Download Roots, Radicals, and Square Root Equations

OpenStax-CNX module: m18879
1
Roots, Radicals, and Square Root
∗
Equations: Square Root Expressions
Wade Ellis
Denny Burzynski
This work is produced by OpenStax-CNX and licensed under the
Creative Commons Attribution License 2.0†
Abstract
This module is from Elementary Algebra by Denny Burzynski and Wade Ellis, Jr. The distinction
between the principal square root of the number x and the secondary square root of the number x is
made by explanation and by example. The simplication of the radical expressions that both involve
and do not involve fractions is shown in many detailed examples; this is followed by an explanation of
how and why radicals are eliminated from the denominator of a radical expression. Real-life applications
of radical equations have been included, such as problems involving daily output, daily sales, electronic
resonance frequency, and kinetic energy. Objectives of this module: understand the concept of square
root, be able to distinguish between the principal and secondary square roots of a number, be able to
relate square roots and meaningful expressions and to simplify a square root expression.
1 Overview
•
•
•
•
Square Roots
Principal and Secondary Square Roots
Meaningful Expressions
Simplifying Square Roots
2 Square Roots
2
When we studied exponents in Section , we noted that 42 = 16 and (−4) = 16. We can see that 16 is the
square of both 4 and −4. Since 16 comes from squaring 4 or −4, 4 and −4 are called the square roots
of 16. Thus 16 has two square roots, 4 and −4. Notice that these two square roots are opposites of each other.
We can say that
Square Root
The square root of a positive number x is a number such that when it is squared the number x results.
Every positive number has two square roots, one positive square root and one negative square root.
Furthermore, the two square roots of a positive number are opposites of each other. The square root of 0 is
0.
∗
†
Version 1.5: Jun 1, 2009 11:48 am +0000
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3 Sample Set A
Example 1
The two square roots of 49 are 7 and −7 since
2
and
72 = 49
(−7) = 49
Example 2
The two square roots of
7 2
8
=
7
8
·
7
8
=
49
64
49
64
are
and
7
8
and
−7 2
8
−7
8
=
since
−7
8
·
−7
8
=
49
64
4 Practice Set A
Name both square roots of each of the following numbers.
Exercise 1
(Solution on p. 12.)
Exercise 2
(Solution on p. 12.)
Exercise 3
(Solution on p. 12.)
Exercise 4
(Solution on p. 12.)
Exercise 5
(Solution on p. 12.)
Exercise 6
(Solution on p. 12.)
36
25
100
64
1
1
4
Exercise 7
(Solution on p. 12.)
9
16
Exercise 8
(Solution on p. 12.)
0.1
Exercise 9
(Solution on p. 12.)
0.09
5 Principal and Secondary Square Roots
There is a notation for distinguishing the positive square root of a number x from the negative square root
of x.
√
Principal Square Root:
x
If x is a positive real number, then
√
x represents the positive square root of x. The positive square root of a number is called the princi-
pal square root of the number.
√
Secondary Square Root: − x
√
− x represents the negative square root of x. The negative square root of a number is called the secondary
square root of the number.
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√
− x indicates the secondary square root of x.
Radical Sign, Radicand, and Radical
√
In the expression x,
q
is called a radical sign.
x is called the radicand.
√
x is called a radical.
The horizontal bar that appears attached to the radical sign,
the radicand.
Because
√
q
, is a grouping symbol that species
√
x and − x are the two square root of x,
√
√
( x) ( x) = x
and
√
√
(− x) (− x) = x
6 Sample Set B
Write the principal and secondary square roots of each number.
Example 3
Principal square root is
√
9 = 3.
√
Secondary square root is − 9 = −3.
9.
Example 4
15.
Principal square root is
√
15.
√
Secondary square root is − 15.
Example 5
Use a calculator to obtain a decimal approximation for the two square roots of 34. Round to two
decimal places.
On the Calculater
Type
34
√
Press
Display reads:
x
5.8309519
Round to 5.83.
q
Notice that the square root symbol on the calculator is
. This means, of course, that a calculator will produce only the positive square root. We must supply the negative square root ourselves.
√
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34 ≈ 5.83
and
√
− 34 ≈ −5.83
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Note:
The symbol ≈ means "approximately equal to."
