... or at least without using the square root button or exponent button on a calculator:
To find the square root of N :
1. Make a rough estimate or guess.
2. Divide N by your estimate.
3. Take the average of the answer from 2. and your estimate
....this will become your new estimate.
(Repeat steps 2 and 3 until you reach the accuracy you want.)
N is the number you want to find the square root of,
G1 is your initial estimate,
G2 is your first calculated improved estimate,
G3 is your next calculated improved estimate,
...
G1 = initial estimate or guess
G2 = (N/G1 + G1) /2
G3 = (N/G2 + G2) /2
G4 = (N/G3 + G3) /2
...
Find the square root of 5 :
G1 = 2
G2 = (5/2 + 2) /2 = 2.25
G3 = (5/2.25 + 2.25) /2 = 2.2361
G4 = (5/2.2361 + 2.2361) /2 = 2.23607
Check progress: 2.23607 x 2.23607 = 5.00001
If you need more accuracy, keep repeating the steps above.
To estimate G1, I ask myself:
(what number) x (what number) = 5 ...
2 x 2 = 4 and
3 x 3 = 9 thus
the square root of 5 must be between the numbers 2 and 3. I chose to start with 2, but you could also choose your initial estimate to be 2.5 or 3 and you will end up with a similar solution for the square root of 5.
Find the square root of 72 :
G1 = 8
G2 = (72/8 + 8) /2 = 8.5
G3 = (72/8.5 + 8.5) /2 = 8.4853
G4 = (72/8.4853 + 8.4853) /2 = 8.48528
Check progress: 8.48528 x 8.48528 = 71.99998
If you need more accuracy, keep repeating the steps above.
To estimate G1, I ask myself:
(what number) x (what number) = 72 ...
8 x 8 = 64 and
9 x 9 = 81 thus
the square root of 72 must be between the numbers 8 and 9. I chose to start with 8, but you could also choose your initial estimate to be 8.5 or 9 and you will end up with a similar solution for the square root of 72.
If you are on a job interview and are asked to find the square root of a number without using the square root button or exponential function on a calculator, this method (called the Babylonian method) will come in handy.
If you need to know the square root of a number...the square root calculator is here to help you.