If C + 1/C = √3 (with C ≠ 0), then find the exact value of: C^3 + 1/C^3 Use algebraic identities to simplify without solving for C explicitly.

Introduction / Context:
This question tests a classic algebraic identity involving powers of (C + 1/C). Many simplification problems avoid finding C directly and instead compute higher-power expressions using identities. The key identity here is: (C + 1/C)^3 = C^3 + 1/C^3 + 3(C + 1/C). Rearranging this gives: C^3 + 1/C^3 = (C + 1/C)^3 − 3(C + 1/C). Because the value of (C + 1/C) is already given as √3, the entire problem becomes straightforward substitution and arithmetic with surds. This is more efficient than trying to solve for C (which would lead to a quadratic equation). These problems are common in aptitude because they test whether you recognize the identity quickly and avoid unnecessary computations. The final result is exact and often turns out to be a neat integer or zero after cancellation.

Given Data / Assumptions:

• C + 1/C = √3 • C ≠ 0 • Required: C^3 + 1/C^3

Concept / Approach:
Use the cube identity: (C + 1/C)^3 = C^3 + 1/C^3 + 3(C + 1/C). Rearrange to isolate C^3 + 1/C^3, then substitute √3 and simplify carefully, paying attention to surd multiplication (√3)^3 = 3√3.

Step-by-Step Solution:
1) Start with the identity: (C + 1/C)^3 = C^3 + 1/C^3 + 3(C + 1/C) 2) Rearrange: C^3 + 1/C^3 = (C + 1/C)^3 − 3(C + 1/C) 3) Substitute C + 1/C = √3: C^3 + 1/C^3 = (√3)^3 − 3(√3) 4) Compute (√3)^3: (√3)^3 = (√3)*(√3)*(√3) = 3√3 5) Subtract: 3√3 − 3√3 = 0

Verification / Alternative check:
A quick consistency check: if C + 1/C = √3, then C satisfies C^2 − √3 C + 1 = 0, meaning C is real (discriminant is positive). Regardless of which root C is, the identity method must give the same value for C^3 + 1/C^3 because it depends only on C + 1/C. Since the computation cancels perfectly, 0 is the only possible exact value.

Why Other Options Are Wrong:
• 3√3 and 6√3: come from forgetting the “− 3(C + 1/C)” part. • √3 or 1/√3: result from incorrect evaluation of (√3)^3. • Any nonzero value contradicts the exact cancellation shown by the identity.

Common Pitfalls:
• Using the wrong identity (mixing with (C − 1/C)^3). • Computing (√3)^3 as √27 but not simplifying correctly to 3√3. • Dropping the factor 3 in the identity term 3(C + 1/C).

Final Answer:
0