We can see the process to manually calculate square roots with an example.
Calculating the square root of 1369.
[tex]\sqrt{1369}[/tex]
step 1: separate the number every two digits with a comma, starting from the right.
step 2: look for a number that multiplied by itself is has a close result to the two-digit number on the left, and write it on the right
The number on the right is 3 because 3*3=9 which is close to 13 (it is not 4 because 4*4=16 and this is greater than 13)
step 3: Multiply that number (3) by itself and substract it from the two digits on the left
step 4: Bring down the other 2 digits, put a line under the previous one, and write the double of the previus number (3*2=6):
step 5: Now we look for a number to write along side the 3 and the 7.
The criterion for choosing this number is that by multiplying said number by the number that is formed in the second line from the right, it gets as close as possible to the number formed by the subtraction and the two numbers that we brought down.
In this case that number is 7.
because with a 7, the number on the second line would be 67, and 67*7=469, which is the number we are trying to get close to
And when we substract the result of 67*7 which is 469 to the 469 that was already in the expression, we get a residue of 0.
Step 6: the squere root is the number on the first line: 37
[tex]\sqrt{1369}=37[/tex]
