Evaluate a^3 + 1/a^3 from a squared sum: If (a + 1/a)^2 = 3, find the value of a^3 + 1/a^3.

Introduction / Context:Power-sum identities allow computation of higher symmetric expressions from lower ones. Given (a + 1/a)^2, we can find (a + 1/a) and then use the identity for a^3 + 1/a^3 in terms of (a + 1/a). This avoids solving explicitly for a.

Given Data / Assumptions:

• (a + 1/a)^2 = 3
• a ≠ 0

Concept / Approach:Use two identities: (a + 1/a)^2 = a^2 + 2 + 1/a^2 ⇒ a^2 + 1/a^2 = 1. Also, a^3 + 1/a^3 = (a + 1/a)^3 − 3(a + 1/a). Knowing S = a + 1/a from S^2, we can compute the cube expression directly.

Step-by-Step Solution:

Verification / Alternative check:From S^2 = 3 ⇒ a^2 + 1/a^2 = 1. Then (a + 1/a)(a^2 − 1 + 1/a^2) = a^3 + 1/a^3, which also evaluates to 0 when S = ±√3.

Why Other Options Are Wrong:

• 10√3/3, 3√3, 6√3, 9: None align with the identity S^3 − 3S when S^2 = 3.

Common Pitfalls:Assuming S = √3 only; even S = −√3 yields the same final value. Also, confusing a^3 + 1/a^3 with (a + 1/a)^3 directly, forgetting the −3(a + 1/a) term.

Final Answer:0