Evaluate the fractional power precisely: Compute (0.00032)^(2/5) and express your answer as a simple fraction.

Introduction / Context:
This question checks fluency with fractional exponents and scientific notation. Rewriting decimals as powers of 10 and small integers as powers of 2 helps simplify the evaluation of (0.00032)^(2/5).

Given Data / Assumptions:

• Decimal: 0.00032
• Exponent: 2/5
• Answer required as a reduced fraction.

Concept / Approach:
Express 0.00032 as a ratio using prime bases. Note 0.00032 = 32/100000 with 32 = 2^5 and 100000 = 10^5. Then apply (a/b)^(2/5) = a^(2/5) / b^(2/5) and use exponent rules to collapse powers cleanly.

Step-by-Step Solution:
0.00032 = 32 / 100000 = 2^5 / 10^5(0.00032)^(2/5) = (2^5 / 10^5)^(2/5)= 2^(5*(2/5)) / 10^(5*(2/5))= 2^2 / 10^2 = 4 / 100 = 1/25

Verification / Alternative check:
Compute the fifth root first: (0.00032)^(1/5) = 0.2 (since 0.2^5 = 0.00032), then square: 0.2^2 = 0.04 = 1/25.

Why Other Options Are Wrong:
1/125 and 1/225 arise from misapplying the root or squaring steps; 1/625 would be (1/5)^4, not applicable here; 1/20 is a decimal rounding error.

Common Pitfalls:
Treating 0.00032 as 3.2×10^-4 instead of 3.2×10^-4 is fine, but ensure consistent exponent distribution; the prime-power route avoids slips.

Final Answer:
1/25