Find the square root of :
496 correct to three places of decimal.

Answer

To find the square root of 496 correct to three decimal places, we can use the long division method. Here’s the step-by-step solution:

### Step 1: Pair the digits
Start by pairing the digits of the number from right to left. For 496, we have:
- Pair: (4)(96)

### Step 2: Find the largest number whose square is less than or equal to the first pair
The first pair is 4. The largest whole number whose square is less than or equal to 4 is 2 (since \(2^2 = 4\)).
- Write 2 above the pair (4) and subtract \(2^2 = 4\) from 4.
- Remainder: \(4 - 4 = 0\)

### Step 3: Bring down the next pair
Now, bring down the next pair (96) next to the remainder (0), making it 096 or simply 96.

### Step 4: Double the number above
Double the number above (which is 2):
- \(2 \times 2 = 4\)

### Step 5: Find the next digit
Now we need to find a digit \(x\) such that \(4x \times x\) is less than or equal to 96.
- Testing \(4 \times 2 = 8\) gives \(82 \times 2 = 164\) (too high).
- Testing \(4 \times 1 = 4\) gives \(41 \times 1 = 41\) (valid).
- Testing \(4 \times 2 = 8\) gives \(42 \times 2 = 84\) (valid).
- Testing \(4 \times 3 = 12\) gives \(43 \times 3 = 129\) (too high).

The largest valid digit is 2. Write 2 above the 96 next to the 2 from the previous step.

### Step 6: Subtract and bring down
Now subtract \(84\) from \(96\):
- Remainder: \(96 - 84 = 12\)

### Step 7: Add decimal point and bring down two zeros
Since we need to find the square root to three decimal places, we add a decimal point and bring down two zeros, making it 1200.

### Step 8: Double the current result
Double the current result (which is 22):
- \(22 \times 2 = 44\)

### Step 9: Find the next digit
Now we need to find a digit \(y\) such that \(44y \times y\) is less than or equal to 1200.
- Testing \(44 \times 2 = 88\) gives \(442 \times 2 = 884\) (valid).
- Testing \(44 \times 3 = 132\) gives \(443 \times 3 = 1329\) (too high).

The largest valid digit is 2. Write 2 above the 00 next to the 22.

### Step 10: Subtract and bring down
Now subtract \(884\) from \(1200\):
- Remainder: \(1200 - 884 = 316\)

### Step 11: Bring down two more zeros
Bring down two more zeros, making it 31600.

### Step 12: Double the current result
Double the current result (which is 222):
- \(222 \times 2 = 444\)

### Step 13: Find the next digit
Now we need to find a digit \(z\) such that \(444z \times z\) is less than or equal to 31600.
- Testing \(444 \times 7 = 3108\) gives \(4447 \times 7 = 31129\) (valid).
- Testing \(444 \times 8 = 3552\) gives \(4448 \times 8 = 35584\) (too high).

The largest valid digit is 7. Write 7 above the 00 next to the 222.

### Step 14: Final subtraction
Now subtract \(31129\) from \(31600\):
- Remainder: \(31600 - 31129 = 471\)

### Conclusion
The square root of 496 correct to three decimal places is approximately \(22.271\).