Example 6 √
The number
Since 72 = 49,
Since 82 = 64,
7<
√
Thus,
4
50 is between what two whole numbers?
√
√
49 = 7.
64 = 8. Thus,
50 < 8
√
50 is a number between 7 and 8.
7 Practice Set B
Write the principal and secondary square roots of each number.
Exercise 10
(Solution on p. 12.)
Exercise 11
(Solution on p. 12.)
Exercise 12
(Solution on p. 12.)
Exercise 13
(Solution on p. 12.)
100
121
35
Use a calculator to obtain a decimal approximation for the two square roots of 35. Round to two
decimal places.
8 Meaningful Expressions
Since we√know that the square of any real number is a positive number or zero, we can see that expressions
such as −16
√ do not describe real numbers. There is no real number that can be squared that will produce
−16. For x to be a real number, we must have x ≥ 0. In our study of algebra, we will assume that all
variables and all expressions in radicands represent nonnegative numbers (numbers greater than or equal to
zero).
9 Sample Set C
Write the proper restrictions that must be placed on the variable so that each expression represents a real
number.
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Example
7
√
For
x − 3 to be a real number, we must have
or
x−3≥0
x≥3
Example
8
√
For
2m + 7 to be a real number, we must have
2m + 7 ≥ 0
or
2m ≥ −7
or
m≥
−7
2
10 Practice Set C
Write the proper restrictions that must be placed on the variable so that each expression represents a real
number.
Exercise
14
√
(Solution on p. 12.)
Exercise 15
(Solution on p. 12.)
x+5
√
y−8
Exercise
16
√
(Solution on p. 12.)
Exercise
17
√
(Solution on p. 12.)
3a + 2
5m − 6
11 Simplifying Square Roots
When variables occur in the radicand, we can often simplify the expression by removing the radical sign.
We can do so by keeping in mind that the radicand is the square of some other expression. We can simplify
a radical by seeking an expression whose square is the radicand. The following observations will help us nd
the square root of a variable quantity.
Example 9
Since x3
2
= x3 ·
2
= x6 , x3 is a square root of x6 . Also
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Example 10
Since x4
2
= x4·2 = x8 , x4 is a square root of x8 . Also
Example 11
Since x6
2
= x6·2 = x12 , x6 is a square root of x12 . Also
These examples suggest the following rule:
If a variable has an even exponent, its square root can be found by dividing that exponent by 2.
The examples of Sample Set B illustrate the use of this rule.
12 Sample Set D
Simplify each expression by removing the radical sign. Assume each variable is
nonnegative
.
Example 12
√
√
2
a2 .
We seek an expression whose square is a2 . Since (a) = a2 ,
a2 = a
Notice that 2 ÷ 2 = 1.
Example 13
p
y8 .
p
y8 = y4
We seek an expression whose square is y 8 . Since y 4
2
= y8 ,
Notice that 8 ÷ 2 = 4.
Example 14
√
√
25m2 n6 .
We seek an expression whose square is 25m2 n6 . Since 5mn3
25m2 n6 = 5mn3
Notice that 2 ÷ 2 = 1 and 6 ÷ 2 = 3.
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2
= 25m2 n6 ,
OpenStax-CNX module: m18879
Example
15
q
7
4
− 121a10 (b − 1) .
h
i2
2
11a5 (b − 1)
q
4
121a10 (b − 1)
q
4
Then, − 121a10 (b − 1)
4
We seek an expression whose square is 121a10 (b − 1) . Since
4
=
121a10 (b − 1) ,
=
11a5 (b − 1)
=
−11a5 (b − 1)
2
2
Notice that 10 ÷ 2 = 5 and 4 ÷ 2 = 2.
13 Practice Set D
Simplify each expression by removing the radical sign. Assume each variable is nonnegative.
Exercise
18
p
(Solution on p. 12.)
Exercise
19
√
(Solution on p. 12.)
Exercise
20
p
(Solution on p. 12.)
Exercise
21
p
(Solution on p. 12.)
y8
16a4
49x4 y 6
−
100x8 y 12 z 2
Exercise
22
q
−
(Solution on p. 12.)
36(a + 5)
Exercise
23
q
225w4 (z 2
4
(Solution on p. 12.)
− 1)
2
Exercise
24
p
(Solution on p. 12.)
Exercise
25
√
(Solution on p. 12.)
