Fractional exponents disguised: A fraction x is multiplied by itself and the product is then divided by its reciprocal. The resulting value is 18 26/27. Determine the original fraction x.

Introduction / Context:
Although the description seems long, it simply asks for a cube root in fractional form. Multiplying a fraction by itself gives x^2, and dividing that by the reciprocal (1/x) gives x^3. The given mixed fraction must be converted to an improper fraction before taking the cube root.

Given Data / Assumptions:

• Operation described: (x * x) / (1/x) = x^3.
• Result equals 18 26/27.
• We must find x as a positive rational in simplest terms.

Concept / Approach:
Convert 18 26/27 to an improper fraction, then equate x^3 to that value. Finally, take the cube root by recognizing perfect cubes in numerator and denominator.

Step-by-Step Solution:
Convert 18 26/27: 18 + 26/27 = (18*27 + 26)/27 = (486 + 26)/27 = 512/27.Thus x^3 = 512/27.Notice 512 = 8^3 and 27 = 3^3.Therefore x = (8/3).

Verification / Alternative check:
Compute x^3 with x = 8/3: (8/3)^3 = 512/27, which matches the given result exactly.

Why Other Options Are Wrong:
8/27 is the reciprocal cube, not the cube root; 22/3 and 11/3 are much larger than 8/3; thus none of the provided options matches the correct 8/3.

Common Pitfalls:
Forgetting that dividing by a reciprocal multiplies by the original fraction; misreading the mixed number; not recognizing perfect cubes.

Final Answer:
None of these (the correct fraction is 8/3).