Evaluate √(0.289 / 0.00121) and express the result as a simplified fraction.

Introduction / Context:
This problem uses exact decimal-to-fraction conversion before taking a square root. Spotting perfect squares in the numerators and denominators greatly streamlines the solution.

Given Data / Assumptions:

• Compute S = √(0.289 / 0.00121).
• All numbers are positive.

Concept / Approach:
Rewrite decimals as fractions with powers of 10 and then simplify using known squares. In particular, 0.289 = 289/1000 and 0.00121 = 121/100000.

Step-by-Step Solution:
0.289 = 289/1000, 0.00121 = 121/100000.Ratio = (289/1000) / (121/100000) = (289 × 100000) / (121 × 1000) = 28900 / 121.Observe: 289 = 17^2 and 121 = 11^2, so 28900/121 = ( (17^2) × 100 ) / (11^2) = ( (170)^2 ) / (11^2 ).Therefore, √(28900/121) = 170/11.

Verification / Alternative check:
Compute decimal approximation: 170/11 ≈ 15.4545..., consistent with √(0.289 / 0.00121) since the ratio is about 238.842..., and the square root of a number near 238.8 is about 15.45.

Why Other Options Are Wrong:
17/110 and 17/1100: Invert the scaling; results are far too small.17/11: Misses the factor of 10 inside the square root.10/11: Unrelated to the derived square-root structure.

Common Pitfalls:
Mistaking 0.289 for 0.0289; misplacing decimal powers when converting to fractions; taking the square root before simplifying to perfect squares.

Final Answer:
170 / 11