Post on 22-Feb-2016
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Simplifying Radical Expressions
Product Property of Radicals
For any numbers a and b where and , a0
ab a b b0
Product Property of Radicals Examples
72 362 36 2 6 2
163 16 3 48 4 3
Examples:
1. 30a34 a34 30 a17 30
2. 54x4 y5z7 9x4 y4z6 6yz
3x2 y2 z3 6yz
Examples:
27a3b73 2b3
4y2 15xy
2 y 15xy
3. 54a3b73
4. 60xy3 3ab2 2b3
Quotient Property of Radicals For any numbers a and b where and , a0 b0
ab
ab
Examples:
1. 716
2. 3225
716
7
4
3225
325
4 2
5
Examples:
483 16
454
452
3 52
3. 483
4. 454
4
Rationalizing the denominator
53
Rationalizing the denominator means to remove any radicals from the denominator.
Ex: Simplify
53
33
5 39
153
5 33
Simplest Radical Form•No perfect nth power factors other than 1.
•No fractions in the radicand.
•No radicals in the denominator.
Examples:
1. 54
2. 20 82 2
54
5
2
10 8
2 10 4 102 20
Examples:
3. 5
2 2
22
5 222
4 35x49x2
4 5
7x
5 24
5 22 4
7x7x
4 35x7x
4. 4 5
7x
Adding radicals
6 7 5 7 3 7
65 3 7
We can only combine terms with radicals if we have like radicals
8 7Reverse of the Distributive Property
Examples:
1. 23+5+7 3-2 =2 3+7 3+5-2 =9 3+3
Examples:
2. 56 3 24 150 =5 6 3 4 6 25 6 =5 6 6 65 6 =4 6
Multiplying radicals - Distributive Property
3 24 3 3 2 34 3
612
Multiplying radicals - FOIL
3 5 24 3
612 104 15
3 2 34 3
5 2 54 3
F O
I L
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
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
Conjugates
Binomials of the form
where a, b, c, d are rational numbers.
a b c d and a b c d
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
Examples:
1. 32
3 5
3535
3 3 5 32 3 253 2 52
37 3103 25
137 3
22
Examples:
6 56 5
2. 1 2 56 5
6 512 51062 5 2
1613 531