The square root of 824464 is-

Options

A. 686
B. 868
C. 908
D. None of these

Answer

To find the square root of 824464 using the division method, follow these steps:

### Step 1: Pair the Digits
Start from the right and pair the digits of the number. For 824464, we can pair them as follows:
- (64)(64)(82)

### Step 2: Find the Largest Square
Next, we need to find the largest square number that is less than or equal to the first pair (82). The largest square number less than 82 is \(9^2 = 81\).

### Step 3: Subtract and Bring Down
Subtract 81 from 82:
\[ 82 - 81 = 1 \]
Now, bring down the next pair (64), making it 164.

### Step 4: Determine the Next Digit
We need to find a digit \(x\) such that:
\[ (18 + x) \cdot x \leq 164 \]
Here, \(18\) is obtained from doubling the previous quotient (9). Testing \(x = 8\):
\[ (18 + 8) \cdot 8 = 26 \cdot 8 = 208 \]
This is too high. Now testing \(x = 7\):
\[ (18 + 7) \cdot 7 = 25 \cdot 7 = 175 \]
Still too high. Testing \(x = 6\):
\[ (18 + 6) \cdot 6 = 24 \cdot 6 = 144 \]
This works. So, we place \(6\) in the quotient.

### Step 5: Subtract Again
Now, subtract \(144\) from \(164\):
\[ 164 - 144 = 20 \]
Bring down the next pair (64), making it 2064.

### Step 6: Repeat the Process
Now we need to find \(y\) such that:
\[ (186 + y) \cdot y \leq 2064 \]
Testing \(y = 9\):
\[ (186 + 9) \cdot 9 = 195 \cdot 9 = 1755 \]
This works. Place \(9\) in the quotient.

### Step 7: Final Subtraction
Subtract \(1755\) from \(2064\):
\[ 2064 - 1755 = 309 \]
Since there are no more pairs to bring down, we conclude here.

### Final Answer
The square root of 824464 is \(908\).