9.2
Simplifying Radical Expressions
OBJECTIVES
9.2
1. Simplify expressions involving numeric radicals
2. Simplify expressions involving algebraic radicals
In Section 9.1, we introduced the radical notation. For most applications, we will want to
make sure that all radical expressions are in simplest form. To accomplish this, the following three conditions must be satisfied.
Rules and Properties: Square Root Expressions in
Simplest Form
An expression involving square roots is in simplest form if
1. There are no perfect-square factors in a radical.
2. No fraction appears inside a radical.
3. No radical appears in the denominator.
For instance, considering condition 1,
117
is in simplest form because 17 has no perfect-square factors
whereas
112
is not in simplest form
because it does contain a perfect-square factor.
112 14 3
A perfect square
To simplify radical expressions, we’ll need to develop two important properties. First, look
at the following expressions:
14 9 136 6
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14 19 2 3 6
Because this tells us that 14 9 14 19, the following general rule for radicals is
suggested.
Rules and Properties: Property 1 of Radicals
For any positive real numbers a and b,
1ab 1a 1b
In words, the square root of a product is the product of the square roots.
707
708
CHAPTER 9
EXPONENTS AND RADICALS
Let’s see how this property is applied in simplifying expressions when radicals are
involved.
Example 1
Simplifying Radical Expressions
NOTE Perfect-square factors
are 1, 4, 9, 16, 25, 36, 49, 64, 81,
100, and so on.
Simplify each expression.
(a) 112 14 3
A perfect square
NOTE Apply Property 1.
14 13
NOTE Notice that we have
213
removed the perfect-square
factor from inside the radical,
so the expression is in simplest
form.
NOTE It would not have
helped to write
(b) 145 19 5
A perfect square
19 15
145 115 3
because neither factor is a
perfect square.
NOTE We look for the largest
perfect-square factor, here 36.
3 15
(c) 172 136 2
A perfect square
136 12
NOTE Then apply Property 1.
6 12
(d) 5118 5 19 2
A perfect square
5 19 12 5 3 12 1512
Be Careful! Even though
1a b 1a 1b
1a b
is not the same as
1a 1b
Let a 4 and b 9, and substitute.
1a b 14 9 113
1a 1b 14 19 2 3 5
Because 113 5, we see that the expressions 1a b and 1a 1b are not in general
the same.
CHECK YOURSELF 1
Simplify.
(a) 120
(b) 175
(c) 198
(d) 148
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C A U TI ON
SIMPLIFYING RADICAL EXPRESSIONS
SECTION 9.2
709
The process is the same if variables are involved in a radical expression. In our
remaining work with radicals, we will assume that all variables represent positive real
numbers.
Example 2
Simplifying Radical Expressions
Simplify each of the following radicals.
(a) 2x3 2x2 x
A perfect square
2x2 1x
NOTE By our first rule for
radicals.
x1x
NOTE 2x2 x (as long as x is
positive).
(b) 24b3 24 b2 b
Perfect squares
24b2 1b
2b1b
NOTE Notice that we want the
(c) 218a5 29 a4 2a
perfect-square factor to have
the largest possible even
exponent, here 4. Keep in mind
that
Perfect squares
29a4 12a
a2 a2 a4
3a2 12a
CHECK YOURSELF 2
Simplify.
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(a) 29x3
(b) 227m3
(c) 250b5
To develop a second property for radicals, look at the following expressions:
16
14 2
A4
116
4
2
14
2
Because
116
16
, a second general rule for radicals is suggested.
14
A4
710
CHAPTER 9
EXPONENTS AND RADICALS
Rules and Properties: Property 2 of Radicals
For any positive real numbers a and b,
1a
a
1b
Ab
In words, the square root of a quotient is the quotient of the square roots.
This property is used in a fashion similar to Property 1 in simplifying radical expressions. Remember that our second condition for a radical expression to be in simplest form
states that no fraction should appear inside a radical. Example 3 illustrates how expressions
that violate that condition are simplified.
Example 3
Simplifying Radical Expressions
Write each expression in simplest form.
write the numerator and
denominator as separate
radicals.
NOTE Apply Property 2.
(b)
3
2
2
12
A 25
125
NOTE Apply Property 2.
(c)
Remove any
perfect squares
from the radical.
12
5
8x2
28x2
B 9
19
NOTE Factor 8x2 as 4x2 2.
24x2 2
3
NOTE Apply Property 1 in the
24x2 12
3
2x12
3
numerator.
CHECK YOURSELF 3
Simplify.
(a)
25
A 16
(b)
7
A9
(c)
12x2
B 49
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NOTE Apply Property 2 to
9
19
(a)
A4
14
SIMPLIFYING RADICAL EXPRESSIONS
SECTION 9.2
711
In our previous examples, the denominator of the fraction appearing in the radical was
a perfect square, and we were able to write each expression in simplest radical form by
removing that perfect square from the denominator.
If the denominator of the fraction in the radical is not a perfect square, we can still apply
Property 2 of radicals. As we will see in Example 4, the third condition for a radical to be
in simplest form is then violated, and a new technique is necessary.
Example 4
Simplifying Radical Expressions
Write each expression in simplest form.
NOTE We begin by applying
(a)
Property 2.
