
(E) ⫺(2x)4/7
兹u
3 2
1 3 2
2
2
u ⫹v
(B) 2兹3
(C) (兹x )/2 or 2兹x
4 2
3
3
3x2yz兹2xz
(B) 3a2b兹
b ⫽ 3a2b兹b
(C) 兹
2x2y
(D) 兹
2
5
3
(B) ⫺5兹
(C) 3兹
8兹2
2x2y3
mn2 ⫹ 4兹mn
3 2
3
(B) 2兹3 ⫺ 5
(C) y ⫹ 2兹y ⫺ 8
(D) x ⫹ 兹
3兹2 ⫺ 4兹3
xy⫺兹
xy2 ⫺ y
3
3 2
3兹2x
2x ⫹ 兹x ⫺ 6
1 ⫹ 兹y ⫹ 兹y
3
6. (A)
(B) 5x2兹
2x2
(C)
(D)
x
4x ⫺ 9
1⫺y
1.
2.
3.
4.
5.
(A)
(A)
(A)
(A)
(A)
(F) (x3 ⫹ y3)1/3
A-64
Appendix A
A BASIC ALGEBRA REVIEW
EXERCISE A-7
3 3
51. 兹
a ⫹ b3
Unless stated to the contrary, all variables are restricted so
that all quantities involved are real numbers.
In Problems 53–64, rationalize denominators and write in
simplified form.
A
53.
兹2m 兹5
兹20m
54.
兹6 兹8c
兹18c
55.
4a3b2
3
兹2ab2
56.
8x3y5
3
兹4x2y
57.
兹
3y3
4x3
58.
兹
59.
3兹y
2兹y ⫺ 3
60.
5兹x
3 ⫺ 2兹x
61.
2兹5 ⫹ 3兹2
5兹5 ⫹ 2兹2
62.
3兹2 ⫺ 2兹3
3兹3 ⫺ 2兹2
63.
x2
兹x ⫹ 9 ⫺ 3
64.
⫺y2
2 ⫺ 兹y2 ⫹ 4
In Problems 1–8, change to radical form. Do not simplify.
1. m2/3
2. n4/5
3. 6x3/5
4. 7y2/5
5. (4xy3)2/5
6. (7x2y)5/7
7. (x ⫹ y)1/2
8. x1/2 ⫹ y1/2
In Problems 9–16, change to rational exponent form. Do not
simplify.
5
9. 兹
b
10. 兹c
4 3
11. 5兹
x
5 2
12. 7m兹
n
5
13. 兹
(2x2y)3
9
14. 兹
(3m4n)2
3
3
15. 兹
x⫹兹
y
3
16. 兹
x⫹y
3
18. 兹⫺27
19. 兹9x8y4
20. 兹16m4y8
4
21. 兹16m4n8
5
22. 兹32a15b10
23. 兹8a3b5
24. 兹27m2n7
3
25. 兹24x4y7
26. 兹24x5y8
27. 兹m2
28. 兹n6
29. 兹兹xy
30. 兹兹5x
3
3
31. 兹9x2 兹9x
4
5
3
4
10
4
32. 兹2x 兹8xy
1
兹5
34.
2
1
兹7
35.
36.
12y
兹6y
37.
2
兹2 ⫺ 1
39.
兹2
兹6 ⫹ 2
40.
兹2
兹10 ⫺ 2
38.
2
4x2
16y3
兹t ⫺ 兹x
t⫺x
66.
兹x ⫺ 兹y
兹x ⫹ 兹y
67.
兹x ⫹ h ⫺ 兹x
h
68.
兹2 ⫹ h ⫹ 兹2
h
In Problems 69–80, evaluate to four significant digits using a
calculator. (Read the instruction booklet accompanying your
n
calculator for the process required to evaluate 兹x.)
69. 兹0.049 375
70. 兹306.721
71. 兹27.0635
8
72. 兹0.070 144
7
73. 兹0.000 000 008 066
12
74. 兹6,423,000,000,000
3
3
75. 兹7 ⫹ 兹7
5
5
76. 兹4 ⫹ 兹4
6x
兹3x
77. 兹兹2 and 兹2
4
兹6 ⫺ 2
79.
3
4
12
3
1
兹2
and
2
兹4
3
78. 兹兹5 and 兹5
3
80.
6
3
1
兹25
and
5
兹5
3
C
For what real numbers are Problems 81–84 true?
B
In Problems 41–52, write in simplified form.
5 6 7 11
41. x兹
3xy
44.
5
65.
5
In Problems 33–40, rationalize denominators, and write in
simplified form.
33.
4
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.)
In Problems 17–32, write in simplified form.
3
17. 兹⫺8
52. 兹x2 ⫹ y2
5
兹32u12v 8
uv
4
兹32m7n9
2mn
3
42. 2a兹
8a8b13
43.
6
45. 兹a4(b ⫺ a)2
8
46. 兹36(u ⫹ v)6
47. 兹兹a9b3
48. 兹兹x8y6
3
3
49. 兹2x2y4 兹3x5y
4
4
50. 兹4m5n 兹6m3n4
3
4
6
81. 兹x2 ⫽ ⫺x
82. 兹x2 ⫽ x
3
83. 兹x3 ⫽ x
3
84. 兹x3 ⫽ ⫺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) 兹3 ⫹ 兹5
(B) 兹2 ⫹ 兹3 ⫹ 兹2 ⫺ 兹3
(C) 1 ⫹ 兹3
3
(D) 兹10 ⫹ 6兹3
(E) 兹8 ⫹ 兹60
(F) 兹6
A-8 Linear Equations and Inequalities
3
86. (A) 2兹
2 ⫹ 兹5
(B) 兹8
(C) 兹3 ⫹ 兹7
(D) 兹3 ⫹ 兹8 ⫹ 兹3 ⫺ 兹8
(E) 兹10 ⫹ 兹84
(F) 1 ⫹ 兹5
87.
1
3
3
兹a ⫺ 兹b
88.
1
3
3
兹m ⫹ 兹n
89.
1
兹x ⫺ 兹y ⫹ 兹z
90.
1
兹x ⫹ 兹y ⫺ 兹z
[Hint for Problem 89: Start by multiplying numerator and
denominator by (兹x ⫺ 兹y) ⫺ 兹z.]
Problems 91 and 92 are calculus-related. Rationalize
numerators.
91.
3
3
兹x ⫹ h ⫺ 兹x
h
92.
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⫽
In Problems 87–90, rationalize denominators.
M0c兹c2 ⫺ 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
兹
T ⫽ 2␲
L
g
where g is the gravitational constant. Show that T can be
written in the form
T⫽
3
3
兹t ⫺ 兹x
t⫺x
A-65
2␲兹gL
g
kn km
n m
93. Show that 兹
x ⫽兹
x for k, m, and n natural numbers
greater than 1.
mn
n
94. Show that 兹兹
x ⫽ 兹x for m and n natural numbers
greater than 1.
m
APPLICATIONS
_Relativistic Mass. The mass M of an object
95. Physics_
moving at a velocity v is given by
M⫽
M0
兹
1⫺
v2
c2
Section A-8 Linear Equations and Inequalities
Equations
Solving Linear Equations
Inequality Relations and Interval Notation
Solving Linear Inequalities
Equations
An algebraic equation is a mathematical statement that relates two algebraic
expressions involving at least one variable. Some examples of equations with x
as a variable are
3x ⫺ 2 ⫽ 7
1
x
⫽
1⫹x x⫺2
2x2 ⫺ 3x ⫹ 5 ⫽ 0
兹x ⫹ 4 ⫽ x ⫺ 1