Question

How to find square root of any number e.g. 645.

Answer 1

Let us understand this through an example:
Find $\sqrt{645}$ to one decimal place.
First group the numbers under the root in pairs from right to left, leaving either one or two digits on the left (6 in this case). For each pair of numbers you will get one digit in the square root.
To start, find a number whose square is less than or equal to the first pair or first number, and write it above the square root line (2):
$\stackrel{2}{\sqrt{6.45}}$
Then continue this way:

Square the 2, giving, 4, write that underneath the 6, and subtract. Bring down the next pair of digits.

Then double the number above the square root symbol line (highlighted), and write it down in parenthesis with an empty line next to it as shown.

Next think what single - digit number something could go opn the empty line so taht forty - something times something would be less than or equal to 245.
45 $\times$ 5 = 225
46 $\times$ 6 = 276, so 5 works.

Write 5 on top of line. Calculate 5 $\times$ 45, write that below 245, subtract, bring down the next pair of digits (in this case the decimal digits 00).

The double the number above the line (25), and write the doubled number (50) in parenthesis with an empty line next to it as indicated:

Think what single digit number something could go on the empty line so that five hundred - something times something would be less than or equal to 2000. 503 $\times$ 3 = 1509.
504 $\times$ 4 = 2016, so 3 works.

Calculate 3 $\times$ 503, write that below 2000, subtract, bring down the next digits.

Then double the 'number' 253 which is above the line (ignoring the decimal point), and write the doubled number 506 in parenthesis with an empty line.