1
11
1
A3
13
13
1
is still not in simplest form because of the radical in the denominator?
13
To solve this problem, we multiply the numerator and denominator by 13. Note that the
denominator will become
Do you see that
13 13 19 3
We then have
NOTE We can do this because
we are multiplying the fraction
13
by
or 1, which does not
13
change its value.
1
1 13
13
13
13 13
3
The expression
(b)
NOTE
12 15 12 5 110
15 15 5
13
is now in simplest form because all three of our conditions are satisfied.
3
2
12
A5
15
12 15
15 15
110
5
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and the expression is in simplest form because again our three conditions are satisfied.
(c)
NOTE We multiply numerator
and denominator by 17 to
“clear” the denominator of the
radical. This is also known as
“rationalizing” the
denominator.
3x
13x
A7
17
13x 17
17 17
121x
7
The expression is in simplest form.
CHAPTER 9
EXPONENTS AND RADICALS
CHECK YOURSELF 4
Simplify.
(a)
1
A2
(b)
2
A3
(c)
2y
A5
Both of the properties of radicals given in this section are true for cube roots, fourth
roots, and so on. Here we have limited ourselves to simplifying expressions involving
square roots.
CHECK YOURSELF ANSWERS
1. (a) 215; (b) 513; (c) 712; (d) 413
(c) 5b2 12b
5
17
2x13
3. (a) ; (b)
; (c)
4
3
7
2. (a) 3x1x; (b) 3m13m;
4. (a)
110y
12
16
; (b)
; (c)
2
3
5
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712
Name
Exercises
9.2
Section
Use Property 1 to simplify each of the following radical expressions. Assume that all
variables represent positive real numbers.
1. 118
2. 150
Date
ANSWERS
1.
2.
3. 128
4. 1108
3.
4.
5. 145
5.
6. 180
6.
7. 148
8. 1125
7.
8.
9.
9. 1200
10. 196
10.
11.
11. 1147
12. 1300
12.
13.
13. 3112
14. 5124
14
15.
2
15. 25x
2
16. 27a
16.
17.
4
17. 23y
6
18. 210x
18.
19.
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20.
19. 22r
3
20. 25a
5
21.
22.
2
21. 227b
4
22. 298m
23.
24
4
23. 224x
3
24. 272x
713
ANSWERS
25.
25. 254a5
26. 2200y6
3 2
27. 2x y
2 5
28. 2a b
26.
27.
28.
Use Property 2 to simplify each of the following radical expressions.
29.
30.
29.
4
A 25
30.
64
A9
31.
9
A 16
32.
49
A 25
33.
3
A4
34.
5
A9
35.
5
A 36
36.
10
A 49
31.
32.
33.
34.
35.
36.
Use the properties for radicals to simplify each of the following expressions. Assume that
all variables represent positive real numbers.
37.
38.
37.
8a2
B 25
38.
12y2
B 49
39.
1
A5
40.
1
A7
41.
3
A2
42.
5
A3
43.
3a
A5
44.
2x
A7
45.
2x2
B 3
46.
5m2
B 2
47.
8s3
B 7
48.
12x3
B 5
39.
40.
41.
42.
44.
45.
46.
47.
48.
714
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43.
ANSWERS
49.
Decide whether each of the following is already written in simplest form. If it is not,
explain what needs to be done.
50.
49. 110mn
51.
50. 118ab
52.
51.
98x2y
B 7x
52.
16xy
3x
53.
54.
53. Find the area and perimeter of this square:
3
3
One of these measures, the area, is a rational number, and the other, the perimeter, is
an irrational number. Explain how this happened. Will the area always be a rational
number? Explain.
n2 1
n2 1
, n,
using odd values of n:
2
2
1, 3, 5, 7, etc. Make a chart like the one below and complete it.
54. (a) Evaluate the three expressions
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n
a
n2 1
2
bn
c
n2 1
2
a2
b2
c2
1
3
5
7
9
11
13
15
(b) Check for each of these sets of three numbers to see if this statement is true:
2a2 b2 2c2. For how many of your sets of three did this work? Sets of
three numbers for which this statement is true are called “Pythagorean triples”
because a2 b2 c2. Can the radical equation be written in this way:
2a2 b2 a b? Explain your answer.
715
ANSWERS
a.
Getting Ready for Section 9.3 [Section 1.6]
b.
Use the distributive property to combine the like terms in each of the following
expressions.
c.
d.
(a)
(c)
(e)
(g)
e.
f.
5x 6x
10y 12y
9a 7a 12a
12m 3n 6m
(b)
(d)
(f)
(h)
8a 3a
7m 10m
5s 8s 4s
8x 5y 4x
g.
Answers
h.
1. 312
13. 613
23. 2x2 16
3. 217
15. x15
5. 315
7. 413
9. 1012
11. 713
17. y2 13
19. r12r
21. 3b13
25. 3a2 16a
27. xy1x
29.
2
5
3
4
115a
43.
5
31.
15
2a12
15
16
37.
39.
41.
6
5
5
2
2s114s
47.
49. Simplest form
7
51. Remove the perfect-square factors from the radical and simplify.
35.
c. 2y
d. 17m
e. 4a
f. s
53.
g. 6m 3n
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b. 5a
a. 11x
h. 4x 5y
13
2
x16
45.
3
33.
716