Find the cube root: Determine the integer n such that n^3 = 103823.
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A49
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B51
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C45
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D47
Answer
Correct Answer: 47
Explanation
Introduction / Context:Here you must recognize 103823 as a perfect cube and identify its cube root. This is a standard indices skill used in number theory and quick arithmetic estimation.Given Data / Assumptions:
- We need n such that n^3 = 103823.
- Options are nearby integers (45–51).
Concept / Approach:Estimate by bracketing with known cubes. Compute 46^3 and 47^3 (or 48^3) mentally or via incremental methods to see which equals 103823 exactly.Step-by-Step Solution:
46^3 = 46 * 46 * 46 = 2116 * 46 = 97,336 (not enough).47^3 = 47 * 47 * 47 = 2209 * 47 = 103,823 (exact match).Therefore, ∛103823 = 47.Verification / Alternative check:Use (n+1)^3 − n^3 = 3n^2 + 3n + 1. From 46^3 = 97,336, add 3*46^2 + 3*46 + 1 = 3*2116 + 138 + 1 = 635 *? Wait carefully: 3*2116=6348; 6348+138+1=6487; 97336+6487=103,823, confirming 47^3.
Why Other Options Are Wrong:
- 45, 49, 51: Their cubes do not equal 103,823 (they are significantly lower or higher).
Common Pitfalls:Mixing up 47^2 (2209) with 47^3, or estimating too loosely without checking parity and growth between consecutive cubes.
Final Answer:
47