If x = 3^(1/3) − 3^(−1/3), where 3^(1/3) denotes the real cube root of 3, then what is the exact value of the expression 3x³ + 9x?

Introduction / Context:
This question involves simplifying an expression built from cube roots of 3. It checks your ability to use algebraic identities with terms like t − 1/t and then substitute back into a polynomial expression, here 3x³ + 9x. Such manipulations are common in aptitude and algebra questions involving surds and roots.

Given Data / Assumptions:

• x = 3^(1/3) − 3^(−1/3).
• All roots are real; let 3^(1/3) be the real cube root of 3.
• The expression to evaluate is 3x³ + 9x.
• Basic algebraic expansion rules can be applied safely.

Concept / Approach:
The key technique is to introduce a simpler symbol for the cube root of 3. Let t = 3^(1/3). Then 3^(−1/3) = 1/t. This allows us to rewrite x as t − 1/t. We then expand (t − 1/t)³ carefully, group like terms, and express x³ in terms of x itself. Finally, we compute 3x³ + 9x using this relation, which removes the surds completely and leaves a rational number.

Step-by-Step Solution:
Let t = 3^(1/3). Then 3^(−1/3) = 1/t. So x = t − 1/t. Compute x³ = (t − 1/t)³. Expand: (t − 1/t)³ = t³ − 3t(1/t)(t − 1/t) − 1/t³. More directly: (t − 1/t)³ = t³ − 3t + 3(1/t) − 1/t³. But t³ = 3 and 1/t³ = 1/3. So x³ = 3 − 3t + 3/t − 1/3. Combine 3 − 1/3 = 8/3. Notice that t − 1/t = x, so −3t + 3/t = −3(t − 1/t) = −3x. Thus x³ = 8/3 − 3x. Multiply by 3: 3x³ = 8 − 9x. Therefore 3x³ + 9x = 8 − 9x + 9x = 8.

Verification / Alternative check:
As a rough numerical check, approximate t ≈ 3^(1/3) ≈ 1.442. Then 1/t ≈ 0.693, so x ≈ 1.442 − 0.693 ≈ 0.749. Compute 3x³ + 9x ≈ 3(0.749³) + 9(0.749). This is approximately 3(0.42) + 6.741 ≈ 1.26 + 6.74 ≈ 8.00, which confirms the exact result of 8.

Why Other Options Are Wrong:
9, 16 and 27 are common distractors that might arise from incorrectly cubing 3^(1/3) or from ignoring the 3^(−1/3) term. Zero would require 3x³ and 9x to cancel exactly, which does not happen given the algebraic relation derived. Only 8 is consistent with both algebraic derivation and numerical verification.

Common Pitfalls:
Typical mistakes include expanding (t − 1/t)³ incorrectly, forgetting that 3^(−1/3) = 1/t, or trying to compute everything numerically without recognizing the pattern t − 1/t. Another error is to treat t³ as 1 instead of 3. Always expand step by step and simplify using the definitions of t and 1/t carefully.

Final Answer:
Hence, the exact value of 3x³ + 9x is 8.