Evaluate `sqrt (11025).`

Options

A. `115.`
B. `105.`
C. `104`
D. `125.`

Answer

To evaluate the square root of 11025 using the division method, we can follow these steps:

### Step-by-Step Solution:

1. **Pair the Digits**: Start from the right and make pairs of digits. For 11025, we pair as (11)(02)(25).

2. **Find the Largest Square**: Look at the first pair (11). The largest square less than or equal to 11 is 3, since \(3^2 = 9\). Write 3 above the line.

3. **Subtract and Bring Down**: Subtract \(9\) from \(11\) to get \(2\). Bring down the next pair (02) to make it \(202\).

4. **Double the Quotient**: Double the number above the line (3) to get \(6\). Write \(6\) down.

5. **Find the Next Digit**: Now we need to find a digit \(x\) such that \(6x \times x\) is less than or equal to \(202\). Testing \(x = 3\):
- \(63 \times 3 = 189\), which is less than \(202\).
- Testing \(x = 4\):
- \(64 \times 4 = 256\), which is greater than \(202\).
- So, we take \(x = 3\).

6. **Write the Next Digit**: Write \(3\) next to \(3\) above the line, making it \(33\).

7. **Subtract Again**: Subtract \(189\) from \(202\) to get \(13\). Bring down the next pair (25) to make it \(1325\).

8. **Double the New Quotient**: Double \(33\) to get \(66\). Write \(66\) down.

9. **Find the Next Digit Again**: Now we need to find a digit \(y\) such that \(66y \times y\) is less than or equal to \(1325\). Testing \(y = 2\):
- \(662 \times 2 = 1324\), which is less than \(1325\).
- Testing \(y = 3\):
- \(663 \times 3 = 1989\), which is greater than \(1325\).
- So, we take \(y = 2\).

10. **Final Result**: Write \(2\) next to \(33\) above the line, making it \(102\).

The square root of \(11025\) is \(105\).

### Final Answer:
\[ \sqrt{11025} = 105 \]