Simplifying Radicals
A square root is simplified when the radicand, the number under the radical sign, contains no perfect
square factors.
Example: 20  4  5  4  5  2 5
To simplify a square root, factor the radicand into as many perfect squares as possible.
Examples: Simplify.
1. 54  9  6  9  6  3 6
2. 12  4  3  4  3  2 3
3. 200  100  2  100  2  10 2
4. 3 8  3 4  2  3 4  2  3  2 2  6 2
White Board Activity
Practice: Simplify.
1. 40
2.
3.
18
2 10
500
10 5
3 2
4. 7 75
35 3
Examples: Simplify.
25
25 5


1.
36
36 6
2.
3

64
3.
40

81
3
64
40
81


White Board Activity:
Practice: Simplify.
16
1.
81
3
8
4  10

9
4  10 2 10

9
9
2.
4
9
2
25
3.
2
5
45
49
3 5
7
Multiplying Radicals
To multiply radicals, multiply all “outside” numbers by “outside” numbers and “inside” numbers by
“inside” numbers, then simplify.
Examples:
1.
2.
3.
Multiply, then simplify if possible.
7  3  21
3 3  9 3
3  15  45  9  5  9  5  3 5
White Board Activity:
Practice: Multiply, then simplify if possible.
1. 5  2
2. 7  7
3.
6 3
7
10
3 2
Examples: Multiply, then simplify if possible.
1. 5 5  2  5  5  5  2  25  10  5  10
2.
3.




 5  7  5 5  7 5  7
7  2   7  2 7  2  7 
5 7
5  7  7  25  49  5  49  44
2
 49  4 7  4  7  4 7  4  11  4 7
White Board Activity:
Practice: Multiply, then simplify if possible.
1.
7  2 7  2 7  22

7 3 7

2.
21  7

3 8

3 8


3. 4  5
Examples: Divide, then simplify if possible.
14
14

 7
1.
2
2
100
5

100
 20  4  5  2 5
5
White Board Activity:
Practice: Divide, then simplify if possible.
15
90
1.
2.
2
3
5
9 5
Examples: Simplify.
12  2 5
1
 3
5
1.
4
2
2
21  8 5
 61
Dividing Radicals
2.

2.
24  9 6
83 6
3
White Board Activity:
Practice: Simplify.
15  5 5
1.
10
2 1

5
3 3
2.
14  6 6
2
73 6
Rationalizing Denominators
If is often easier to work with radical expressions if the denominator does not contain a radical. The
process for rewriting an expression without a radical in the denominator is called rationalizing. If
involves multiplying the radical expression by a radical version of 1.
Examples: Rationalize the denominator of each of the following.
2
2
7 2 7 2 7




1.
7
7
7 7
48
2.
5
12

5
43

5
4 3

5
2 3

3
3

15
2 9

15
15

23
6
White Board Activity:
Practice: Rationalize the denominator of each of the following.
7
5
1.
2.
20
3
5 3
35
3
10
Assessment:
Question Student Pairs
Independent Practice:
Radical Assignment