Find the square root of 10404.

Answer

To find the square root of 10404 using the long division method, follow these steps:

### Step 1: Pair the digits
Start by pairing the digits of the number from right to left. For 10404, we can pair them as follows:
- 10 | 40 | 4

### Step 2: Find the largest square
Look at the first pair (10). The largest square less than or equal to 10 is 3 (since \(3^2 = 9\)). Write 3 above the line.

### Step 3: Subtract and bring down the next pair
Subtract \(9\) from \(10\) to get \(1\). Then bring down the next pair (40) to get \(140\).

### Step 4: Double the number above the line
Double the number above the line (which is 3) to get \(6\).

### Step 5: Find the next digit
Now, we need to find a digit \(x\) such that \(6x \times x\) is less than or equal to \(140\). Testing \(x = 2\):
- \(62 \times 2 = 124\) (which is less than 140)

So, we write \(2\) next to \(3\) above the line, making it \(32\).

### Step 6: Subtract again
Subtract \(124\) from \(140\) to get \(16\). Bring down the next pair (04) to get \(1604\).

### Step 7: Double the number above the line again
Double \(32\) to get \(64\).

### Step 8: Find the next digit
Now, we need to find a digit \(y\) such that \(64y \times y\) is less than or equal to \(1604\). Testing \(y = 2\):
- \(642 \times 2 = 1284\) (which is less than 1604)

So, we write \(2\) next to \(32\) above the line, making it \(102\).

### Step 9: Final subtraction
Subtract \(1284\) from \(1604\) to get \(320\). Bring down the next pair (00) to get \(3200\).

### Step 10: Double the number above the line again
Double \(102\) to get \(204\).

### Step 11: Find the last digit
Now, we need to find a digit \(z\) such that \(204z \times z\) is less than or equal to \(3200\). Testing \(z = 1\):
- \(2041 \times 1 = 2041\) (which is less than 3200)

So, we write \(1\) next to \(102\) above the line, making it \(1021\).

### Conclusion
The square root of \(10404\) is \(102\).