1. How to Find the Square Root of a Non-Perfect Square

2. Perfect Squares 25, 16 and 81 are called perfect squares.
This means that if each of these numbers were the area of a square, the length of one side would be a whole number.
Area = 25
Area = 81 9 5
Area = 16 4 5 4 9

3. Perfect Squares 12 = 1 22 = 4 32 = 9 42 = 16 52 = 25 62 = 36 72 = 49
82 = 64
92 = 81
102 = 100
112 = 121
122 = 144
132 = 169
142 = 196
152 = 225
162 = 256
172 = 289
182 = 324
192 = 361
202 = 400
All of the numbers in red (1 , 4 , 9 , 16, 25 etc) are considered “perfect” because their square root is an integer (and rational)

4. Non-Perfect Squares
What about the numbers in between all of the perfect squares?
Why isn’t 20 a perfect square?
20 can’t make a square with whole numbers. (Area) 20 1
Area = 20 4 5
Area = 20 2 10
Area = 20
The square root of 20 is irrational because 20 isn’t “perfect”. It will end up being a number somewhere between 4 and 5.

5. Non-Perfect Squares
The square root of any non-perfect square is an irrational number that lies between two integers
We can approximate its value to a decimal to know how far between those two integers it falls
(although it is an approximation)

6. How to find an approximation of the square root of 20…
What two perfect squares does 20 lie between?
16 and 25
The square root of 16 is 4, so the square root of 20 must be a little more than 4.
How to find the “little more”
Is the “non-perfect square” 20 closer to 16 or 25?
It seems to be right in the middle. So pick a number in between 4 and 5.
Multiply 4.4 times What do you get?
19.36
20 – = 0.64
Lets see if we can get closer to 20. Multiply 4.5 times What do you get?
20.25
20 – = -0.25
4.5 is the best estimate for the square root of 20.

7. How to find an approximation of the square root of 150…
What two perfect squares does 150 lie between?
144 and 169
The square root of 144 is 12, so the square root of 150 must be a little more than 12.
How to find the “little more”
Is the “non-perfect square” 150 closer to 144 or 169?
It seems to be closer to So pick a number closer to 12.
Multiply 12.2 times What do you get?
148.84
150 – = 1.16
Lets see if we can get closer to Multiply 12.3 times What do you get?
151.29
150 – = -1.29
12.2 is the best estimate for the square root of 150.

8. How to find an approximation of the square root of 200…
What two perfect squares does 200 lie between?
196 and 225
The square root of 196 is 14, so the square root of 200 must be a little more than 14.
How to find the “little more”
Is the “non-perfect square” 200 closer to 196 or 225?
It seems to be really close to So pick a number close to 14.
Multiply 14.1 times What do you get?
198.81
200 – = 1.19
Lets see if we can get closer to Multiply 14.2 times What do you get?
201.64
200 – = -1.64
14.1 is the best estimate for the square root of 200.