Rational Square Roots
Lesson 9-3
You need to
memorize at least the
first 15 perfect
squares.
Square
root
Square
root
1 = 1
81 = 9
4 = 2
100 = 10
9 = 3
121 = 11
16 = 4
144 = 12
25 = 5
169 = 13
36 = 6
196 = 14
49 = 7
225 = 15
64 = 8
Radical Sign
The is called
 the radical
sign.
The number under the
radical is called the
radicand. In 25 ,25 is the
radicand.
Steps

1.Simplify if you know
the square root.
2.Use the Hockey Stick
Method if necessary.
Factor Tree Method
900

For every
pair (square)
you pull out
1.
Hockey Stick
1296

For every
pair (square)
you pull out
1.
64
169

64  64
169 169
8

13
576


1225
2  82
576
3


1225
52  72
  38
5 7
  24
35
This means the answer could be
24
24
or 
35
35
25
64
625
 144

 676
729

3364
10

6

5 4
5  4
2
 
6
2
2
2
2
2
2
Irrational numbers cannot
be written as a fraction.
They are non-terminating,
non-repeating decimals.
Example: √67
Irrational Numbers
Find two consecutive whole numbers that
the given square root is between. (Try to
do this without using the table).
18
16 = 4 and 25 = 5
so
18 is between 4 and 5
115
100 = 10 and 121 = 11
so
115 is between 10 and 11
Steps for Simplifying
1. Look for perfect square factors
using a factor tree.
2. Pull out all squares
3. Leave factors with no square
root under the radical.
216
For every pair
(square) you
pull out 1.
3
125
For every pair
(square) you
pull out 1.
1920
For every pair
(square) you
pull out 1.
Simplified Radical Form
No factor inside the radical should be a perfect square.
18 =
9 2 = 9 2
= 3 2
108 =
36 3
= 36 3
= 6 3
96 =
16 6
= 16 6
= 4 6
18
3 12
196
960
8 2592
12
3 4
49  7
23
Rational
24
46
2 6
8
24
2 2
125
Irrational
5  25
Irrational
5 5
Irrational
Irrational
Simplify the following expressions
-4 =
-2
764 + 9
=
7 8 + 9
=
5
25
56 + 9 = 65
+ 49 =
=
5 5 + 7
25 + 7 = 32
Simplify the following expressions


4
81
=
1
–
36
4
81

=
1
144
2
9
1
=
6
2
=
12
1
=
12
–
–
1
12
1
12
Pop Quiz!!!
Simplify lowest radical
12
8
49
125
24