How to find square root manually?

Steps to Solve

1. Choose a method to estimate the square root or use long division method

One common manual method for finding the square root is the long division method.

2. Set up the number in pairs of digits

Group the digits of the number to the left and right of the decimal point. This helps in aligning the calculations properly.

For example, if finding the square root of 152.2756, pair them as (1 52. 27 56)

3. Find the largest number whose square is less than or equal to the first pair or single digit

For the first digit/first pair, find the largest number whose square is less than or equal to the number. This will be the first digit of the square root.

For 152.2756, the first digit is 1 because $1^2$ is 1.

4. Subtract and bring down the next pair

Subtract the square of the first digit from the first pair, then bring down the next pair of digits next to the remainder.

For 152.2756: $$ ext{First pair is } 1. onumber \ 1^2 = 1 onumber \ 1-1 = 0 onumber \ ext{Bring down next pair }52\Rightarrow 052.$$

5. Double the current quotient and determine the next digit

Double the current quotient, write it down, and append a digit 'x' to it to form a number 2a (where a is the current quotient). Find x such that 2ax multiplied by x is less than or equal to the new number.

For 152.2756: $$0 + 52 = 52 \ ext{Double the quotient is }2
onumber \text{2ax = }20 * x onumber \ ext{choose x = 2} \nonumber 2*2 = 4 \

Subtract 44 from 52, answer is 8.

Bring down the next pair 27 making it 827. $$22+x =222 =4 $$

6. Repeat the steps

Repeat the process: bring down the next pair of digits, double the current quotient, append 'x', multiply 2ax by x, and continue until you reach desired decimal places.

So the steps are; \begin{align*} & 2+827=40. add digit from 2 as '4'

3*2=24
52-44 =8, bring down 27 making it 827

So the square root 152.2756 is around 12.36