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```					Simplifying Radical
Expressions
Product Property of

For any numbers a and
b where a  0 and b  0 ,

ab  a b
Product Property of
72    36 2  36 2
6 2
48  16 3  16 3
4 3
Examples:
30a  a  30
34         34
1.
 a
17
30

54x y z  9x y z  6yz
4 5 7            4 4 6
2.
 3x y z
2 2   3
6yz
Examples:
3
54a b  27a b  2b
3 7    3        3 7   3
3.
 3ab  2b
2 3

60xy  4 y  15xy
3          2
4.
 2 y 15xy
Quotient Property of
For any numbers a and
b where a  0 and b  0 ,
a   a

b   b
Examples:
7    7     7
1.          
16   16   4

32     32       32 4 2
2.                 
25     25       5   5
Examples:
48       48
3.               16  4
3        3

45       45       45 3 5
4.                     
4        4        2   2
Rationalizing the
denominator
Rationalizing the denominator means
to remove any radicals from the
denominator.
Ex: Simplify

5     5       3       5 3       5 3     15
                              
3     3       3       9          3      3
•No perfect nth power factors
other than 1.

Examples:
5      5    5
1.           
4      4   2

20 8  8
2.      10  10 4  10 2
2 2     2        20
Examples:
5    2     5 2 5 2
5 2
3.               
2 2 2 2 4   2 2   4

5 4 5 7x 4 35x
4. 4            
7x   7x   7x    49x 2

4 35x

7x
We can only combine terms with 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
Distributive Property
 2  4 3
3

       3 2  3 4 3

       6 12
   3 5
F

   24 3
O

       3 2  3 4 3
I           L
 5 2  5 4 3
    6 12 10 4 15
Examples:
   F

1. 2 3  4 5 3  6 5
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:
       5 4  2 7
2. 5 4  2 7
=  2 2 7 2  2 7
5        5
F        O
 1010 10 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 anda 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?

The product of conjugates is a
rational number. Therefore, we can
rationalize denominator of a fraction
by multiplying by its conjugate.
Examples:
32 35
1.         
35 35
3  3  5 3  2 3  2 5

 
2
3 5 2

3  7 3  10 13  7 3
               
3 25        22
Examples:
1 2 5 6 5
2.       
6 5 6 5
6  5  12 5  10

 
2
6 
2
5         16  13 5

31

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