Transcript Simplify radicals by using the quotient rule.

Chapter 8
Section 2
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8.2
1
2
3
4
5
Multiplying, Dividing, and Simplifying
Radicals
Multiply square root radicals.
Simplify radicals by using the product rule.
Simplify radicals by using the quotient rule.
Simplify radicals involving variables.
Simplify other roots.
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Objective 1
Multiply square root radicals.
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Slide 8.2 - 3
Multiply square root radicals.
For nonnegative real numbers a and b,
a  b  a  b.
a  b  a  b and
That is, the product of two square roots is the square root of
the product, and the square root of a product is the product of
the square roots.
It is important to note that the radicands not be negative numbers in the
product rule. Also, in general, x  y  x  y .
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 8.2 - 4
EXAMPLE 1
Using the Product Rule to
Multiply Radicals
Find each product. Assume that x  0.
Solution:
3 5
 3 5
 15
6  11
 6 11
 66
13  x
 13  x
 13x
10  10
 10 10
 100
 10
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Slide 8.2 - 5
Objective 2
Simplify radicals by using the
product rule.
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Slide 8.2 - 6
Simplify radicals using the product rule.
A square root radical is simplified when no perfect
square factor remains under the radical sign.
This can be accomplished by using the product rule:
a b  a  b
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Slide 8.2 - 7
EXAMPLE 2
Using the Product Rule to
Simplify Radicals
Simplify each radical.
Solution:
60
 4  15
 2 15
500
 100  5
 10 5
17
It cannot be simplified further.
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Slide 8.2 - 8
EXAMPLE 3
Multiplying and Simplifying
Radicals
Find each product and simplify.
Solution:
10  50
6 2
 10  50
 500
 100  5
 62
 12
2 3
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 10 5
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Objective 3
Simplify radicals by using the
quotient rule.
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Slide 8.2 - 10
Simplify radicals by using the quotient rule.
The quotient rule for radicals is similar to the product
rule.
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 8.2 - 11
EXAMPLE 4
Using the Quotient Rule to
Simply Radicals
Simplify each radical.
Solution:
4
49
4

49
48
3

48
3
 16
5
36
5

36
5

6

2
7
4
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Slide 8.2 - 12
EXAMPLE 5
Using the Quotient Rule to
Divide Radicals
Simplify.
Solution:
8 50
4 5
8 50
 
4
5
50
 2
5
 2  10
 2 10
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Slide 8.2 - 13
EXAMPLE 6
Using Both the Product
and Quotient Rules
Simplify.
Solution:
3 7

8 2

3 7

8 2

21
16

21
16

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21
4
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Objective 4
Simplify radicals involving
variables.
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Slide 8.2 - 15
Simplify radicals involving variables.
Radicals can also involve variables.
The square root of a squared number is always
nonnegative. The absolute value is used to express this.
For any real number a,
a2  a .
The product and quotient rules apply when variables
appear under the radical sign, as long as the variables
represent only nonnegative real numbers
x  0, x  x.
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Slide 8.2 - 16
EXAMPLE 7
Simplifying Radicals Involving
Variables
Simplify each radical. Assume that all variables
represent positive real numbers.
Solution:
x
6
x
3
100 p 8
 100  p8
7
y4

7
y4
Since  x

3 2
 x6
 10 p 4
7
 2
y
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Slide 8.2 - 17
Objective 5
Simplify other roots.
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Slide 8.2 - 18
Simplify other roots.
To simplify cube roots, look for factors that are perfect
cubes. A perfect cube is a number with a rational cube root.
For example, 3 64  4, and because 4 is a rational
number, 64 is a perfect cube.
For all real number for which the indicated roots exist,
n
a  n b  n ab and
n
a na

b  0 .
n
b
b
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Slide 8.2 - 19
EXAMPLE 8
Simplifying Other Roots
Simplify each radical.
Solution:
3
108
 3 27  3 4
 33 4
4
160
 4 16 10
 4 16  4 10
4
16
625
4
16
4
625
 2 4 10
2

5
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Slide 8.2 - 20
Simplify other roots. (cont’d)
Other roots of radicals involving variables can also
be simplified. To simplify cube roots with variables,
use the fact that for any real number a,
3
a3  a.
This is true whether a is positive or negative.
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Slide 8.2 - 21
EXAMPLE 9
Simplifying Cube Roots Involving
Variables
Simplify each radical.
Solution:
3
z
9
 z3
3
8x 6
 3 8  3 x6
3
54t 5
 3 27t 3  2t 2
15
3
a
64
3
15
a
 3
64
 2x 2
 3 27t 3  3 2t 2
 3t 3 2t 2
a5

4
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Slide 8.2 - 22