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