Exercise
26
√
(Solution on p. 13.)
0.25y 6 z 14
x2n , where n is a natural number.
x4n , where n is a natural number.
14 Exercises
Exercise 27
How many square roots does every positive real number have?
Exercise 28 q
The symbol
Exercise 29 q
The symbol (Solution on p. 13.)
represents which square root of a number?
(Solution on p. 13.)
represents which square root of a number?
For the following problems, nd the two square roots of the given number.
Exercise 30
64
Exercise 31
81
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(Solution on p. 13.)
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Exercise 32
25
Exercise 33
121
(Solution on p. 13.)
Exercise 34
144
Exercise 35
225
(Solution on p. 13.)
Exercise 36
10,000
Exercise 37
(Solution on p. 13.)
1
16
Exercise 38
1
49
Exercise 39
(Solution on p. 13.)
25
36
Exercise 40
121
225
Exercise 41
(Solution on p. 13.)
0.04
Exercise 42
0.16
Exercise 43
(Solution on p. 13.)
1.21
For the following problems, evaluate each expression. If the expression does not represent a real number,
write "not a real number."
Exercise
44
√
49
Exercise
45
√
(Solution on p. 13.)
64
Exercise
46
√
− 36
Exercise
47
√
(Solution on p. 13.)
− 100
Exercise
48
√
− 169
Exercise
49
q
−
(Solution on p. 13.)
36
81
Exercise
50
q
−
121
169
Exercise
51
√
−225
Exercise
52
√
−36
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(Solution on p. 13.)
OpenStax-CNX module: m18879
Exercise
53
√
9
(Solution on p. 13.)
− −1
Exercise
54
√
− −5
Exercise
√ 55
(Solution on p. 13.)
− − 9
Exercise
√ 56
− − 0.81
For the following problems, write the proper restrictions that must be placed on the variable so that the
expression represents a real number.
Exercise 57
√
(Solution on p. 13.)
y + 10
Exercise
58
√
x+4
Exercise
59
√
(Solution on p. 13.)
a − 16
Exercise
60
√
h − 11
Exercise
61
√
(Solution on p. 13.)
2k − 1
Exercise
62
√
7x + 8
Exercise
63
√
(Solution on p. 13.)
−2x − 8
Exercise 64
√
−5y + 15
For the following problems, simplify each expression by removing the radical sign.
Exercise
65
√
(Solution on p. 13.)
m6
Exercise
66
√
k 10
Exercise
67
√
(Solution on p. 13.)
a8
Exercise
68
√
h16
Exercise
69
p
(Solution on p. 13.)
x4 y 10
Exercise
70
√
a6 b20
Exercise
71
√
(Solution on p. 13.)
a4 b6
Exercise
72
p
x8 y 14
Exercise
73
√
81a2 b2
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(Solution on p. 13.)
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Exercise
74
p
49x6 y 4
Exercise
75
√
(Solution on p. 14.)
100m8 n2
Exercise
76
p
225p14 r16
Exercise
77
p
(Solution on p. 14.)
36x22 y 44
Exercise
78
q
169w4 z 6 (m − 1)
Exercise
79
q
4
Exercise
80
q
14
(Solution on p. 14.)
25x12 (y − 1)
64a10 (a + 4)
Exercise
81
q
9m6 n4 (m
2
(Solution on p. 14.)
18
+ n)
Exercise
82
√
25m26 n42 r66 s84
Exercise
83
q
(Solution on p. 14.)
2
(f − 2) (g + 6)
4
Exercise
84
q
6
(2c − 3) + (5c + 1)
2
Exercise
85
√
(Solution on p. 14.)
− 64r4 s22
Exercise
86
q
−
121a6 (a − 4)
8
Exercise
87
(Solution on p. 14.)
q
2
− − (w + 6)
Exercise
q 88
− −
4a2 b2 (c2
+ 8)
2
Exercise
89
√
(Solution on p. 14.)
1.21h4 k 4
Exercise
90
p
2.25m6 p6
Exercise
91
q
−
Exercise
q 92
−
(Solution on p. 14.)
169a2 b4 c6
196x4 y 6 z 8
81y 4 (z−1)2
225x8 z 4 w6
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15 Exercised for Review
Exercise 93
(Solution on p. 14.)
() Find the quotient.
