Evaluate `sqrt3` correct up to two places of decimal.

Answer

To evaluate \(\sqrt{3}\) correct up to two decimal places, we can use the long division method. Here’s a step-by-step solution:

### Step 1: Set Up the Problem
We want to find the square root of 3. We can start by writing it as \(\sqrt{3.000000}\) to allow for decimal places.

### Step 2: Pair the Digits
Since we are working with decimals, we can pair the digits starting from the decimal point. This gives us pairs: \(3.00\) (the first pair is \(3\) and the second pair is \(00\)).

### Step 3: Find the Largest Integer
The largest integer whose square is less than or equal to \(3\) is \(1\) (since \(1^2 = 1\)). We write \(1\) above the square root symbol.

### Step 4: Subtract and Bring Down the Next Pair
Now, we subtract \(1^2 = 1\) from \(3\):
\[ 3 - 1 = 2 \]
Next, we bring down the next pair, which is \(00\), making it \(200\).

### Step 5: Double the Quotient
Double the quotient we have so far (which is \(1\)):
\[ 2 \times 1 = 2 \]
Now we need to find a digit \(x\) such that \(2x \times x\) is less than or equal to \(200\).

### Step 6: Find the Next Digit
We try different values for \(x\):
- For \(x = 5\): \(25 \times 5 = 125\) (this works)
- For \(x = 6\): \(26 \times 6 = 156\) (this works)
- For \(x = 7\): \(27 \times 7 = 189\) (this works)
- For \(x = 8\): \(28 \times 8 = 224\) (this does not work)

So, the largest digit \(x\) we can use is \(7\). We write \(7\) next to \(1\) in the quotient.

### Step 7: Subtract Again
Now we subtract \(189\) from \(200\):
\[ 200 - 189 = 11 \]
Next, we bring down the next pair of zeros, making it \(1100\).

### Step 8: Double the New Quotient
Double the current quotient \(17\):
\[ 2 \times 17 = 34 \]
Now we need to find a digit \(y\) such that \(34y \times y\) is less than or equal to \(1100\).

### Step 9: Find the Next Digit
We try different values for \(y\):
- For \(y = 3\): \(343 \times 3 = 1029\) (this works)
- For \(y = 4\): \(344 \times 4 = 1376\) (this does not work)

So, the largest digit \(y\) we can use is \(3\). We write \(3\) next to \(17\) in the quotient.

### Step 10: Final Subtraction
Now we subtract \(1029\) from \(1100\):
\[ 1100 - 1029 = 71 \]
Next, we bring down the next pair of zeros, making it \(7100\).

### Step 11: Round Off
At this point, we have \(1.73\) as our result. Since we only need two decimal places, we check the next digit (which is \(0\)). Since \(0 < 5\), we do not round up.

### Final Answer
Thus, \(\sqrt{3} \approx 1.73\) correct up to two decimal places.

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