Find the square root of the following by long division method.
5625

Options

A. 75
B. 80
C. 57
D. 145

Answer

To find the square root of 5625 using the long division method, follow these steps:

### Step-by-Step Solution:

1. **Pair the Digits**: Start from the right and pair the digits of the number. For 5625, we have pairs: (56)(25).

2. **Find the Largest Square**: Look at the first pair (56). Find the largest integer whose square is less than or equal to 56. The largest integer is 7, since \(7^2 = 49\) and \(8^2 = 64\) (which is too large).

3. **Subtract and Bring Down**: Write 7 above the pair (56) and also write it below. Now, subtract \(49\) from \(56\):
\[ 56 - 49 = 7 \]
Now, bring down the next pair (25) next to 7, making it 725.

4. **Double the Quotient**: Double the number you wrote above (which is 7) to get 14. This will be the starting point for the next divisor.

5. **Find the Next Digit**: We need to find a digit (let's call it \(x\)) such that when we append it to 14 (making it 14x) and multiply it by \(x\), the result is less than or equal to 725.
\[ (140 + x) \times x \leq 725 \]
Testing \(x = 5\):
\[ 145 \times 5 = 725 \]
This works perfectly.

6. **Complete the Division**: Write 5 above the pair (25) next to 7, making it 75. Now, subtract \(725\) from \(725\):
\[ 725 - 725 = 0 \]

7. **Final Result**: Since there is no remainder, the square root of 5625 is the quotient we have obtained, which is 75.

### Final Answer:
The square root of 5625 is **75**.