To find the square root of 4489 by the long division method, we will go step by step.
Step 1: Pair the digits starting from the right.
We pair the digits of 4489 into two groups: (44) and (89).
Step 2: Find the largest number whose square is less than or equal to the leftmost pair. The largest number whose square is less than or equal to 44 is 6 because 6^2 = 36 is less than 44, and 7^2 = 49 is more than 44.
So we write 6 at the top, and 6^2 below 44.
```
__6__
√ 4489
- 36
-----
889
```
Step 3: Bring down the next group of digits, which is 89, next to the remainder, which is 8, making it 889.
Step 4: Double the divisor (6) and write it down with a space on the right for the next digit. The new divisor is 12_.
```
_6_
√4489
- 36
-----
1289
```
Step 5: Find a digit (X) to fill in the blank that makes (12X) x X less than or equal to 889. The digit X in the divisor is 7 because 127 x 7 = 889, which is exactly our dividend.
So we write 7 at the top.
```
_67_
√4489
- 36
-----
1289
- 889
-----
0
```
Step 6: Subtract the result of the multiplication from the new dividend to find the new remainder. This new remainder is 0, which means we have found that 67 is the square root of 4489 (because we have no remainder).
So, the square root of 4489 is 67.
