Find the square root of :
`4.2025`

Answer

To find the square root of 4.2025, we can use the long division method for square roots. Here’s a step-by-step solution:

### Step 1: Pair the digits
First, we need to pair the digits of the number starting from the decimal point. For the number 4.2025, we can pair it as follows:
- Before the decimal: 4
- After the decimal: 20 | 25

### Step 2: Find the largest square
Now, we find the largest square less than or equal to the first pair (4):
- The largest square is \(2^2 = 4\).
- Write 2 above the square root symbol.

### Step 3: Subtract and bring down the next pair
Subtract 4 from 4:
- \(4 - 4 = 0\)
- Now bring down the next pair (20), making it 020 (or just 20).

### Step 4: Double the number above the root
Double the number above the root (which is 2):
- \(2 \times 2 = 4\)
- Write this as 4_ (where _ is a blank space for the next digit).

### Step 5: Find the next digit
Now we need to find a digit \(x\) such that:
- \(4x \times x \leq 20\)
- Testing \(x = 4\):
- \(44 \times 4 = 176\) (too high)
- Testing \(x = 3\):
- \(43 \times 3 = 129\) (too high)
- Testing \(x = 2\):
- \(42 \times 2 = 84\) (too high)
- Testing \(x = 1\):
- \(41 \times 1 = 41\) (too high)
- Testing \(x = 0\):
- \(40 \times 0 = 0\) (works)

So, we put 0 in the blank space.

### Step 6: Subtract and bring down the next pair
Now we have:
- \(20 - 0 = 20\)
- Bring down the next pair (25), making it 2025.

### Step 7: Repeat the process
Now, we double the number above the root (which is still 2):
- \(2 \times 2 = 4\)
- Write this as 40_.

Now we need to find a digit \(y\) such that:
- \(40y \times y \leq 2025\)
- Testing \(y = 5\):
- \(405 \times 5 = 2025\) (exact match)

### Step 8: Final subtraction
Now we subtract:
- \(2025 - 2025 = 0\)

### Conclusion
Since we have reached 0, the square root of 4.2025 is:
\[ \sqrt{4.2025} = 2.05 \]