CLASS-7
FINDING SQUARE ROOT BY DIVISION METHOD

FINDING SQUARE ROOT BY DIVISION METHOD -

To find the square root of a number using the division method, here's a step-by-step guide:-

Step-by-Step Process:-

  1. Group the digits:-

       Start by grouping the digits of the number in pairs, starting from the decimal point (for decimal numbers) or the right-most digit (for whole numbers).

       For example, for the number 2025, you would group it as (20)(25).

   2. Find the largest divisor:-

       Find the largest integer whose square is less than or equal to the first group (from the left). This will be the first digit of the square root.

      For example, with 2025, the first group is 20. The square of 4 (since 4²= 16) is the largest number less than or equal to 20.

   3. Divide and subtract:-  Subtract the square of the digit from the first group, then bring down the next pair of digits.

      For example, from 20, subtract 16 (which is 4²), leaving 4. Now, bring down the next pair (25), so you have 425.

   4. Double the current result:-   

       Double the number you found in the previous step. This becomes the divisor for the next part of the division.

       For example, 4 doubled is 8.

   5. Find the next digit:-   

       Now, divide the new number (in this case, 425) by the doubled result (8). You add the result to your square root and continue the process.

   6. Repeat the steps:- 

       Repeat the process of subtracting, bringing down the next pair of digits, and doubling the current result until you get the desired level of precision.


General Steps:-

  1. Pair digits from right to left.
  2. Find the largest square number less than or equal to the first pair.
  3. Subtract and bring down the next pair of digits.
  4. Double the quotient and determine the next divisor.
  5. Repeat until all digits are processed.


Example.1) Find the square root of 529

Ans.)

  1. Pair the digits from right to left:-

       529 → (5) (29)

  2. Find the largest number whose square is ≤ the first pair (5):-

      2³= 4 (This is the closest perfect square, since 3³= 9 is too big) Write 2 as the first digit of the square root. Subtract 4 from 5, leaving 1. Bring down the next pair (29), making it 129.

  3. Find the next digit:-

      Double the quotient (2) → 2 × 2 = 4.

      Find the largest digit x such that (40 + x) × x ≤ 129.

      If x = 3, then (40+3) × 3 = 43 × 3 = 129. 

      So, the next digit is 3.

   4. Final result:-

      The square root of 529 is 23.     (Ans.)


Example.2) Square Root of 2025

  1. Group the digits:-   (20) (25)
  2. Find the first digit:-  The largest number whose square is ≤ 20 is 4, because 4² = 16. Subtract 16 from 20:- 20 - 16 = 4. Bring down the next pair, which is 25, making 425.
  3. Double the first digit:-  4 × 2 = 8.
  4. Divide:-  425 ÷ 8 = 53. Add 3 to 4 (since 53 is close to 52, it means the next digit of the square root is 5). Subtract (53 × 5) from 425. Continue this process until you have the desired accuracy.


Example.3) Find the square root of 55696.

Ans.)

Roots


Example.4) Find the square root of 288369

Ans.)

Roots

5. Find the largest possible digit to fill the blank which will also become the new digit in the quotient. Such that when the new divisor is multiplied by new digit in the quotient the product is less than or equal dividend. In this case 103 x 3 = 309, so we choose the new digit as 3. Place 309 under under 383 and get the reminder 74.

6. Repeat the process 3, 4, and 5. Reminder is zero (0).