Evaluate `sqrt(66049).`

Options

A. 325
B. 257
C. 125
D. 115

Answer

To evaluate \( \sqrt{66049} \) using the long division method, follow these steps:

### Step-by-Step Solution:

1. **Pair the Digits**:
Start from the right, pair the digits of the number. For \( 66049 \), we have pairs as follows: \( 66 | 04 | 9 \).

2. **Find the Largest Square**:
Look at the leftmost pair (66). The largest integer whose square is less than or equal to 66 is 8, since \( 8^2 = 64 \). Write 8 above the line.

3. **Subtract and Bring Down**:
Subtract \( 64 \) from \( 66 \) to get \( 2 \). Then, bring down the next pair (04), making it \( 204 \).

4. **Double the Quotient**:
Double the number above the line (8) to get 16. Now, we need to find a digit \( x \) such that \( (160 + x) \times x \) is less than or equal to \( 204 \).

5. **Find the Suitable Digit**:
Testing \( x = 1 \):
\[ (160 + 1) \times 1 = 161 \quad \text{(which is less than 204)} \]
Testing \( x = 2 \):
\[ (160 + 2) \times 2 = 162 \times 2 = 324 \quad \text{(which is greater than 204)} \]
So, \( x = 1 \) is the suitable digit. Write 1 above the line.

6. **Subtract Again**:
Subtract \( 161 \) from \( 204 \) to get \( 43 \). Bring down the next pair (09), making it \( 4309 \).

7. **Double the Quotient Again**:
Double the current quotient (81) to get 162. Now, we need to find a digit \( y \) such that \( (1620 + y) \times y \) is less than or equal to \( 4309 \).

8. **Find the Suitable Digit Again**:
Testing \( y = 2 \):
\[ (1620 + 2) \times 2 = 1622 \times 2 = 3244 \quad \text{(which is less than 4309)} \]
Testing \( y = 3 \):
\[ (1620 + 3) \times 3 = 1623 \times 3 = 4869 \quad \text{(which is greater than 4309)} \]
So, \( y = 2 \) is the suitable digit. Write 2 above the line.

9. **Final Subtraction**:
Subtract \( 3244 \) from \( 4309 \) to get \( 1065 \). Bring down the next pair (00), making it \( 106500 \).

10. **Double the Quotient Again**:
Double the current quotient (812) to get 1624. Now, we need to find a digit \( z \) such that \( (16240 + z) \times z \) is less than or equal to \( 106500 \).

11. **Find the Suitable Digit Again**:
Testing \( z = 6 \):
\[ (16240 + 6) \times 6 = 16246 \times 6 = 97476 \quad \text{(which is less than 106500)} \]
Testing \( z = 7 \):
\[ (16240 + 7) \times 7 = 16247 \times 7 = 113729 \quad \text{(which is greater than 106500)} \]
So, \( z = 6 \) is the suitable digit. Write 6 above the line.

12. **Final Subtraction**:
Subtract \( 97476 \) from \( 106500 \) to get \( 9000 \).

13. **Conclusion**:
Since we have no remainder, we conclude that \( \sqrt{66049} = 257 \).

### Final Answer:
\[ \sqrt{66049} = 257 \]