Find the cube root: Determine the integer n such that n^3 = 103823.

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:

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: