Find square root of each of the following numbers by division method:
(i) 2304
(ii) 4489
(iii) 3481
(iv) 529
(v) 3249
(vi) 1369
(vii) 5776
(viii) 7921
(ix) 576
(x) 1024
(xi) 3136
(xii) 900

Solution:

We will find the square root of each number using the long division method.

(i) √2304

1. Group the digits in pairs starting from the right: 23 04
2. Find the largest integer whose square is less than or equal to 23. That's 4 (4² = 16).
3. Subtract 16 from 23, leaving 7.
4. Bring down the next pair, 04.
5. Double the quotient (4) to get 8. Now we need to find a digit 'x' such that (80 + x) * x ≤ 704. That digit is 8 (80+8 = 88, 88 * 8 = 704).
6. Subtract 704 from 704, leaving 0.
   Therefore, √2304 = 48.

(ii) √4489

1. Group the digits: 44 89
2. Find the largest integer whose square is less than or equal to 44. That's 6 (6² = 36).
3. Subtract 36 from 44, leaving 8.
4. Bring down the next pair, 89.
5. Double the quotient (6) to get 12. Now we need to find a digit 'x' such that (120 + x) * x ≤ 889. That digit is 7 (127 * 7 = 889).
6. Subtract 889 from 889, leaving 0.
   Therefore, √4489 = 67.

(iii) √3481

1. Group the digits: 34 81
2. Find the largest integer whose square is less than or equal to 34. That's 5 (5² = 25).
3. Subtract 25 from 34, leaving 9.
4. Bring down the next pair, 81.
5. Double the quotient (5) to get 10. Now we need to find a digit 'x' such that (100 + x) * x ≤ 981. That digit is 9 (109 * 9 = 981).
6. Subtract 981 from 981, leaving 0.
   Therefore, √3481 = 59.

(iv) √529

1. Group the digits: 5 29
2. Find the largest integer whose square is less than or equal to 5. That's 2 (2² = 4).
3. Subtract 4 from 5, leaving 1.
4. Bring down the next pair, 29.
5. Double the quotient (2) to get 4. Now we need to find a digit 'x' such that (40 + x) * x ≤ 129. That digit is 3 (43 * 3 = 129).
6. Subtract 129 from 129, leaving 0.
   Therefore, √529 = 23.

(v) √3249

1. Group the digits: 32 49
2. Find the largest integer whose square is less than or equal to 32. That's 5 (5² = 25).
3. Subtract 25 from 32, leaving 7.
4. Bring down the next pair, 49.
5. Double the quotient (5) to get 10. Now we need to find a digit 'x' such that (100 + x) * x ≤ 749. That digit is 7 (107 * 7 = 749).
6. Subtract 749 from 749, leaving 0.
   Therefore, √3249 = 57.

(vi) √1369

1. Group the digits: 13 69
2. Find the largest integer whose square is less than or equal to 13. That's 3 (3² = 9).
3. Subtract 9 from 13, leaving 4.
4. Bring down the next pair, 69.
5. Double the quotient (3) to get 6. Now we need to find a digit 'x' such that (60 + x) * x ≤ 469. That digit is 7 (67 * 7 = 469).
6. Subtract 469 from 469, leaving 0.
   Therefore, √1369 = 37.

(vii) √5776

1. Group the digits: 57 76
2. Find the largest integer whose square is less than or equal to 57. That's 7 (7² = 49).
3. Subtract 49 from 57, leaving 8.
4. Bring down the next pair, 76.
5. Double the quotient (7) to get 14. Now we need to find a digit 'x' such that (140 + x) * x ≤ 876. That digit is 6 (146 * 6 = 876).
6. Subtract 876 from 876, leaving 0.
   Therefore, √5776 = 76.

(viii) √7921

1. Group the digits: 79 21
2. Find the largest integer whose square is less than or equal to 79. That's 8 (8² = 64).
3. Subtract 64 from 79, leaving 15.
4. Bring down the next pair, 21.
5. Double the quotient (8) to get 16. Now we need to find a digit 'x' such that (160 + x) * x ≤ 1521. That digit is 9 (169 * 9 = 1521).
6. Subtract 1521 from 1521, leaving 0.
   Therefore, √7921 = 89.

(ix) √576

1. Group the digits: 5 76
2. Find the largest integer whose square is less than or equal to 5. That's 2 (2² = 4).
3. Subtract 4 from 5, leaving 1.
4. Bring down the next pair, 76.
5. Double the quotient (2) to get 4. Now we need to find a digit 'x' such that (40 + x) * x ≤ 176. That digit is 4 (44 * 4 = 176).
6. Subtract 176 from 176, leaving 0.
   Therefore, √576 = 24.

(x) √1024

1. Group the digits: 10 24
2. Find the largest integer whose square is less than or equal to 10. That's 3 (3² = 9).
3. Subtract 9 from 10, leaving 1.
4. Bring down the next pair, 24.
5. Double the quotient (3) to get 6. Now we need to find a digit 'x' such that (60 + x) * x ≤ 124. That digit is 2 (62 * 2 = 124).
6. Subtract 124 from 124, leaving 0.
   Therefore, √1024 = 32.

(xi) √3136

1. Group the digits: 31 36
2. Find the largest integer whose square is less than or equal to 31. That's 5 (5² = 25).
3. Subtract 25 from 31, leaving 6.
4. Bring down the next pair, 36.
5. Double the quotient (5) to get 10. Now we need to find a digit 'x' such that (100 + x) * x ≤ 636. That digit is 6 (106 * 6 = 636).
6. Subtract 636 from 636, leaving 0.
   Therefore, √3136 = 56.

(xii) √900

1. Group the digits: 9 00
2. Find the largest integer whose square is less than or equal to 9. That's 3 (3² = 9).
3. Subtract 9 from 9, leaving 0.
4. Bring down the next pair, 00.
5. Double the quotient (3) to get 6. Now we need to find a digit 'x' such that (60 + x) * x ≤ 0. That digit is 0 (60 * 0 = 0).
6. Subtract 0 from 0, leaving 0.
   Therefore, √900 = 30.