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

Answer

To find the square root of 82.6 correct to two decimal places, we can use the long division method for square roots. Here’s a step-by-step solution:

### Step 1: Prepare the Number
We start by writing the number 82.6 in a way that makes it easier to find the square root. We can consider it as 82.60 (adding two zeros after the decimal to facilitate the calculation).

### Step 2: Pair the Digits
Next, we pair the digits starting from the decimal point. For 82.60, we have:
- Pair 1: 82
- Pair 2: 60

### Step 3: Find the Largest Square
Now we find the largest square less than or equal to the first pair (82). The largest perfect square less than 82 is 81, which is \(9^2\).

### Step 4: Subtract and Bring Down the Next Pair
We subtract 81 from 82:
\[ 82 - 81 = 1 \]
Now, we bring down the next pair (60), making it 160.

### Step 5: Double the Quotient
Next, we double the quotient we have so far (which is 9):
\[ 2 \times 9 = 18 \]

### Step 6: Find the Next Digit
We need to find a digit (let's call it \(x\)) such that:
\[ (180 + x) \times x \leq 1600 \]
We will try different values for \(x\):
- For \(x = 8\):
\[ 188 \times 8 = 1504 \] (this works)
- For \(x = 9\):
\[ 189 \times 9 = 1701 \] (this is too high)

So, we take \(x = 8\).

### Step 7: Subtract Again
Now we subtract:
\[ 1600 - 1504 = 96 \]
We now have a new remainder of 96.

### Step 8: Bring Down the Next Pair
Next, we bring down another pair of zeros, making it 9600.

### Step 9: Repeat the Process
We double the current quotient (which is 98):
\[ 2 \times 98 = 196 \]
Now we need to find a digit \(y\) such that:
\[ (1960 + y) \times y \leq 9600 \]
- For \(y = 4\):
\[ 1964 \times 4 = 7856 \] (this works)
- For \(y = 5\):
\[ 1965 \times 5 = 9825 \] (this is too high)

So, we take \(y = 4\).

### Step 10: Final Calculation
Now we subtract:
\[ 9600 - 7856 = 1044 \]
We can stop here since we only need two decimal places.

### Final Result
Thus, the square root of 82.6 correct to two decimal places is:
\[ 9.04 \]