Find the square root of :
`245` correct to two places of decimal.

Answer

To find the square root of 245 correct to two decimal places, we can use the long division method. Here’s a step-by-step solution:

### Step 1: Set up the number for square root extraction
We will treat 245 as 245.0000 to facilitate finding the square root to two decimal places.

### Step 2: Pair the digits
Starting from the decimal point, pair the digits:
- The first pair is 24 (from 245)
- The second pair is 50 (from 00)

So we have: 24 | 50

### Step 3: Find the largest number whose square is less than or equal to the first pair (24)
The largest integer whose square is less than or equal to 24 is 4, since \(4^2 = 16\).

### Step 4: Subtract and bring down the next pair
- Subtract \(16\) from \(24\):
\[ 24 - 16 = 8 \]
- Bring down the next pair (50):
\[ 8 | 50 \rightarrow 850 \]

### Step 5: Double the quotient and find the next digit
Double the current quotient (which is 4):
\[ 2 \times 4 = 8 \]
Now we need to find a digit \(x\) such that:
\[ (80 + x) \times x \leq 850 \]

### Step 6: Test values for \(x\)
- For \(x = 1\):
\[ (80 + 1) \times 1 = 81 \quad (\text{too low}) \]
- For \(x = 2\):
\[ (80 + 2) \times 2 = 82 \times 2 = 164 \quad (\text{too low}) \]
- For \(x = 3\):
\[ (80 + 3) \times 3 = 83 \times 3 = 249 \quad (\text{too low}) \]
- For \(x = 4\):
\[ (80 + 4) \times 4 = 84 \times 4 = 336 \quad (\text{too low}) \]
- For \(x = 5\):
\[ (80 + 5) \times 5 = 85 \times 5 = 425 \quad (\text{too low}) \]
- For \(x = 6\):
\[ (80 + 6) \times 6 = 86 \times 6 = 516 \quad (\text{too low}) \]
- For \(x = 7\):
\[ (80 + 7) \times 7 = 87 \times 7 = 609 \quad (\text{too low}) \]
- For \(x = 8\):
\[ (80 + 8) \times 8 = 88 \times 8 = 704 \quad (\text{too low}) \]
- For \(x = 9\):
\[ (80 + 9) \times 9 = 89 \times 9 = 801 \quad (\text{too low}) \]
- For \(x = 10\):
\[ (80 + 10) \times 10 = 90 \times 10 = 900 \quad (\text{too high}) \]

The largest \(x\) that works is \(9\).

### Step 7: Subtract and bring down the next pair
- Subtract \(801\) from \(850\):
\[ 850 - 801 = 49 \]
- Bring down the next pair (00):
\[ 49 | 00 \rightarrow 4900 \]

### Step 8: Double the quotient and find the next digit
Now the quotient is \(49\), double it:
\[ 2 \times 49 = 98 \]
We need to find \(y\) such that:
\[ (980 + y) \times y \leq 4900 \]

### Step 9: Test values for \(y\)
- For \(y = 4\):
\[ (980 + 4) \times 4 = 984 \times 4 = 3936 \quad (\text{too low}) \]
- For \(y = 5\):
\[ (980 + 5) \times 5 = 985 \times 5 = 4925 \quad (\text{too high}) \]
- For \(y = 3\):
\[ (980 + 3) \times 3 = 983 \times 3 = 2949 \quad (\text{too low}) \]

The largest \(y\) that works is \(4\).

### Step 10: Finalize the answer
Combining the results, we have:
\[ \sqrt{245} \approx 15.65 \]

### Final Answer
The square root of 245 correct to two decimal places is \(15.65\).