Exercise 94
() Find the sum.
1
x+1
x2 −1
4x2 −1
+
3
x+1
÷
+
x−1
2x+1 .
2
x2 −1 .
Exercise 95
() Solve the equation, if possible:
Exercise 96
() Perform the division:
Exercise 97
() Perform the division:
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(Solution on p. 14.)
1
x−2
=
3
x2 −x−2
−
3
x+1 .
15x3 −5x2 +10x
.
5x
(Solution on p. 14.)
x3 −5x2 +13x−21
.
x−3
OpenStax-CNX module: m18879
Solutions to Exercises in this Module
Solution to Exercise (p. 2)
6 and −6
Solution to Exercise (p. 2)
5 and −5
Solution to Exercise (p. 2)
10 and −10
Solution to Exercise (p. 2)
8 and −8
Solution to Exercise (p. 2)
1 and −1
Solution to Exercise (p. 2)
1
2
and −
3
4
and −
1
2
Solution to Exercise (p. 2)
3
4
Solution to Exercise (p. 2)
0.1 and −0.1
Solution to Exercise (p. 2)
0.03 and −0.03
Solution
to Exercise
(p. 4)
√
√
100 = 10 and − 100 = −10
Solution
to Exercise
(p. 4)
√
√
121 = 11 and − 121 = −11
Solution
to√Exercise (p. 4)
√
35 and − 35
Solution to Exercise (p. 4)
5.92 and − 5.92
Solution to Exercise (p. 5)
x ≥ −5
Solution to Exercise (p. 5)
y≥8
Solution to Exercise (p. 5)
a ≥ − 23
Solution to Exercise (p. 5)
m≥
6
5
Solution to Exercise (p. 7)
y4
Solution to Exercise (p. 7)
4a2
Solution to Exercise (p. 7)
7x2 y 3
Solution to Exercise (p. 7)
−10x4 y 6 z
Solution to Exercise (p. 7)
−6(a + 5)
2
Solution to Exercise (p. 7)
15w2 z 2 − 1
Solution to Exercise (p. 7)
0.5y 3 z 7
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OpenStax-CNX module: m18879
Solution to Exercise (p. 7)
xn
Solution to Exercise (p. 7)
x2n
Solution to Exercise (p. 7)
two
Solution to Exercise (p. 7)
secondary
Solution to Exercise (p. 7)
9 and −9
Solution to Exercise (p. 8)
11 and −11
Solution to Exercise (p. 8)
15 and −15
Solution to Exercise (p. 8)
1
4
and −
5
6
and −
1
4
Solution to Exercise (p. 8)
5
6
Solution to Exercise (p. 8)
0.2 and − 0.2
Solution to Exercise (p. 8)
1.1 and − 1.1
Solution to Exercise (p. 8)
8
Solution to Exercise (p. 8)
−10
Solution to Exercise (p. 8)
− 23
Solution to Exercise (p. 8)
not a real number
Solution to Exercise (p. 8)
not a real number
Solution to Exercise (p. 9)
3
Solution to Exercise (p. 9)
y ≥ −10
Solution to Exercise (p. 9)
a ≥ 16
Solution to Exercise (p. 9)
k≥
1
2
Solution to Exercise (p. 9)
x ≤ −4
Solution to Exercise (p. 9)
m3
Solution to Exercise (p. 9)
a4
Solution to Exercise (p. 9)
x2 y 5
Solution to Exercise (p. 9)
a2 b3
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Solution to Exercise (p. 9)
9ab
Solution to Exercise (p. 10)
10m4 n
Solution to Exercise (p. 10)
6x11 y 22
Solution to Exercise (p. 10)
5x6 (y − 1)
2
Solution to Exercise (p. 10)
9
3m3 n2 (m + n)
Solution to Exercise (p. 10)
(f − 2) (g + 6)
4
Solution to Exercise (p. 10)
−8r2 s11
Solution to Exercise (p. 10)
w+6
Solution to Exercise (p. 10)
1.1h2 k 2
Solution to Exercise (p. 10)
2 3
13ab c
− 14x
2 y3 z4
Solution to Exercise (p. 11)
x+1
2x−1
Solution to Exercise (p. 11)
No solution; x = 2 is excluded.
Solution to Exercise (p. 11)
x2 − 2x + 7
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