Compute a decimal expression with exponents and a linear denominator: Evaluate (0.5^3 × 0.6^3) / (0.5 × 0.5 − 0.3 + 0.6 × 0.6) accurately, simplifying powers first.
Aptitude
Decimal Fraction
Difficulty: Medium
Choose an option
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A0.0871
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B0.3
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C0.1
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D0.61
Answer
Correct Answer: 0.0871
Explanation
Introduction / Context:This item blends small decimal powers with a simple quadratic-like denominator. Calculating the powers first, then simplifying the denominator, and finally performing the division keeps errors in check. It is a good test of decimal place-value discipline and exponent handling.
Given Data / Assumptions:
- Numerator: 0.5^3 × 0.6^3.
- Denominator: 0.5 × 0.5 − 0.3 + 0.6 × 0.6.
- No rounding is needed until the final step.
Concept / Approach:Compute 0.5^3 and 0.6^3 exactly, multiply for the numerator, then evaluate each term in the denominator and combine. The final division can be expressed as a fraction to keep track of place values, then converted to a decimal with appropriate precision.
Step-by-Step Solution:
0.5^3 = 0.125; 0.6^3 = 0.216.Numerator = 0.125 × 0.216 = 0.027.Denominator = (0.5 × 0.5) − 0.3 + (0.6 × 0.6) = 0.25 − 0.3 + 0.36 = 0.31.Quotient = 0.027 / 0.31 ≈ 0.08709677…Rounded to four significant figures (or to four decimal places), this is ≈ 0.0871.Verification / Alternative check:
As a fraction, 0.027/0.31 = 27/3100 ≈ 0.008709677? Note the misplaced zero; correct decimal is 0.08709677… (31 × 0.0871 ≈ 2.7001).Why Other Options Are Wrong:
- 0.3 and 0.61: Far too large relative to the small numerator 0.027.
- 0.1: Close but still an overestimate; reflects rounding 0.027/0.27 by mistake.
Common Pitfalls:
- Dropping a zero in the decimal when dividing by 0.31.
- Miscomputing 0.6^3 (some mistakenly take 0.6^2 = 0.36 and stop).
Final Answer:
0.0871