1
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Objectives
2
6
5
3
4
Simplifying Radical Expressions
Use the product rule for radicals.
Use the quotient rule for radicals.
Simplify radicals.
Simplify products and quotients of radicals with different indexes.
Use the Pythagorean theorem.
Use the distance formula.
8.3
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Use the product rule for radicals.
Objective 1
Slide 8.3- 3
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Product Rule for Radicals
If and are real numbers and n is a natural number, then
That is, the product of two nth roots is the nth root of the product.
n a n b
.n n na b ab
Slide 8.3- 4
Use the product rule for radicals.
Use the product rule only when the radicals have the same index.
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Multiply. Assume that all variables represent positive real numbers.
5 13 5 13
7 xy 7xy
65
Slide 8.3- 5
CLASSROOM EXAMPLE 1
Using the Product Rule
Solution:
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Multiply. Assume that all variables represent positive real numbers.
3 32 7 3 2 7
66 2 38 2r r 6 516r
2 25 59 8y x xy 4 25 72y x
37 5
3 14
This expression cannot be simplified by using the product rule.
Slide 8.3- 6
CLASSROOM EXAMPLE 2
Using the Product Rule
Solution:
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Use the quotient rule for radicals.
Objective 2
Slide 8.3- 7
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Quotient Rule for Radicals
If and are real numbers, b ≠ 0, and n is a natural number,
then
That is, the nth root of a quotient is the quotient of the nth roots.
n a n b
.n
nn
a a
b b
Slide 8.3- 8
Use the quotient rule for radicals.
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Simplify. Assume that all variables represent positive real numbers.
100
81
10
9
11
2511
5
8
16
y3
18
125
3
3
18
125
2
312
x
r
3 2
3 12
x
r
3 18
5
8
16
y
4
4
y
3 2
4
x
r
Slide 8.3- 9
CLASSROOM EXAMPLE 3
Using the Quotient Rule
Solution:
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Simplify.
32 216 216 4 2
300 3100 3100 10 3
35 Cannot be simplified further.
Slide 8.3- 11
CLASSROOM EXAMPLE 4
Simplifying Roots of Numbers
Solution:
3 54 3 227 33 27 2 33 2
4 243 44 3 3 43 3
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Conditions for a Simplified Radical
1. The radicand has no factor raised to a power greater than or equal to the index.
2. The radicand has no fractions.
3. No denominator contains a radical.
4. Exponents in the radicand and the index of the radical have greatest common factor 1.
Slide 8.3- 12
Simplify radicals.
Be careful with which factors belong outside the radical sign and which belong inside.
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Simplify. Assume that all variables represent positive real numbers.
725p 3 225 ( )p p 35p p
372y x 2236 y y x 6 2y yx
7 5 63 27y x z3 6 3 2 63 3 y y x x z 2 2 233y xz yx
5 74 32a b4 4 4 34 2 2 a a b b 342 2ab ab
Slide 8.3- 13
CLASSROOM EXAMPLE 5
Simplifying Radicals Involving Variables
Solution:
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Simplify. Assume that all variables represent positive real numbers.
12 32 1/1232 1/ 42
6 2t 1/ 62t 3 t
4 2
1/3t
Slide 8.3- 14
CLASSROOM EXAMPLE 6
Simplifying Radicals by Using Smaller Indexes
Solution:
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
If m is an integer, n and k are natural numbers, and all indicated roots exist, then
.kn nkm ma a
Slide 8.3- 15
Simplify radicals.
kn kma
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Simplify products and quotients of radicals with different indexes.
Objective 4
Slide 8.3- 16
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Simplify.
35 4
1/ 2 3/ 6 6 3 6 125 5 5 5
1/3 2/ 6 6 2 6 164 4 4
35 4 6 6125 16 6 2000
The indexes, 2 and 3, have a least common index of 6, use rational exponents to write each radical as a sixth root.
Slide 8.3- 17
CLASSROOM EXAMPLE 7
Multiplying Radicals with Different Indexes
Solution:
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Use the Pythagorean theorem.
Objective 5
Slide 8.3- 18
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Pythagorean Theorem
If c is the length of the longest side of a right triangle and a and b are lengths of the shorter sides, then
c2 = a2 + b2.
The longest side is the hypotenuse, and the two shorter sides are the legs, of the triangle. The hypotenuse is the side opposite the right angle.
Slide 8.3- 19
Use the Pythagorean theorem.
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Find the length of the unknown side of the triangle.
c2 = a2 + b2
c2 = 142 + 82
c2 = 196 + 64
c2 = 260
260c
4 65c
4 65c
2 65c The length of the hypotenuse is 2 65.
Slide 8.3- 20
CLASSROOM EXAMPLE 8
Using the Pythagorean Theorem
Solution:
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Find the length of the unknown side of the triangle.
c2 = a2 + b2
62 = 42 + b2
36 = 16 + b2
20 = b2
20 b
4 5 b
4 5 b
2 5 bThe length of the leg is 2 5.
Slide 8.3- 21
CLASSROOM EXAMPLE 8
Using the Pythagorean Theorem (cont’d)
Solution:
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Use the distance formula.
Objective 6
Slide 8.3- 22
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Distance Formula
The distance between points (x1, y1) and (x2, y2) is
2 22 1 2 1( ) ( )d x x y y
Slide 8.3- 23
Use the distance formula.
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Find the distance between each pair of points.
(2, –1) and (5, 3)
Designate which points are (x1, y1) and (x2, y2).
(x1, y1) = (2, –1) and (x2, y2) = (5, 3)
2 22 1 2 1( ) ( )d x x y y
2 2( ) (5 32 ( ))1d
2 2(3) (4)d
9 16d
5 52d
Start with the x-value and y-value of the same point.
Slide 8.3- 24
CLASSROOM EXAMPLE 9
Using the Distance Formula
Solution:
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Find the distance between each pairs of points.
(–3, 2) and (0, 4)
Designate which points are (x1, y1) and (x2, y2).
(x1, y1) = (–3, 2) and (x2, y2) = (0, 4)
2 22 1 2 2( ) ( )d x x y y
2 2( ( )) (3 )0 4 2d
2 2(3) ( 6)d
9 36d
3 545d Slide 8.3- 25
CLASSROOM EXAMPLE 9
Using the Distance Formula (cont’d)
Solution:
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