5.5 roots of real numbers and radical expressions
TRANSCRIPT
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5.5 Roots of Real Numbers and
Radical Expressions
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Definition of nth RootFor any real numbers a and b and any positive integers n,
if an = b,
then a is the nth root of b.
** For a square root the value of n is 2.
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Notation
814index
radical
radicand
Note: An index of 2 is understood but not written in a square root sign.
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Simplify 814
To simplify means to find x in the equation:
x4 = 81
Solution: = 3 814
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Principal Root
The nonnegative root of a number
64
64
64
Principal square root
Opposite of principal square root
Both square roots
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Summary of Roots
even
odd
one + root one - root
one + root no - roots
no real roots
no + roots one - root
one real root, 0
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Examples
1. 169x4
2. - 8x-3 4
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Examples
3. 125x63
4. m3n33
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Taking nth roots of variable expressions
Using absolute value signs
If the index (n) of the radical is even, the power under the radical
sign is even, and the resulting power is odd, then we must use
an absolute value sign.
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Examples
1. an 44
2. xy2 66
EvenEven
EvenEven
anOdd
=x y2
Odd
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3. x6
4. 3 y2 186
EvenEven
x3
Odd
= 3- y2 3
Odd
Even
Even
2
= 3- y2 3
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Product Property of Radicals
For any numbers a and b where and , a0
ab a b
b0
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Product Property of Radicals Examples
72 362 36 2
6 2
163 16 3 48 4 3
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What to do when the index will not divide evenly into the radical????• Smartboard Examples
..\..\Algebra II Honors 2007-2008\Chapter 5\5.5 Simplifying Radicals\Simplifying Radicals.notebook
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Examples:
1. 30a34 a34 30
a17 30
2. 54x4 y5z7 9x4 y4z6 6yz
3x2 y2 z3 6yz
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Examples:
27a3b73 2b3
4y2 15xy
2 y 15xy
3. 54a3b73
4. 60xy3
3ab2 2b3
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Quotient Property of Radicals For any numbers a and b where and , a0 b0
ab
a
b
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Examples:
1. 716
2. 3225
7
16
74
32
25
325
4 2
5
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Examples:
483 16
45
4
452
3 52
3. 48
3
4. 454
4
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Rationalizing the denominator
53
Rationalizing the denominator means to remove any radicals from the denominator.
Ex: Simplify
5
3
3
3
5 3
9
153
5 33
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Simplest Radical Form
•No perfect nth power factors other than 1.
•No fractions in the radicand.
•No radicals in the denominator.
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Examples:
1. 54
2. 20 8
2 2
5
4
52
10
82 10 4 102
20
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Examples:
3.
5
2 2
2
2
5 222
4 35x
49x2
4 5
7x
5 24
5 2
2 4
7x
7x
4 35x7x
4. 4
57x
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Adding radicals
6 7 5 7 3 7
65 3 7
We can only combine terms with radicals if we have like radicals
8 7
Reverse of the Distributive Property
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Examples:
1. 23+5+7 3-2
=2 3+7 3+5-2
=9 3+3
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Examples:
2. 56 3 24 150
=5 6 3 4 6 25 6
=5 6 6 65 6
=4 6
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Multiplying radicals - Distributive Property
3 24 3
3 2 34 3
612
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Multiplying radicals - FOIL
3 5 24 3
612 104 15
3 2 34 3
5 2 54 3
F O
I L
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Examples:
1. 2 3 4 5 36 5
612 15 4 15120
2 3 3 2 36 5
4 5 3 4 56 5
F O
I L
16 15126
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Examples:
2. 5 4 2 7 5 4 2 7
1010 102 7
2 710 2 72 7
F O
I L
= 52 2 7 52 2 7
100 20 7 20 7 4 49 100 4772
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Conjugates
Binomials of the form
where a, b, c, d are rational numbers.
a b c d and a b c d
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The product of conjugates is a rational number. Therefore, we can
rationalize denominator of a fraction by multiplying by its conjugate.
Ex: 5 6 Conjugate: 5 6
3 2 2 Conjugate: 3 2 2
What is conjugate of 2 7 3?
Answer: 2 7 3
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Examples:
1.
32
3 5
35
35
3 3 5 32 3 25
3 2 52
37 3 103 25
137 3
22
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Examples:
6 5
6 5 2.
1 2 5
6 5
6 512 510
62 5 2
1613 531