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

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

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)

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.

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)

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 4.4. What do you get? 19.36 20 – 19.36 = 0.64 Lets see if we can get closer to 20. Multiply 4.5 times 4.5. What do you get? 20.25 20 – 20.25 = -0.25 4.5 is the best estimate for the square root of 20.

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 144. So pick a number closer to 12. Multiply 12.2 times 12.2. What do you get? 148.84 150 – 148.84 = 1.16 Lets see if we can get closer to 150. Multiply 12.3 times 12.3. What do you get? 151.29 150 – 151.29 = -1.29 12.2 is the best estimate for the square root of 150.

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 196. So pick a number close to 14. Multiply 14.1 times 14.1. What do you get? 198.81 200 – 198.81 = 1.19 Lets see if we can get closer to 200. Multiply 14.2 times 14.2. What do you get? 201.64 200 – 201.64 = -1.64 14.1 is the best estimate for the square root of 200.