Find the square root of 149.597361.

Answer

To find the square root of 149.597361 using the long division method, follow these steps:

### Step-by-Step Solution:

1. **Pair the Digits**: Start from the decimal point and pair the digits. For 149.597361, we can pair them as (1)(49)(59)(73)(61).

2. **Find the Largest Square**: Look for the largest square less than or equal to the first pair (1). The largest square is 1 (since 1^2 = 1). Write 1 above the line.

3. **Subtract and Bring Down**: Subtract 1 from 1 to get 0. Bring down the next pair (49) to make it 049.

4. **Double the Quotient**: Double the number above the line (1) to get 2. Now we need to find a digit (let's call it 'x') such that (20 + x) * x is less than or equal to 49.

5. **Find 'x'**: Testing values for 'x':
- For x = 2: (20 + 2) * 2 = 22 * 2 = 44 (this works)
- For x = 3: (20 + 3) * 3 = 23 * 3 = 69 (this is too high)

So, we take x = 2. Write 2 above the line next to 1, making it 12.

6. **Subtract Again**: Subtract 44 from 49 to get 5. Bring down the next pair (59) to make it 559.

7. **Double the New Quotient**: Double 12 to get 24. Now we need to find a digit 'y' such that (240 + y) * y is less than or equal to 559.

8. **Find 'y'**: Testing values for 'y':
- For y = 2: (240 + 2) * 2 = 242 * 2 = 484 (this works)
- For y = 3: (240 + 3) * 3 = 243 * 3 = 729 (this is too high)

So, we take y = 2. Write 2 above the line next to 12, making it 12.2.

9. **Subtract Again**: Subtract 484 from 559 to get 75. Bring down the next pair (73) to make it 7573.

10. **Double the New Quotient**: Double 122 to get 244. Now we need to find a digit 'z' such that (2440 + z) * z is less than or equal to 7573.

11. **Find 'z'**: Testing values for 'z':
- For z = 3: (2440 + 3) * 3 = 2443 * 3 = 7329 (this works)
- For z = 4: (2440 + 4) * 4 = 2444 * 4 = 9776 (this is too high)

So, we take z = 3. Write 3 above the line next to 12.2, making it 12.23.

12. **Subtract Again**: Subtract 7329 from 7573 to get 244. Bring down the next pair (61) to make it 24461.

13. **Double the New Quotient**: Double 1223 to get 2446. Now we need to find a digit 'w' such that (24460 + w) * w is less than or equal to 24461.

14. **Find 'w'**: Testing values for 'w':
- For w = 1: (24460 + 1) * 1 = 24461 * 1 = 24461 (this works)
- For w = 2: (24460 + 2) * 2 = 24462 * 2 = 48924 (this is too high)

So, we take w = 1. Write 1 above the line next to 12.23, making it 12.231.

15. **Final Result**: The square root of 149.597361 is approximately 12.231.