Find the value of [12/(√5 + √3)] + [18/(√5 − √3)] after rationalising the denominators and simplifying the surds to simplest radical form.

Introduction / Context:
This question involves simplifying expressions with surds (square roots) in the denominator. Rationalising denominators is a standard algebraic technique taught in school mathematics, often used in aptitude tests to check comfort with irrational numbers. Here, we must simplify a sum of two rationalised terms and express the final answer in the neatest radical form.

Given Data / Assumptions:

• Expression: [12/(√5 + √3)] + [18/(√5 − √3)].

• The square roots represent positive real numbers.

• We must simplify completely and match the result with one of the structured options.

Concept / Approach:
The main idea is to multiply numerator and denominator of each fraction by the conjugate of its denominator. For terms of the form a/(√5 + √3), the conjugate is (√5 − √3). Multiplying conjugates produces a difference of squares, eliminating the square roots from the denominator. After rationalising both fractions, we expand the numerators, combine like terms, and then factor if possible to match the answer pattern.

Step-by-Step Solution:
Step 1: Rationalise the first fraction: 12/(√5 + √3) × (√5 − √3)/(√5 − √3). Step 2: Denominator becomes (√5)^2 − (√3)^2 = 5 − 3 = 2. Numerator becomes 12(√5 − √3). Step 3: So the first term simplifies to [12(√5 − √3)] / 2 = 6(√5 − √3). Step 4: Rationalise the second fraction: 18/(√5 − √3) × (√5 + √3)/(√5 + √3). Step 5: Denominator becomes (√5)^2 − (√3)^2 = 5 − 3 = 2. Numerator becomes 18(√5 + √3). Step 6: So the second term simplifies to [18(√5 + √3)] / 2 = 9(√5 + √3). Step 7: Add the two simplified expressions: 6(√5 − √3) + 9(√5 + √3). Step 8: Expand and combine: 6√5 − 6√3 + 9√5 + 9√3 = (6√5 + 9√5) + (−6√3 + 9√3) = 15√5 + 3√3. Step 9: Factor out 3: 15√5 + 3√3 = 3(5√5 + √3).

Verification / Alternative check:
We can check correctness by approximating the square roots numerically. Take √5 ≈ 2.236 and √3 ≈ 1.732. Then 12/(√5 + √3) ≈ 12 / (3.968) ≈ 3.025 and 18/(√5 − √3) ≈ 18 / (0.504) ≈ 35.714. Their sum is around 38.739. Evaluate 3(5√5 + √3) with the same approximations: inside the bracket we have 5 × 2.236 + 1.732 ≈ 11.18 + 1.732 = 12.912. Multiply by 3 to get about 38.736, which matches the earlier approximate sum very closely. This confirms that the expression simplifies to 3(5√5 + √3).

Why Other Options Are Wrong:
Options of the form 15(√5 − √3) and 3(5√5 − √3) involve negative coefficients for √3 and would yield much smaller numerical values. Option 15(√5 + √3) has too large a multiplier and gives a value roughly five times the simplified expression. Option 3(3√5 + √3) does not match the exact combination of coefficients obtained. Only 3(5√5 + √3) fits both the algebraic simplification and numerical verification.

Common Pitfalls:
Common mistakes include multiplying only numerator or only denominator by the conjugate, forgetting to use the difference of squares formula correctly, or mismanaging signs when expanding (√5 − √3) and (√5 + √3). Another pitfall is adding the two original fractions by finding a common denominator without first rationalising, which makes the task more complicated. Rationalising each term first and then simplifying is a clean and efficient strategy for surd expressions like this.

Final Answer:
The simplified value of the given surd expression is 3(5√5 + √3).