Find the square root of 7 correct to two decimal places, then use it to find the value of `sqrt((4 + sqrt(7))/(4 - sqrt(7)))` correct to three significant digits.

Answer

To solve the problem step by step, we will first find the square root of 7 correct to two decimal places, and then use that value to evaluate the expression \(\sqrt{\frac{4 + \sqrt{7}}{4 - \sqrt{7}}}\) correct to three significant digits.

### Step 1: Finding the Square Root of 7

1. **Set up the number**: Write 7 as 7.000000.
2. **Pair the digits**: Start from the decimal point and pair the digits: (7)(00)(00)(00).
3. **Find the largest digit**: The largest digit whose square is less than or equal to 7 is 2 (since \(2^2 = 4\)).
4. **Subtract**: \(7 - 4 = 3\).
5. **Bring down the next pair**: Bring down the next pair of zeros to make it 300.
6. **Double the current quotient**: Double the 2 to get 4. Now we need to find a digit \(x\) such that \(4x \cdot x\) is less than or equal to 300.
7. **Trial and error**:
- For \(x = 6\), \(46 \cdot 6 = 276\) (which is less than 300).
- For \(x = 7\), \(47 \cdot 7 = 329\) (which is too high).
8. **Subtract**: \(300 - 276 = 24\).
9. **Bring down the next pair**: Bring down the next pair of zeros to make it 2400.
10. **Double the current quotient**: Now, double 46 to get 92. Find \(y\) such that \(92y \cdot y\) is less than or equal to 2400.
11. **Trial and error**:
- For \(y = 2\), \(922 \cdot 2 = 1844\) (which is less than 2400).
- For \(y = 3\), \(923 \cdot 3 = 2769\) (which is too high).
12. **Subtract**: \(2400 - 1844 = 556\).
13. **Bring down the next pair**: Bring down the next pair of zeros to make it 55600.
14. **Double the current quotient**: Double 462 to get 924. Find \(z\) such that \(924z \cdot z\) is less than or equal to 55600.
15. **Trial and error**:
- For \(z = 6\), \(9246 \cdot 6 = 55476\) (which is less than 55600).
- For \(z = 7\), \(9247 \cdot 7 = 64729\) (which is too high).
16. **Subtract**: \(55600 - 55476 = 124\).
17. **Bring down the next pair**: Bring down the next pair of zeros to make it 12400.
18. **Double the current quotient**: Double 4626 to get 9252. Find \(w\) such that \(9252w \cdot w\) is less than or equal to 12400.
19. **Trial and error**:
- For \(w = 1\), \(92521 \cdot 1 = 92521\) (which is too high).
- For \(w = 0\), \(92520 \cdot 0 = 0\).
20. **Final approximation**: The square root of 7 is approximately 2.645. Since the last digit is 5, we round up to 2.65.

### Step 2: Evaluating the Expression

1. **Substitute \(\sqrt{7}\)**: Now we substitute \(\sqrt{7}\) with 2.65 in the expression \(\sqrt{\frac{4 + \sqrt{7}}{4 - \sqrt{7}}}\).
2. **Calculate the numerator**: \(4 + \sqrt{7} = 4 + 2.65 = 6.65\).
3. **Calculate the denominator**: \(4 - \sqrt{7} = 4 - 2.65 = 1.35\).
4. **Form the fraction**: \(\frac{6.65}{1.35}\).
5. **Perform the division**:
\[ \frac{6.65}{1.35} \approx 4.9259 \]
6. **Take the square root**: \(\sqrt{4.9259} \approx 2.22\) (correct to three significant digits).

### Final Answer

The value of \(\sqrt{\frac{4 + \sqrt{7}}{4 - \sqrt{7}}}\) is approximately **2.22**.