What is the square root of 11236 ?

Options

A. 106
B. 96
C. 114
D. 104

Answer

To find the square root of 11236, we can use the method of pairing digits and estimating the square root step by step.

### Step 1: Pair the digits
First, we start from the right and pair the digits of the number 11236. We can pair them as follows:
- 11 | 23 | 6

### Step 2: Find the largest square
Next, we look at the first pair (11). The largest square less than or equal to 11 is 3 (since \(3^2 = 9\)). We write 3 above the square root symbol.

### Step 3: Subtract and bring down the next pair
Now we subtract \(9\) from \(11\):
\[ 11 - 9 = 2 \]
Next, we bring down the next pair (23) to make it 223.

### Step 4: Double the number above the root
We double the number we have above the square root (which is 3) to get 6. We write this down:
\[ 6\_ \]
Now we need to find a digit \(x\) such that \(6x \times x\) is less than or equal to 223.

### Step 5: Estimate the next digit
We can try \(x = 3\):
\[ 63 \times 3 = 189 \]
This is less than 223. Now, we can try \(x = 4\):
\[ 64 \times 4 = 256 \]
This is greater than 223. So, we take \(x = 3\).

### Step 6: Subtract and bring down the next pair
Now we subtract \(189\) from \(223\):
\[ 223 - 189 = 34 \]
Next, we bring down the last pair (06) to make it 3406.

### Step 7: Double the number above the root
We double the number we have above the square root (which is now 33) to get 66. We write this down:
\[ 66\_ \]
Now we need to find a digit \(y\) such that \(66y \times y\) is less than or equal to 3406.

### Step 8: Estimate the next digit
We can try \(y = 5\):
\[ 665 \times 5 = 3325 \]
This is less than 3406. Now, we can try \(y = 6\):
\[ 666 \times 6 = 3996 \]
This is greater than 3406. So, we take \(y = 5\).

### Step 9: Final calculation
Now we subtract \(3325\) from \(3406\):
\[ 3406 - 3325 = 81 \]
At this point, we have completed our calculation, and the square root of 11236 is:
\[ 335 \]

### Conclusion
The square root of 11236 is 106.