Evaluate the expression carefully using index rules and cube roots: (42 × 229) ÷ (9261)^(1/3) = ?

Introduction / Context:
This problem checks your comfort with order of operations and perfect cubes. Recognizing that 9261 is a perfect cube allows a short and clean computation without a calculator. Once the cube root is simplified, the expression reduces to a straightforward integer division after multiplying two integers.

Given Data / Assumptions:

• Expression: (42 × 229) ÷ (9261)^(1/3).
• Standard order of operations: evaluate the root first, then perform multiplication and division left to right.
• All numbers are integers; 9261 is suspected to be a perfect cube.

Concept / Approach:
Use the identity that if N = a^3, then N^(1/3) = a. Here, 21^3 = 9261. So (9261)^(1/3) = 21. Then multiply 42 by 229 and divide the product by 21. Because 42 is divisible by 21, we can simplify before multiplying to avoid large intermediate numbers.

Step-by-Step Solution:

Verification / Alternative check:
Multiply first then divide: 42 × 229 = 9618; 9618 ÷ 21 = 458. Both paths agree, confirming the result.

Why Other Options Are Wrong:
448, 452, 456, and 462 are near the true value but occur if you mis-evaluate the cube root or make a small multiplication/division slip.

Common Pitfalls:
Treating 9261^(1/3) as 31 or 19 by guesswork, or dividing 229 by 21 instead of simplifying 42 ÷ 21 first. Always simplify where possible to reduce arithmetic errors.

Final Answer:
458