For non zero real x, evaluate the expression (x^(1/3) + x^(−1/3)) / (x^(2/3) − 1 + x^(−2/3)) and choose the equivalent simplified form from the options.

Introduction / Context:
This question involves exponents with fractional powers and requires algebraic simplification. The goal is to express a complicated rational expression in a more compact form using a cleverly chosen substitution and standard identities.

Given Data / Assumptions:

• x is a non zero real number.
• The expression is (x^(1/3) + x^(−1/3)) / (x^(2/3) − 1 + x^(−2/3)).
• We must select an equivalent simplified form from the options.

Concept / Approach:
We introduce t = x^(1/3) to rewrite all powers in simpler terms. Then x^(−1/3) becomes 1/t, x^(2/3) becomes t^2, and x^(−2/3) becomes 1/t^2. The denominator then becomes t^2 − 1 + 1/t^2, which we can recognize in terms of (t + 1/t)^2. This leads to a simplification completely in terms of t + 1/t, which maps back to the original expression x^(1/3) + x^(−1/3).

Step-by-Step Solution:
Step 1: Let t = x^(1/3). Then x^(−1/3) = 1/t, x^(2/3) = t^2, and x^(−2/3) = 1/t^2. Step 2: Rewrite the numerator as t + 1/t. Step 3: Rewrite the denominator as t^2 − 1 + 1/t^2. Step 4: Observe that t^2 + 1/t^2 = (t + 1/t)^2 − 2. Step 5: Therefore t^2 − 1 + 1/t^2 = (t^2 + 1/t^2) − 1 = (t + 1/t)^2 − 2 − 1 = (t + 1/t)^2 − 3. Step 6: The entire expression becomes (t + 1/t) / ((t + 1/t)^2 − 3). Step 7: Now replace t back with x^(1/3) to obtain (x^(1/3) + x^(−1/3)) / ((x^(1/3) + x^(−1/3))^2 − 3).

Verification / Alternative check:
We can test with a simple value like x = 8, so x^(1/3) = 2 and x^(−1/3) = 1/2. The original expression and the simplified form both give the same numerical value when computed carefully, confirming that the algebraic manipulation is correct and that there is no loss of generality for non zero x.

Why Other Options Are Wrong:
Option B is simply the reciprocal of the numerator and ignores the denominator structure. Option C equals the numerator only and does not account for the denominator. Option D equals the denominator alone. Option E gives the inverse of the correct expression, flipping numerator and denominator, which does not match the original fraction.

Common Pitfalls:
A major pitfall is to treat x^(2/3) − 1 + x^(−2/3) as something unrelated to (x^(1/3) + x^(−1/3))^2 and miss the pattern. Students might also make errors in manipulating fractional exponents or forget to subtract 3 instead of 2 when rewriting the denominator. Using a systematic substitution greatly reduces the chance of such mistakes.

Final Answer:

The expression is equal to (x^(1/3) + x^(−1/3)) / ((x^(1/3) + x^(−1/3))^2 − 3).