Rule to find the square root of a number by division method:
In case of numbers where the factorization is not easy or of big numbers or of numbers whose square root is not exact, we find the square root of such numbers by division method.
This method is explained below with the help of a few examples.
Example: Find the square root of the following numbers:
(i) 55696 (ii) 288369 (iii) 7856809
Solution:
(i) Steps
1. Place a bar (or arrow) over every pair of digits from right to left (<—) i.e. starting from unit’s digit.
If the number of digits is odd, then the left most digit too will have a bar. Each pair of digits and then remaining one digit (if any) on the extreme left is called period.
2. Take the first pair of digits or the single digit as the case may be. In this case, it is the digit 5. Find the greatest number whose square is 5 or less than 5. Such a number is 2. Write 2 on the top in the quotient and also in the divisor. Subtract\( { 2 }^{ 2 } \) i.e. 4 from 5. The remainder is 1.
3. Bring down the pair of digits under the next bar (i.e. 56 in this case) to the right of the remainder. So the new dividend is 156.
4. Double the quotient (i.e. 2 in this case) to get 4 and enter it with a blank on its right at the place of new divisor.
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 the new digit in the quotient the product is less than or equal to the dividend.
In this case 43 x 3 = 129, so we choose the new digit as 3. Place 129 under 156. Subtract and get the remainder 27.
6. Bring down the pair of digits under the next bar (i.e. 96 in this case) to the right of the remainder. So the new dividend is 2796.
7. Double the quotient (i.e. 23 in this case) to get 46 and enter it with a blank on its right at the place of new divisor.
8. 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 this new digit in the quotient the product is less than or equal to the dividend.
In this case 466 x 6 = 2796. So we choose the new digit as 6. Place 2796 under 2796. Subtract and get the remainder 0.
\( \therefore \) \( \sqrt { 55696 } \) = 236.
(ii) Steps
1. Place a bar over every pair of digits from right to left (<—).
2. Take the first pair of digits. In this case, it is 28. Find the greatest number whose square is 28 or less than 28. Such a number is 5. Write 5 on the top in the quotient and also in the divisor. Subtract \( { 5 }^{ 2 } \) i.e. 25 from 28. The remainder is 3.
3. Bring down the pair of digits under the next bar (i.e. 83 in this case) to the right of the remainder. So new dividend is 383.
4. Double the quotient (i.e. 5 in this case) to get 10 and enter it with a blank at the place of new divisor.
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 this new digit in the quotient the product is less than or equal to dividend. In this case 103 x 3 = 309, so we choose the new digit as 3. Place 309 under 383 and get the remainder 74.
6. Repeat the process of steps 3, 4 and 5. Remainder is 0.
\( \therefore \) \( \sqrt { 288369 } \) = 537
(iii) Steps
1. Places a bar over every pair of digits from right to left (<—). Here, single digit 7 is left, so put a bar over 7.
2. Take the digit 7. Find the greatest number whose square is 7 or less than 7. Such a number is 2. Write 2 at the top in the quotient and also 2 in the divisor. Subtract 22 i.e. 4 from 7. The remainder is 3.
3. Bring down the pair of digits under the next bar (i.e. 85) to the right of the remainder. So new dividend is 385.
4. Double the quotient (i.e. 2) to get 4 and enter it with a blank on its right at the place of new divisor.
5. Find the largest possible digit to fill the blank which will also become the digit in the quotient, such that when the new divisor is multiplied by this digit in the quotient the product is less than or equal to the dividend. In this case 48 x 8 = 384, so we choose the new digit as 8. Place 384 under Subtract and get the remainder 1.
6. Bring down the pair of digits under the new bar (i.e. 68) to the right of the remainder. So new dividend is 168.
7. Double the quotient (i.e. 28) to get 56 and enter it with a blank on its right at the place of new divisor.
8. Since no non-zero digit can fill the blank such that when the new divisor is multiplied by this new digit in the quotient so that the product is less than or equal to the dividend. Therefore, put 0 at blank in the divisor to get 560 and also put 0 in the quotient to get 280.
9. Bring down the pair of digits under the next bar (i.e. 09) to the right of 168 to get 16809 as the new dividend. Now find the largest possible digit to be put at the right of the divisor 560 which will also become the new digit in the quotient, such that when the new divisor is multiplied by the new digit in the quotient the product is less than or equal to dividend.
In this case 5603 x 3 = 16809, so we choose new digit as 3. Place 16809 under the dividend 16809 and get remainder 0.
\( \therefore \) \( \sqrt { 7856809 } \) = 2803