Product Property of
Radicals
For any numbers a and b where a

0 and b

0 ,
ab
 a
 b

Product Property of
Radicals Examples
72 36

2

36

6 2

2
48

16

3

16

4 3

3

Examples:
1. 30 a
34  a
34
 a
17

30
30
2. 54 x
4 y
5 z
7 
9 x
4 y
4 z
6

3 x
2 y
2 z
3

6 yz
6 yz

Examples:
3
3. 54 a
3 b
7  3
27 a
3 b
7  3
2 b

3 ab
2  3
2 b
4. 60 xy
3 
4 y
2 
15 xy

2 y 15 xy

Quotient Property of
Radicals
For any numbers a and b where a

0 and b

0 , a b
 a b

Examples:
1.
7
16

7

16 4
7
2.
32

25
32

25
32

5
4 2
5

Examples:
3.
48
3

48

16

4
3
4.
45
4

45

4
45

2
3 5
2

Rationalizing the denominator
Rationalizing the denominator means to remove any radicals from the denominator.
Ex: Simplify
5

3
5

3
3

3
5 3
9

5 3
3

15
3

Simplest Radical Form
•No perfect nth power factors other than 1.
•No fractions in the radicand.
•No radicals in the denominator.

Examples:
1.
5
4

5

4 2
5
2.
20 8
2 2

10
8
2

10 4

10

2

20

Examples:
3.
5

2 2
2
2

5 2
2 4

5 2
2

2

5 2
4
4. 4
5
7x

4 5
7 x

7 x
7 x

4 35
49 x 2 x

4 35 x
7 x

Adding radicals
We can only combine terms with radicals if we have like radicals
6 7


5 7

3 7

6

5

3

7

8 7
Reverse of the Distributive Property

Examples:
1. 2 3 + 5 + 7 3 - 2
= 2 3 + 7 3 + 5 - 2
= 9 3 + 3

Examples:
2. 5 6

3 24

150
= 5 6

3 4 6

25 6
= 5 6

6 6

5 6
= 4 6

Multiplying radicals -
Distributive Property
3

2

4 3


3

2

3

4 3

6

12

Multiplying radicals - FOIL

3

5
F
 
2

O
4 3


3

2

3

4 3
I
L

5

2

5

4 3

6

12

10

4 15

1. 2 3
Examples:


4 5
 
3

6 5
F O

2 3

3

2 3

6 5

I
L

4 5

3

4 5

6 5

6

12 15

4 15

120

16 15

126

Examples:




F
2
2 7


5 4
2 7

5
O



10

10

10

2 7
2
2 7


2 7
I L

2 7

10

2 7

2 7


100

20 7

20 7

4 49

100

4

7

72

Conjugates
Binomials of the form
a b
 c d and a b
 c d where a, b, c, d are rational numbers.

Ex : 5

6

Conjugate: 5

6
3

2 2

Conjugate: 3

2 2
What is conjugate of 2 7

3?
Answer : 2 7

3
The product of conjugates is a rational number. Therefore, we can rationalize denominator of a fraction by multiplying by its conjugate.

Examples:


1.
3

2
3


5
3

5
3

5
3

3

5

3

2 3

2

5
 
2

5 2
3

7 3

10

3

25
13

7 3

22

Examples:
2.
1

2 5
6

5

6

5
6

5

6

5

12 5

10
6 2 
 
2

16

13 5
31