Given a^2 + 1/a^2 = 17/4, determine the exact value of a^3 − 1/a^3. Assume real a for principal values.

Introduction / Context:This problem explores classic symmetric identities with a and 1/a. Starting from a^2 + 1/a^2, you can recover both a + 1/a and a − 1/a and then compute higher-order expressions such as a^3 − 1/a^3 using factorization formulas.

Given Data / Assumptions:

• a^2 + 1/a^2 = 17/4.
• a is real and non-zero (so reciprocals exist).
• We want a^3 − 1/a^3.

Concept / Approach:Use (a ± 1/a)^2 relations: a^2 + 1/a^2 = (a + 1/a)^2 − 2 = (a − 1/a)^2 + 2. Then use a^3 − 1/a^3 = (a − 1/a)(a^2 + 1/a^2 + 1). Choose the principal positive root for a − 1/a to obtain the standard positive value.

Step-by-Step Solution:

Verification / Alternative check:If a − 1/a is taken negative, the magnitude remains 63/8 but the sign flips. Most exam conventions assume the positive branch unless specified otherwise.

Why Other Options Are Wrong:75/16 and 95/8 come from misusing the identities or adding 1 incorrectly. “None of these” is incorrect because 63/8 is attainable directly.

Common Pitfalls:Confusing a^3 − 1/a^3 with a^3 + 1/a^3, and misapplying the ± when taking square roots. Always plug back to confirm.

Final Answer:63/8