Evaluate the expression using index laws and real-number arithmetic: [(11)^3 × (6)^2] ÷ (4)^3 = ?

Introduction / Context:
This problem tests basic laws of indices and order of operations. We must carefully compute powers first, then multiply and divide. Converting to exact integers as long as possible helps prevent rounding errors, and only at the end should we convert to decimals if needed.

Given Data / Assumptions:

• Compute (11)^3, (6)^2, and (4)^3.
• Multiply 11^3 and 6^2, then divide by 4^3.
• All quantities are real and positive.

Concept / Approach:
Use the exponent definitions a^m * a^n = a^(m+n) (when bases match) and evaluate each power separately when bases differ. Respect BODMAS/PEMDAS: powers → multiplication → division → addition/subtraction.

Step-by-Step Solution:

Verification / Alternative check:
Break 36 as 64 * (36/64) = 64 * 0.5625; then (1331 * 36)/64 = 1331 * 0.5625 = 748.6875, confirming the previous result without long division.

Why Other Options Are Wrong:

• 2994.75 and 4492.125: These are results of multiplying correctly but forgetting to divide by 64 or dividing by a different number.
• 272.25: Too small; suggests dividing by an extra factor or miscomputing 11^3.

Common Pitfalls:
Typical mistakes include computing 11^3 as 121 or 133 instead of 1331, or dividing by 4 rather than 4^3. Always compute exponents before other operations.

Final Answer: