S I M P L I F Y I N... 8.2 I n t h i s

420
(8-8)
Chapter 8
Powers and Roots
8.2
In this
section
SIMPLIFYING SQUARE ROOTS
In Section 8.1 you learned to simplify some radical expressions using the product
rule. In this section you will learn three basic rules to follow for writing expressions
involving square roots in simplest form. These rules can be extended to radicals
with index greater than 2, but we will not do that in this text.
●
Using the Product Rule
●
Rationalizing the
Denominator
Using the Product Rule
●
Simplified Form of a Square
Root
We can use the product rule to simplify square roots of certain numbers. For
example,
5 9
5
4
Factor 45 as 9 5.
9
5
Product rule for radicals
35
9 3
Because 45 is not a perfect square, we cannot write 4
5 without the radical sym is considered a simpler expression that represents the exact
bol. However, 35
value of 45. When simplifying square roots, we can factor the perfect squares out
of the radical and replace them with their square roots. Look for the factors
4,
E X A M P L E
1
calculator
9,
16,
25,
36,
Simplifying radicals using the product rule
Simplify.
2
b) 5
0
a) 1
49,
and so on.
c) 7
2
Solution
a) Because 12 4 3, we can use the product rule to write
12 4
3
23.
close-up
You can use a calculator to see
that 12
and 23 agree for
the first 10 digits (out of infinitely many). Having the same
first 10 digits does not make
12
= 23. The product rule
for radicals guarantees that
they are equal.
b) 5
0 2
5 2 52
c) Note that 4, 9, and 36 are perfect squares and are factors of 72. In factoring out a
perfect square, it is most efficient to use the largest perfect square:
62
72 36 2
If we had factored out 9, we could still get the correct answer as follows:
72 9 8
3 8 3 4
2
3 22 62
■
Rationalizing the Denominator
Radicals such as 2
, 3, and 5 are irrational numbers. So a fraction such
3
as has an irrational denominator. Because fractions with rational denominators
5
are considered simpler than fractions with irrational denominators, we usually
convert fractions with irrational denominators to equivalent ones with rational
denominators. That is, we rationalize the denominator.
8.2
E X A M P L E
2
Simplifying Square Roots
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421
Rationalizing denominators
Simplify each expression by rationalizing its denominator.
3
3
a) b) 7
5
Solution
a) Because 5 5
5, we multiply numerator and denominator by 5:
3 5
3
5 5 5
35
5
Multiply numerator and denominator by 5.
5 5
5
b) Because 7 7
7, multiply the numerator and denominator by 7:
3 3
7
7 7 7
21
7
Multiply numerator and denominator by 7.
Product rule for radicals
■
Simplified Form of a Square Root
When we simplify any expression, we try to write a “simpler” expression that is
equivalent to the original. However, one person’s idea of simpler is sometimes different from another person’s. For a square root the expression must satisfy three
conditions to be in simplified form. These three conditions provide specific rules to
follow for simplifying square roots.
Simplified Form for Square Roots
An expression involving a square root is in simplified form if it has
1. no perfect-square factors inside the radical,
2. no fractions inside the radical, and
3. no radicals in the denominator.
Because a decimal is a form of a fraction, a simplified square root should not
contain any decimal numbers. Also, a simplified expression should use the fewest
rather than 2 3 even though
number of radicals possible. So we write 6
and 3 are both in simplified form.
both 2
E X A M P L E
3
Simplified form for square roots
Write each radical expression in simplified form.
2
a) 3
00
b) 5
10
c) 6
Solution
a) We must remove the perfect square factor of 100 from inside the radical:
003 1
00 3 103
300 1
422
(8-10)
Chapter 8
Powers and Roots
b) We first use the quotient rule to remove the fraction
calculator
5 2
5
2
close-up
from inside the radical:
Quotient rule for radicals
2 5
5 5
10
5
Using a calculator to check
simplification problems will
help you to understand the
concepts.
2
5
Rationalize the denominator.
Product rule for radicals
c) The numerator and denominator have a common factor of 2
:
10 2 5
6
2 3
5
3
5 3
3 3
15
3
Product rule for radicals
Reduce.
Rationalize the denominator.
Product rule for radicals
10
Note that we could have simplified by first using the quotient rule to get
10
6
10
6 and then reducing
10
.
6
6
Another way to simplify
10
6
is to first
. You should try these alternamultiply the numerator and denominator by 6
15
tives. Of course, the simplified form is by any method.
3
■
In the next example we simplify some expressions involving variables. Remember that any exponential expression with an even exponent is a perfect square.
E X A M P L E
4
Radicals containing variables
Simplify each expression. All variables represent nonnegative real numbers.
a) x3
b) 8
a9
c) 1
8a4b7
Solution
a) x3 x2x
x2 x
xx
The largest perfect square factor of x 3 is x 2.
Product rule for radicals
For any nonnegative x, x2 x.
b) 8
a9 4
a8 2
a
2a 42a
The largest perfect square factor of 8a 9 is 4a 8.
4a8 2a 4
c) 1
8a4b7 9
a4b6 2b
3a b 2b
2 3
Factor out the perfect squares.
9a4
b6 3a 2b3
■
8.2
Simplifying Square Roots
(8-11)
423
If square roots of variables appear in the denominator, then we rationalize the
denominator.
E X A M P L E
5
Radicals containing variables
Simplify each expression. All variables represent positive real numbers.
5
a) a
a
b) b
2
c) 6a
helpful
hint
If you are going to compute
the value of a radical expression with a calculator, it
doesn’t matter if the denominator is rational. However,
rationalizing the denominator
provides another opportunity
to practice building up the
denominator of a fraction and
multiplying radicals.
