Chapter Chapter 8 8 Section Section 2 2 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Welcome message from author
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
Chapter Chapter 88Section Section 22
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
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.
11
44
33
22
55
8.28.28.28.2
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Objective 11
Multiply square root radicals.
Slide 8.2 - 3
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Multiply square root radicals.
For nonnegative real numbers a and 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.
Slide 8.2 - 4
a b a b .a b a b
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
EXAMPLE 1
Find each product. Assume that
6 11
Using the Product Rule to Multiply Radicals
Slide 8.2 - 5
Solution:
0.x
3 5
6 11
13 x
10 10
3 5
13 x
10 10
15
66
13x
100 10
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Objective 22
Simplify radicals by using the
product rule.
Slide 8.2 - 6
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
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
Slide 8.2 - 7
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE 2
Simplify each radical.
500
Using the Product Rule to Simplify Radicals
Slide 8.2 - 8
Solution:
60
17
4 15
100 5
It cannot be simplified further.
2 15
10 5
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE 3
Find each product and simplify.
6 2
Multiplying and Simplifying Radicals
Slide 8.2 - 9
Solution:
10 50
6 2
10 50 500 100 5 10 5
12 2 3
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Objective 33
Slide 8.2 - 10
Simplify radicals by using the quotient rule.
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Simplify radicals by using the quotient rule.
The quotient rule for radicals is similar to the product
rule.
Slide 8.2 - 11
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE 4
Simplify each radical.
48
3
Solution:
Using the Quotient Rule to Simply Radicals
Slide 8.2 - 12
4
49
5
36
4
49
2
7
48
3 16 4
5
36
5
6
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE 5
Simplify.
Solution:
Using the Quotient Rule to Divide Radicals
Slide 8.2 - 13
8 50
4 5
8 50
4 5
502
5 2 10
2 10
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Simplify.
EXAMPLE 6Using Both the Product and Quotient Rules
Slide 8.2 - 14
Solution:
3 7
8 2
3 7
8 2
21
16
21
16
21
4
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Objective 44
Slide 8.2 - 15
Simplify radicals involving variables.
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
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.
The product and quotient rules apply when variables appear under the radical sign, as long as the variables represent only nonnegative real numbers
Slide 8.2 - 16
2For any real number , .a a a
, .0x x x
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE 7 Simplifying Radicals Involving Variables
Slide 8.2 - 17
Simplify each radical. Assume that all variables represent positive real numbers.
Solution:6x
8100 p
4
7
y
3x 23 6Since x x
8100 p 410 p
4
7
y
2
7
y
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Objective 55
Simplify other roots.
Slide 8.2 - 18
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
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, , and because 4 is a rational
number, 64 is a perfect cube.
For all real number for which the indicated roots exist,
3 64 4
n a . 0ndn
n n n
n
a aa b ab b
bb
Slide 8.2 - 19
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE 8 Simplifying Other Roots
Slide 8.2 - 20
Simplify each radical.
Solution:3 108
4 160
416
625
33 27 4 33 4
4 16 10 4 416 10 42 10
4
4
16
625
2
5
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
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,
This is true whether a is positive or negative.
3 3 .a a
Slide 8.2 - 21
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE 9 Simplifying Cube Roots Involving Variables
Slide 8.2 - 22
Simplify each radical.
Solution:
3 9z
3 68x
3 554t
15
3a
64
3z
22x3 63 8 x
3 3 227 2t t 3 33 227 2t t 3 23 2t t
3 15
3 64
a
5
4
a
Related Documents
