If a + 1/a = √3 (a ≠ 0), evaluate the expression a^6 − 1/a^6 + 2 exactly, using symmetry and angle-based reasoning.

Introduction / Context:
This problem highlights the power of treating expressions of the form a + 1/a as “trigonometric-like.” When a + 1/a equals a constant close to 2, it often corresponds to 2*cos(θ). That perspective can make high powers surprisingly simple due to periodicity of angles.

Given Data / Assumptions:

• a + 1/a = √3, with a ≠ 0 (complex values allowed).
• Compute a^6 − 1/a^6 + 2.

Concept / Approach:
Set a + 1/a = 2*cos(θ). Here √3 = 2*cos(θ) ⇒ cos(θ) = √3/2 ⇒ θ = 30 degrees. Then a can be represented (up to choice of sign in the quadratic roots) as e^{iθ}. Powers like a^6 then map to e^{i6θ}, which is easy to evaluate. Finally compute the target difference and add 2.

Step-by-Step Solution:
a + 1/a = √3 ⇒ a satisfies a^2 − √3 a + 1 = 0 with roots e^{±iπ/6}.Hence a = e^{iπ/6} (or e^{−iπ/6}).Compute a^6 = e^{iπ} = −1, and 1/a^6 = e^{−iπ} = −1.Therefore a^6 − 1/a^6 = (−1) − (−1) = 0.Finally, a^6 − 1/a^6 + 2 = 0 + 2 = 2.

Verification / Alternative check:
Algebraic route: Let S1 = a + 1/a = √3. Then S2 = S1^2 − 2 = 1, S3 = S1*S2 − S1 = 0. Using the pattern for differences: a^6 − 1/a^6 = (a^3 − 1/a^3)(a^3 + 1/a^3) = (a^3 − 1/a^3)*0 = 0, leading again to +2 overall.

Why Other Options Are Wrong:

• 3√3, 5, 1, 0: These typically arise from confusing sums and differences of powers or assuming a^6 = 1 instead of −1.

Common Pitfalls:
Forgetting that √3 = 2*cos(30°); mixing up a^6 + 1/a^6 with a^6 − 1/a^6; sign mistakes when interpreting e^{±iπ}.

Final Answer:
2