Solution
5
5 a
a) a a a
5a
a
b)
b a
b
a
Multiply numerator and denominator by a
.
a a
a
Quotient rule for radicals
a b
b b
Rationalize the denominator.
ab
b
Product rule for radicals
2 6a
2
c) 6
a 6
a
6a
12a
6a
4 3
a
6a
23a
6a
23a
2 3a
3a
3a
CAUTION
Rationalize the denominator.
Product rule for radicals
Factor out the perfect square.
4 2
Factor the denominator.
Divide out the common factor 2.
■
Do not attempt to reduce an expression like the one in
Example 5(c):
3a
3a
You cannot divide out common factors when one is inside a radical.
424
(8-12)
Chapter 8
WARM-UPS
Powers and Roots
True or false? Explain your answer.
1. 2
0 25
True 2. 18 92
False
3
9 3
1
3. True
4. False
3
4 2
3
3
5. a aa
for any positive value of a. True
3
9
6. a a for any positive value of a. False
7. y17 y 8y for any positive value of y. True
6
8. 3 False
9. 4 2
False 10. 2
83 17
2
8. 2
False
EXERCISES
Reading and Writing After reading this section, write out the
answers to these questions. Use complete sentences.
1. How do we simplify a radical with the product rule?
We use the product rule to factor out a perfect square from
inside a square root.
2. Which integers are perfect squares?
The perfect squares are 1, 4, 9, 16, 25, and so on.
3. What does it mean to rationalize a denominator?
To rationalize a denominator means to rewrite the expression so that the denominator is a rational number.
4. What is simplified form for a square root?
A square root in simplified form has no perfect squares or
fractions inside the radical and no radicals in the
denominator.
5. How do you simplify a square root that contains a variable?
To simplify a square root containing variables, use the same
techniques as we use on square roots of numbers.
6. How can you tell if an exponential expression is a perfect
square?
Any even power of a variable is a perfect square.
Assume that all variables in the exercises represent positive real
numbers.
Simplify each radical. See Example 1.
7. 8
22
8. 20
25
9. 24
26
10. 7
5
53
11. 2
8
27
12. 40
210
13. 9
0
310
14. 200
102
15. 500
105
16. 9
8
72
17. 150
56
18. 120
230
Simplify each expression by rationalizing the denominator. See
Example 2.
1
1
3
19. 20. 21. 5
6
2
5
6
32
5
6
2
4
3
7
23. 24. 22. 3
2
6
43
42
6
3
6
2
10
3
4
26. 27. 25. 17
10
5
45
310
1017
5
10
17
11
10
3
29. 30. 28. 7
3
19
77
30
319
7
3
19
Write each radical expression in simplified form. See Example 3.
3
32. 48
33. 31. 63
2
6
37
43
2
3
5
5
34. 35. 36. 5
8
18
15
10
10
5
4
6
6
12
75
37. 38. 39. 10
20
3
15
15
5
5
5
45
40. 5
3
15
41. 10
6
2
30
42. 21
70
7
8.2
Simplify each expression. See Example 4.
43. a
8
44. y10
45. a9
a4
y5
a4a
46. t11
t 5t
47. 8a6
2a32
48. 18w
9
4
3w 2
w
4b9
49. 2
0a
2a2b45b
50. 12y
x23
2xy3
y
51. 27y
x33
3xy3
xy
52. 4
5y
x53
2
3x y5xy
53. 27a3b8c2
3ab4c3a
9
4
54. 125y
x3z
4 2
5xy z 5xy
Simplify each expression. See Example 5.
1
1
55. 56. 57.
x
2x
x
2x
x
2x
3
5
59. 60.
58. 15y
2b
10b
5y
2b
5y
3x
6
61. 62. 63.
2y
5w
6
30w
xy
2y
5w
64.
6x
4y
xy
6
2y
8yx
3
65.
2x2xy
y
Simplify each expression.
67. 8
0
x3
68. 90
y80
40
4x5x
3y 10
20x6
70. 4
8y
x27
71. x5
5
4xy33y
4x 35x
22p2
30t 5
73. 74. p6pq
t 23t
116pq
3q
3n4b5n2b2c7
76. nbc
3n4b5c3nb
10t23t
4xy2
77. x9y36
xy3
6y
94
3x y
2
3a
6a
3a
5
10x
2x
2x
10y
15x
6
xy
3x
Simplifying Square Roots
(8-13)
425
Solve each problem.
83. Economic order quantity. The formula for economic
order quantity
E
I
2AS
was used in Exercise 83 of Section 8.1.
a) Express the right-hand side in simplified form.
2
AIS
E I
b) Find E when A 23, S $4566, and I $80.
51.2
8ts
5
66.
2s22st
t
69. 9x
y915
4 7
3y x yx
7x7y
72. 7
x9
x 2y7x
a3b7a2b3c4
75. abc
a3b8cac
8m
3n2
78. m3n26m
n
3
23n
3m2n3
Use a calculator to evaluate each expression.
2
2 6
1
79. 80. 2
3
3
2
0
0
20
6
81. 3
82. 2 5
2
0
0
FIGURE FOR EXERCISE 83
84. Landing speed. Aircraft design engineers determine the
proper landing speed V (in ft/sec) by using the formula
V
841L
,
CS
where L is the gross weight of the aircraft in pounds, C is
the coefficient of lift, and S is the wing surface area in
square feet.
a) Express the right-hand side in simplified form.
29L
C
S
V CS
b) Find V when L 8600 pounds, C 2.81, and S 200
square feet.
113.4
FIGURE FOR EXERCISE 84