Solve for a from an index equation, then evaluate a power: If (1/5)^(3a) = 0.008, find the value of (0.25)^a.

Introduction / Context:This question links two exponential expressions through the same unknown exponent a. Recognizing decimal-to-fraction conversions (like 0.008) and simple base conversions makes the computation straightforward.Given Data / Assumptions:

• (1/5)^(3a) = 0.008.
• Compute (0.25)^a after finding a.

Concept / Approach:Convert 0.008 to a power of 5: 0.008 = 8/1000 = 1/125 = 5^(−3). Then equate exponents. Next, compute (0.25)^a by using the found a value and the identity 0.25 = 1/4.Step-by-Step Solution:

Verification / Alternative check:Note 0.25 = 1/4 = 2^(−2). With a = 1, (0.25)^a = 2^(−2) = 0.25, confirming the result independently of decimals.

Why Other Options Are Wrong:

• 6.25, 20.5, 22.5: These correspond to unrelated computations and do not follow from a = 1.

Common Pitfalls:Forgetting that (1/5)^k = 5^(−k), or mis-converting 0.008. Keeping everything as exact fractions prevents rounding mistakes.

Final Answer: