In algebra with powers and reciprocals, suppose x is a real number greater than 1 such that x^4 + 1/x^4 = 34. Using this condition, find the exact value of the expression x^3 − 1/x^3.

Introduction / Context:
This question tests manipulation of expressions involving powers of a variable and its reciprocal. Instead of solving for x directly, you can use algebraic identities that connect x^4 + 1/x^4 with x^2 + 1/x^2 and x − 1/x. These kinds of problems are frequently asked in aptitude tests because they reward a structured approach and knowledge of standard identities over blind computation.

Given Data / Assumptions:
- x is a real number with x greater than 1, so x and 1/x are both positive, and x − 1/x is positive.
- We are given x^4 + 1/x^4 = 34.
- We must determine x^3 − 1/x^3 exactly.
- Only algebraic identities and logical reasoning are needed, no numerical approximation of x itself is required.

Concept / Approach:
First, recall that (x^2 + 1/x^2)^2 = x^4 + 2 + 1/x^4. This identity allows us to compute x^2 + 1/x^2 from the given sum of fourth powers. Next, use the identity (x − 1/x)^2 = x^2 + 1/x^2 − 2 to find x − 1/x. Finally, use the identity x^3 − 1/x^3 = (x − 1/x)(x^2 + 1 + 1/x^2) to get the desired value. The condition x greater than 1 helps us choose the correct sign when taking square roots.

Step-by-Step Solution:
1) Start with the identity (x^2 + 1/x^2)^2 = x^4 + 2 + 1/x^4. 2) Substitute x^4 + 1/x^4 = 34 to get (x^2 + 1/x^2)^2 = 34 + 2 = 36. 3) Therefore x^2 + 1/x^2 = ±6. But x^2 + 1/x^2 is always positive, so x^2 + 1/x^2 = 6. 4) Use the identity (x − 1/x)^2 = x^2 + 1/x^2 − 2, which gives (x − 1/x)^2 = 6 − 2 = 4. 5) Hence x − 1/x = ±2. Since x is greater than 1, x − 1/x is positive, so x − 1/x = 2. 6) Use the identity x^3 − 1/x^3 = (x − 1/x)(x^2 + 1 + 1/x^2). 7) Substitute x − 1/x = 2 and x^2 + 1/x^2 = 6, giving x^3 − 1/x^3 = 2 · (6 + 1) = 2 · 7 = 14.

Verification / Alternative check:
We can check consistency by working backwards. Suppose x − 1/x = 2. Then x^2 − 2 + 1/x^2 = 4, so x^2 + 1/x^2 = 6. Squaring this gives (x^2 + 1/x^2)^2 = 36, which equals x^4 + 2 + 1/x^4, so x^4 + 1/x^4 = 34, matching the given condition. This confirms that the chain of identities is coherent and that x^3 − 1/x^3 = 14 is correct.

Why Other Options Are Wrong:
Option A (6) is the value of x^2 + 1/x^2, not x^3 − 1/x^3. Option C (8) and Option E (−14) arise from mis using signs or forgetting that x − 1/x must be positive when x greater than 1. Option D (0) would mean x^3 = 1/x^3, which implies x^6 = 1, not consistent with the given x^4 + 1/x^4 = 34. Only Option B matches the correctly derived value.

Common Pitfalls:
Learners sometimes try to solve for x directly using a high degree equation, which is unnecessary and much harder. Another common mistake is forgetting to add the middle term 2 when using the identity for x^4 + 1/x^4, or mishandling signs when taking square roots. Writing the relevant identities separately and substituting carefully step by step is the safest way to handle such expressions.

Final Answer:
Using identities for powers of x and its reciprocal, the value of x^3 − 1/x^3 is 14.