Find the square root of :
`0.1764`

Options

A. `0.48`
B. `0.52`
C. `0.42`
D. `0.62`

Answer

To find the square root of 0.1764 using the long division method, follow these steps:

### Step 1: Pair the digits
Start by pairing the digits of the number. For the number 0.1764, we can pair it as follows:
- Before the decimal: 0 (one pair)
- After the decimal: 17 | 64 (two pairs)

So, we have pairs: (0) (17) (64).

### Step 2: Set up the long division
We will set it up for long division. We will find the square root of 0.1764.

### Step 3: Find the largest square
Look for the largest square number less than or equal to the first pair (0). The largest square is 0 (since 0 x 0 = 0).

### Step 4: Subtract and bring down the next pair
Subtract 0 from 0, which gives us 0. Now, bring down the next pair (17). We now have 1.7.

### Step 5: Find the next digit
We need to find a number which, when doubled and multiplied by itself, gives a product less than or equal to 17.
- 1 x 1 = 1 (double of 1 is 2)
- 2 x 2 = 4 (double of 2 is 4)
- 3 x 3 = 9 (double of 3 is 6)
- 4 x 4 = 16 (double of 4 is 8)

The largest square less than or equal to 17 is 16 (4 x 4).

### Step 6: Subtract and bring down the next pair
Now, subtract 16 from 17, which gives us 1. Bring down the next pair (64), making it 164.

### Step 7: Find the next digit
Now we need to find a number that, when added to 8 (the double of 4), and multiplied by that same number, gives us a product less than or equal to 164.
Let’s denote the next digit as x:
- (8 + x) * x = 164

Testing values:
- For x = 1: (8 + 1) * 1 = 9 (too low)
- For x = 2: (8 + 2) * 2 = 20 (too low)
- For x = 3: (8 + 3) * 3 = 33 (too low)
- For x = 4: (8 + 4) * 4 = 48 (too low)
- For x = 5: (8 + 5) * 5 = 65 (too low)
- For x = 6: (8 + 6) * 6 = 84 (too low)
- For x = 7: (8 + 7) * 7 = 105 (too low)
- For x = 8: (8 + 8) * 8 = 128 (too low)
- For x = 9: (8 + 9) * 9 = 153 (too low)
- For x = 10: (8 + 10) * 10 = 180 (too high)

The largest valid x is 9. So, we take 9.

### Step 8: Final calculation
Now we subtract 153 from 164, which gives us 11. We have now completed the long division.

### Step 9: Write the result
The square root of 0.1764 is 0.42.

### Final Answer
Thus, the square root of 0.1764 is **0.42**.

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