If x + 1/x ≠ 0 and x^3 + 1/x^3 = 0 for a non zero number x, find the value of the fourth power (x + 1/x)^4.

Introduction / Context:
This question probes algebraic identities involving x + 1/x and x^3 + 1/x^3. Such identities are common in algebra and aptitude exams because they allow high powers to be expressed in terms of simpler expressions. Given a condition on x^3 + 1/x^3, we are asked to find the fourth power of x + 1/x, without solving explicitly for x.

Given Data / Assumptions:

• x is a non zero number so that 1/x is defined.

• x + 1/x ≠ 0.

• x^3 + 1/x^3 = 0.

• We need to evaluate (x + 1/x)^4.

Concept / Approach:
We use the identity for the cube of a sum involving reciprocals: (x + 1/x)^3 = x^3 + 3x + 3/x + 1/x^3 = x^3 + 1/x^3 + 3(x + 1/x). This can be written compactly as (x + 1/x)^3 = (x^3 + 1/x^3) + 3(x + 1/x). Since x^3 + 1/x^3 is given as zero, this identity simplifies dramatically. Once we find x + 1/x, we can compute its fourth power using the relation (x + 1/x)^4 = [(x + 1/x)^2]^2.

Step-by-Step Solution:
Step 1: Use the identity: (x + 1/x)^3 = x^3 + 1/x^3 + 3(x + 1/x). Step 2: Substitute x^3 + 1/x^3 = 0 into the identity to get (x + 1/x)^3 = 0 + 3(x + 1/x) = 3(x + 1/x). Step 3: Let t = x + 1/x. Then the equation becomes t^3 = 3t. Step 4: Rearrange: t^3 − 3t = 0, so t(t^2 − 3) = 0. Step 5: Since x + 1/x ≠ 0, t ≠ 0. Therefore t^2 − 3 = 0 and t^2 = 3. Step 6: We need (x + 1/x)^4, which is t^4. With t^2 = 3, we have t^4 = (t^2)^2 = 3^2 = 9.

Verification / Alternative check:
We can check consistency by working backwards. If t^2 = 3, then t = √3 or t = −√3. In each case t^3 = t × t^2 = t × 3 = 3t, so t^3 − 3t = 0, which matches the identity derived from x^3 + 1/x^3 = 0. Therefore the value t^2 = 3 and t^4 = 9 is consistent with the condition. The exact sign of t is not needed because the fourth power t^4 is the same for both √3 and −√3.

Why Other Options Are Wrong:
Option 12 and 15 would require t^2 to be non integer or would not follow from t^2 = 3. Option 16 corresponds to t^2 = 4, which would make t^3 − 3t equal to 16t − 3t = 13t, not zero unless t = 0, contradicting x + 1/x ≠ 0. Option 3 is equal to t^2 and not t^4. Only 9 matches the correct computation of the fourth power based on the given condition.

Common Pitfalls:
A frequent mistake is to try to solve for x directly from x^3 + 1/x^3 = 0, which is unnecessarily complex and may lead to confusion with complex numbers. Another pitfall is to misapply the identity for (x + 1/x)^3, forgetting the middle term 3(x + 1/x). Some candidates also plug in numerical guesses for x instead of using algebra. Focusing on the symmetric form t = x + 1/x and applying identities in terms of t is a much cleaner method.

Final Answer:
The value of the fourth power (x + 1/x)^4 is 9.