Eight Standard >> Method of finding square root | Part-2

Square Root by long division

Here's the step-by-step procedure to find the square root of a perfect square number using the long division method, starting with a different approach:

Step 1: Write down the perfect square number "N" for which you want to find the square root.

Step 2: Group the digits in pairs from right to left, starting with the decimal point. If there's an odd number of digits, the leftmost group will contain only one digit.

Step 3: Now, let's find the first digit of the square root. Consider the leftmost group of digits and find the largest perfect square less than or equal to this group. Write down the square root of that perfect square as the first digit of the square root.

Step 4: Subtract the value of the perfect square found in step 3 from the leftmost group. This will give you the remainder.

Step 5: Bring down the next pair of digits to the right of the remainder obtained in step 4.

Step 6: Double the current digit of the square root (the one found in step 3) and write it as the divisor.

Step 7: Find the largest single-digit number (0 to 9) that, when added after the divisor, gives a product less than or equal to the current remainder (the two digits brought down in step 5).

Step 8: Append the found digit to the current partial square root.

Step 9: Multiply the divisor (the doubled digit found in step 6) by the digit found in step 7, and write the product below the remainder.

Step 10: Subtract the product obtained in step 9 from the remainder to get a new remainder.

Step 11: Repeat steps 5 to 10 until you have brought down all the pairs of digits and found the digits of the square root.

Step 12: Continue this process until you get the desired level of accuracy for the square root.

Let's illustrate the method with an example:

Example: Find the square root of the perfect square number 7056.

Step 1: The perfect square number is 7056.

Step 2: Group the digits in pairs: 70|56

Step 3: The largest perfect square less than or equal to 70 is 64 (8^2 = 64). So, the first digit of the square root is 8.

Step 4: Subtract 64 from 70: 70 - 64 = 6

Step 5: Bring down the next pair of digits: 6|56

Step 6: Double the current digit (8) and use it as the divisor: Divisor = 2 * 8 = 16

Step 7: Find the largest single-digit number we can add after 16 to get a value less than or equal to 656. The largest digit is 4 (16 + 4 = 20).

Step 8: Append 4 to the current partial square root: 84

Step 9: Multiply the divisor (16) by the digit found in step 7 (4): 16 * 4 = 64

Step 10: Subtract 64 from 656: 656 - 64 = 592

Step 11: Bring down the next pair of digits: 59|2

Step 12: Double the current partial square root (84) and consider it as the new divisor: Divisor = 2 * 84 = 168

Step 13: Find the largest single-digit number we can add after 168 to get a value less than or equal to 592. The largest digit is 3 (168 + 3 = 171).

Step 14: Append 3 to the current partial square root: 843

Step 15: Multiply the divisor (168) by the digit found in step 13 (3): 168 * 3 = 504

Step 16: Subtract 504 from 592: 592 - 504 = 88

Step 17: Bring down the next pair of digits: 88

Step 18: Since we have brought down all the digits, and there are no more digits to bring down, we stop here.

The square root of 7056 is 84.