Solve for a from an index equation, then evaluate a power: If (1/5)^(3a) = 0.008, find the value of (0.25)^a.
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A20.5
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B22.5
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C0.25
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D6.25
Answer
Correct Answer: 0.25
Explanation
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:
(1/5)^(3a) = 5^(−3a).0.008 = 1/125 = 5^(−3).Therefore −3a = −3 ⇒ a = 1.Compute (0.25)^a = (0.25)^1 = 0.25.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:
0.25