Find the value of `sqrt(5)` correct to `2` decimal places, then use it to find the square root of `(3-sqrt(5))/(3+sqrt(5))` correct to `2` significant digits.

Answer

To solve the problem step by step, we will first find the value of \( \sqrt{5} \) correct to 2 decimal places, and then use that value to find the square root of \( \frac{3 - \sqrt{5}}{3 + \sqrt{5}} \) correct to 2 significant digits.

### Step 1: Finding \( \sqrt{5} \) Correct to 2 Decimal Places

1. **Set up the number**: Write \( 5 \) as \( 5.00000 \) to facilitate finding the square root.
2. **Make pairs**: Starting from the decimal point, group the digits into pairs: \( 5 | 00 | 00 \).
3. **Find the largest integer**: The largest integer whose square is less than or equal to \( 5 \) is \( 2 \) (since \( 2^2 = 4 \)).
- **Calculation**: \( 5 - 4 = 1 \).
4. **Bring down the next pair**: Bring down the next pair of zeros, making it \( 100 \).
5. **Double the current result**: Double \( 2 \) to get \( 4 \) (this will be the first part of our next divisor).
6. **Find the next digit**: We need to find a digit \( x \) such that \( 40x \times x \) is less than or equal to \( 100 \).
- Testing \( x = 2 \): \( 42 \times 2 = 84 \) (valid).
- Testing \( x = 3 \): \( 43 \times 3 = 129 \) (too high).
- **Calculation**: \( 100 - 84 = 16 \).
7. **Bring down the next pair**: Bring down another pair of zeros, making it \( 1600 \).
8. **Double the current result**: Now double \( 42 \) to get \( 84 \).
9. **Find the next digit**: We need to find a digit \( y \) such that \( 840y \times y \) is less than or equal to \( 1600 \).
- Testing \( y = 1 \): \( 841 \times 1 = 841 \) (valid).
- Testing \( y = 2 \): \( 842 \times 2 = 1684 \) (too high).
- **Calculation**: \( 1600 - 841 = 759 \).
10. **Bring down the next pair**: Bring down another pair of zeros, making it \( 75900 \).
11. **Double the current result**: Double \( 841 \) to get \( 1682 \).
12. **Find the next digit**: We need to find a digit \( z \) such that \( 16820z \times z \) is less than or equal to \( 75900 \).
- Testing \( z = 4 \): \( 16824 \times 4 = 67376 \) (valid).
- Testing \( z = 5 \): \( 16825 \times 5 = 84125 \) (too high).
- **Calculation**: \( 75900 - 67376 = 8524 \).

At this point, we have \( \sqrt{5} \approx 2.236 \). Rounding this to 2 decimal places gives us \( 2.24 \).

### Step 2: Finding \( \sqrt{\frac{3 - \sqrt{5}}{3 + \sqrt{5}}} \)

1. **Substituting the value**: Substitute \( \sqrt{5} \approx 2.24 \) into the expression:
\[ \frac{3 - \sqrt{5}}{3 + \sqrt{5}} = \frac{3 - 2.24}{3 + 2.24} = \frac{0.76}{5.24} \]
2. **Rationalizing the denominator**: Multiply the numerator and denominator by \( 3 - \sqrt{5} \):
\[ \frac{(3 - \sqrt{5})^2}{(3 + \sqrt{5})(3 - \sqrt{5})} = \frac{(3 - 2.24)^2}{3^2 - (\sqrt{5})^2} = \frac{0.76^2}{9 - 5} = \frac{0.5776}{4} \]
3. **Calculating the result**:
\[ \frac{0.5776}{4} = 0.1444 \]
4. **Taking the square root**:
\[ \sqrt{0.1444} \approx 0.38 \]

### Final Answer:
The value of \( \sqrt{\frac{3 - \sqrt{5}}{3 + \sqrt{5}}} \) correct to 2 significant digits is \( 0.38 \).