Find the square root of 120409.

Answer

To find the square root of 120409 using the long division method, we can follow these steps:

### Step-by-Step Solution:

1. **Pair the Digits**: Start from the right and pair the digits of the number. For 120409, we can pair them as (12)(04)(09).

2. **Find the Largest Square**: Look for the largest square number less than or equal to the first pair (12). The largest square is \(3^2 = 9\). Write 3 above the line.

3. **Subtract and Bring Down**: Subtract \(9\) from \(12\) to get \(3\). Bring down the next pair (04) next to \(3\) to make it \(304\).

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

5. **Find the Next Digit**: Now, we need to find a digit \(x\) such that \( (60 + x) \times x \) is less than or equal to \(304\). Testing \(4\):
- \(64 \times 4 = 256\) (this works).
- Write \(4\) next to \(3\) above the line.

6. **Subtract Again**: Subtract \(256\) from \(304\) to get \(48\). Bring down the next pair (09) to make it \(4809\).

7. **Double the New Quotient**: Double \(34\) (the number above the line) to get \(68\).

8. **Find the Next Digit Again**: Now, find a digit \(y\) such that \( (680 + y) \times y \) is less than or equal to \(4809\). Testing \(7\):
- \(687 \times 7 = 4809\) (this works perfectly).
- Write \(7\) next to \(34\) above the line.

9. **Final Result**: Since we have no remainder, the square root of \(120409\) is \(347\).

### Final Answer:
The square root of \(120409\) is \(347\).

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