Simplify (√5 − √3) / (√5 + √3) to an exact radical expression.

Introduction / Context:
Rationalizing a fraction with radicals in numerator and denominator can often be simplified using the conjugate. This tests algebraic manipulation of surds.

Given Data / Assumptions:

• Expression: (√5 − √3) / (√5 + √3).
• We seek an exact simplified form.

Concept / Approach:
Multiply numerator and denominator by the conjugate of the denominator (√5 − √3). Use (a − b)^2 and the difference of squares identity in the denominator.

Step-by-Step Solution:
[(√5 − √3) / (√5 + √3)] * [(√5 − √3) / (√5 − √3)] = (√5 − √3)^2 / (5 − 3). (√5 − √3)^2 = 5 − 2√15 + 3 = 8 − 2√15. Denominator: 5 − 3 = 2. Result = (8 − 2√15) / 2 = 4 − √15.

Verification / Alternative check:
Substitute numerical values (√5 ≈ 2.236, √3 ≈ 1.732) to confirm both sides evaluate close to the same decimal (~0.127..., which equals 4 − √15 numerically).

Why Other Options Are Wrong:
4 + √15 would arise from (√5 + √3)^2 / (5 − 3), not our expression. 1/2 and 1 do not match the exact simplification.

Common Pitfalls:
Forgetting to square the entire numerator when using the conjugate or mixing up signs in (a − b)^2.

Final Answer:
4 − √15