Evaluate negative fractional exponents and add: Compute (16)^(−3/4) + 2^(−3) + (8)^(−2/3).
Aptitude
Surds and Indices
Difficulty: Easy
Choose an option
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A1/2
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B3/2
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C9/2
-
D5/4
Answer
Correct Answer: 1/2
Explanation
Introduction / Context:This tests comfort with fractional and negative exponents. Using base-2 representations makes evaluation quick and exact without calculators.Given Data / Assumptions:
- (16)^(−3/4), 2^(−3), (8)^(−2/3).
Concept / Approach:Write 16 = 2^4 and 8 = 2^3, then apply (a^m)^n = a^(mn). Negative exponents give reciprocals: a^(−k) = 1/a^k.Step-by-Step Solution:
(16)^(−3/4) = (2^4)^(−3/4) = 2^(−3) = 1/8.2^(−3) = 1/8.(8)^(−2/3) = (2^3)^(−2/3) = 2^(−2) = 1/4.Sum = 1/8 + 1/8 + 1/4 = 1/4 + 1/4 = 1/2.Verification / Alternative check:Compute as decimals: 0.125 + 0.125 + 0.25 = 0.5, which is 1/2.
Why Other Options Are Wrong:
- 3/2, 9/2, 5/4: Each implies at least one exponent was evaluated with the wrong sign or power.
Common Pitfalls:Confusing (a^m)^n with a^(m/n), or forgetting that negative exponents produce reciprocals.
Final Answer:
1/2