What is the square root of 11881 ?

Options

A. 111
B. 109
C. 119
D. 101

Answer

To find the square root of 11881 using the long division method, follow these steps:

### Step 1: Pair the Digits
Start from the right side and pair the digits of the number. For 11881, we can pair them as follows:
- (11)(88)(1)

### Step 2: Find the Largest Square
Look at the first pair (11). The largest integer whose square is less than or equal to 11 is 3, since \(3^2 = 9\).

### Step 3: Subtract and Bring Down
Subtract \(9\) from \(11\):
\[ 11 - 9 = 2 \]
Now, bring down the next pair (88) to the remainder:
\[ 2\text{ (remainder)} \rightarrow 288 \]

### Step 4: Double the Quotient
Double the quotient (which is currently 3):
\[ 2 \times 3 = 6 \]

### Step 5: Find the Next Digit
Now, we need to find a digit \(x\) such that:
\[ (60 + x) \times x \leq 288 \]
Testing \(x = 4\):
\[ 64 \times 4 = 256 \quad (\text{valid}) \]
Testing \(x = 5\):
\[ 65 \times 5 = 325 \quad (\text{not valid}) \]
So, \(x = 4\) is the correct digit.

### Step 6: Subtract Again
Now, subtract \(256\) from \(288\):
\[ 288 - 256 = 32 \]
Bring down the next pair (01):
\[ 32 \rightarrow 3201 \]

### Step 7: Double the Quotient Again
The current quotient is \(34\) (from 3 and 4), so double it:
\[ 2 \times 34 = 68 \]

### Step 8: Find the Next Digit Again
Now, find a digit \(y\) such that:
\[ (680 + y) \times y \leq 3201 \]
Testing \(y = 4\):
\[ 684 \times 4 = 2736 \quad (\text{valid}) \]
Testing \(y = 5\):
\[ 685 \times 5 = 3425 \quad (\text{not valid}) \]
So, \(y = 4\) is the correct digit.

### Step 9: Final Subtraction
Subtract \(2736\) from \(3201\):
\[ 3201 - 2736 = 465 \]
Since we have no more pairs to bring down, we conclude here.

### Final Result
The square root of 11881 is:
\[ 34 \text{ (from the digits 3, 4)} \]

Thus, the square root of 11881 is \(109\).

---