What is the square root of 151321 ?

Answer

To find the square root of 151321 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 151321. We have:
- Pair 1: 21
- Pair 2: 51
- Pair 3: 1
Thus, we can write it as: 1 | 51 | 21.

2. **Find the Largest Square**:
Look for the largest square number less than or equal to the first pair (1). The largest square is \(1^2 = 1\).
- Write 1 above the line (the square root) and subtract \(1^2\) from 1:
\[ 1 - 1 = 0. \]

3. **Bring Down the Next Pair**:
Bring down the next pair (51) next to the remainder (0):
\[ 0 | 51 \rightarrow 051. \]

4. **Double the Root**:
Double the current root (1) to get 2. Now we need to find a digit \(x\) such that \(2x \cdot x\) is less than or equal to 51.
- Testing \(x = 2\):
\[ 22 \cdot 2 = 44 \quad (\text{which is less than 51}). \]
- Testing \(x = 3\):
\[ 23 \cdot 3 = 69 \quad (\text{which is greater than 51}). \]
So, we take \(x = 2\).

5. **Subtract and Bring Down the Next Pair**:
Write 2 above the line, subtract \(44\) from \(51\):
\[ 51 - 44 = 7. \]
Now bring down the next pair (21) to get:
\[ 721. \]

6. **Double the Root Again**:
Double the current root (12) to get 24. Now we need to find a digit \(y\) such that \(24y \cdot y\) is less than or equal to 721.
- Testing \(y = 2\):
\[ 242 \cdot 2 = 484 \quad (\text{which is less than 721}). \]
- Testing \(y = 3\):
\[ 243 \cdot 3 = 729 \quad (\text{which is greater than 721}). \]
So, we take \(y = 2\).

7. **Final Subtraction**:
Write 2 above the line, subtract \(484\) from \(721\):
\[ 721 - 484 = 237. \]

8. **Combine the Results**:
The digits above the line give us the square root:
\[ \sqrt{151321} = 382. \]

### Final Answer:
The square root of 151321 is **389**.