(sqrt(5) + sqrt(3))/(sqrt(5) - sqrt(3)) =

1. Translate the problem requirements: We need to simplify the fraction with square roots in both numerator and denominator to match one of the given answer choices
2. Identify the rationalization strategy: Recognize that we have a fraction with radicals in the denominator, requiring multiplication by the conjugate to eliminate the radicals from the denominator
3. Apply conjugate multiplication: Multiply both numerator and denominator by the conjugate of the denominator to rationalize
4. Expand and simplify: Use the difference of squares pattern in the denominator and FOIL method in the numerator to reach the final simplified form

Execution of Strategic Approach

1. Translate the problem requirements

We need to simplify the fraction \(\frac{\sqrt{5} + \sqrt{3}}{\sqrt{5} - \sqrt{3}}\) to match one of the given answer choices.

Looking at our answer choices, we see that some are simple numbers (like 2, 4), one involves \(\sqrt{15}\), and one is a sum involving \(\sqrt{15}\). This tells us our final answer should be in a simplified form without radicals in the denominator.

Process Skill: TRANSLATE

2. Identify the rationalization strategy

We have a fraction with square roots in the denominator: \(\sqrt{5} - \sqrt{3}\). To eliminate radicals from the denominator, we use the rationalization technique.

The key insight is to multiply by the conjugate of the denominator. The conjugate of \(\sqrt{5} - \sqrt{3}\) is \(\sqrt{5} + \sqrt{3}\) (we change the sign between the terms).

We multiply both numerator and denominator by this conjugate:
\(\frac{\sqrt{5} + \sqrt{3}}{\sqrt{5} - \sqrt{3}} \cdot \frac{\sqrt{5} + \sqrt{3}}{\sqrt{5} + \sqrt{3}}\)

3. Apply conjugate multiplication

Now we multiply:
\(\frac{\sqrt{5} + \sqrt{3}}{\sqrt{5} - \sqrt{3}} \cdot \frac{\sqrt{5} + \sqrt{3}}{\sqrt{5} + \sqrt{3}} = \frac{(\sqrt{5} + \sqrt{3})(\sqrt{5} + \sqrt{3})}{(\sqrt{5} - \sqrt{3})(\sqrt{5} + \sqrt{3})}\)

This gives us:
\(\frac{(\sqrt{5} + \sqrt{3})^2}{(\sqrt{5} - \sqrt{3})(\sqrt{5} + \sqrt{3})}\)

The denominator uses the difference of squares pattern: \((a-b)(a+b) = a^2 - b^2\)
The numerator requires expanding the square: \((a+b)^2 = a^2 + 2ab + b^2\)

4. Expand and simplify

Denominator calculation:
\((\sqrt{5} - \sqrt{3})(\sqrt{5} + \sqrt{3}) = (\sqrt{5})^2 - (\sqrt{3})^2 = 5 - 3 = 2\)

Numerator calculation:
\((\sqrt{5} + \sqrt{3})^2 = (\sqrt{5})^2 + 2(\sqrt{5})(\sqrt{3}) + (\sqrt{3})^2\)
\(= 5 + 2\sqrt{15} + 3\)
\(= 8 + 2\sqrt{15}\)

Final simplification:
\(\frac{8 + 2\sqrt{15}}{2} = \frac{8}{2} + \frac{2\sqrt{15}}{2} = 4 + \sqrt{15}\)

Process Skill: MANIPULATE

5. Final Answer

\(\frac{\sqrt{5} + \sqrt{3}}{\sqrt{5} - \sqrt{3}} = 4 + \sqrt{15}\)

This matches answer choice E. We can verify this is correct by checking that our rationalization process eliminated all radicals from the denominator and produced a result that appears in our answer choices.

Common Faltering Points

Errors while devising the approach

1. Not recognizing the need for rationalization: Students may attempt to directly combine or simplify the radicals in the numerator and denominator without realizing that the proper approach is to rationalize the denominator. They might try to factor out common terms or perform invalid operations like canceling √5 from both numerator and denominator.

2. Choosing the wrong conjugate: Students may incorrectly identify the conjugate of the denominator. Instead of using (√5 + √3) as the conjugate of (√5 - √3), they might use incorrect expressions like (√3 - √5) or attempt to rationalize using the same expression as the denominator.

Errors while executing the approach

1. Algebraic expansion errors: When expanding (√5 + √3)², students commonly forget the middle term 2√15 and incorrectly calculate it as just 5 + 3 = 8. The correct expansion is (√5)² + 2(√5)(√3) + (√3)² = 5 + 2√15 + 3.

2. Difference of squares calculation mistakes: In the denominator, students may incorrectly apply the difference of squares formula (a-b)(a+b) = a² - b². They might calculate (√5 - √3)(√5 + √3) as something other than 5 - 3 = 2, possibly getting confused about which terms to square.

3. Final fraction simplification errors: When simplifying (8 + 2√15)/2, students may incorrectly distribute the division, getting results like 4 + 2√15 instead of the correct 4 + √15. They might forget to divide both terms by 2 or make errors in the arithmetic.

Errors while selecting the answer

1. Stopping at an intermediate step: Students might select answer choice D (8√15) if they only calculate the middle term 2√15 from the expansion but miss the constant terms. Or they might select answer B (4) if they only consider the constant part and ignore the radical term completely.