Find the fraction from a cube-operation description A fraction f is multiplied by itself and then divided by its reciprocal, giving 18 26/27. Determine the fraction.

Introduction / Context:
This verbal description encodes an algebraic operation on a fraction f. “Multiplied by itself” is f^2, and “divided by its reciprocal” 1/f yields f^2 ÷ (1/f) = f^3. The result is given as a mixed number, which we convert to an improper fraction to solve for f.

Given Data / Assumptions:

• Operation result: 18 26/27.
• Operation on f: f^2 ÷ (1/f) = f^3.
• Find f in simplest form.

Concept / Approach:
Since f^3 equals the given rational, take the rational cube root. Recognize perfect cubes in numerator and denominator for a clean result.

Step-by-Step Solution:
Convert mixed number: 18 26/27 = (18*27 + 26)/27 = (486 + 26)/27 = 512/27.Set f^3 = 512/27.Note 512 = 8^3 and 27 = 3^3.Therefore f = cube_root(512/27) = 8/3.

Verification / Alternative check:
Compute (8/3)^3 = 512/27, which equals 18 26/27, confirming the result.

Why Other Options Are Wrong:
8/27 is far too small; 11/3, 22/3, 32/3 produce cubes much larger than 512/27.

Common Pitfalls:
Misinterpreting “divided by the reciprocal” as division by f (which would give f); converting the mixed number incorrectly.

Final Answer:
8/3