`sqrt(125 ul(44))` = ?

Answer

To solve the expression \( \sqrt{12544} \) using the long division method, we can follow these steps:

### Step 1: Pair the digits
Start from the right and pair the digits in groups of two. For \( 12544 \), we can pair it as follows:
- \( 12 | 54 | 4 \)

### Step 2: Find the largest square
Now, we look at the first pair \( 12 \). The largest square less than or equal to \( 12 \) is \( 3^2 = 9 \).

### Step 3: Subtract and bring down the next pair
Subtract \( 9 \) from \( 12 \):
- \( 12 - 9 = 3 \)

Now, bring down the next pair \( 54 \) to get \( 354 \).

### Step 4: Double the quotient
Double the quotient we have so far, which is \( 3 \):
- \( 3 \times 2 = 6 \)

### Step 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 \( 354 \).
Testing \( x = 5 \):
- \( 65 \times 5 = 325 \) (this works)

Now subtract \( 325 \) from \( 354 \):
- \( 354 - 325 = 29 \)

### Step 6: Bring down the next pair
Bring down the next pair \( 04 \) to get \( 2904 \).

### Step 7: Double the current quotient
Double the current quotient \( 35 \):
- \( 35 \times 2 = 70 \)

### Step 8: Find the next digit
Now find a digit \( y \) such that \( (700 + y) \times y \) is less than or equal to \( 2904 \).
Testing \( y = 4 \):
- \( 704 \times 4 = 2816 \) (this works)

Now subtract \( 2816 \) from \( 2904 \):
- \( 2904 - 2816 = 88 \)

### Step 9: Bring down the next pair
Since there are no more pairs to bring down, we can stop here. The quotient we have found is \( 112 \).

### Final Answer
Thus, the square root of \( 12544 \) is \( 112 \).