Use exponent rules to condense powers of 3: Evaluate 27^3 × 3^4 ÷ 3^10 and choose the correct value.

Introduction / Context: This question tests consolidation of exponents using the same base. The key is to rewrite 27 as a power of 3 and then apply the rules for multiplying and dividing powers with the same base: add exponents when multiplying and subtract when dividing.

Given Data / Assumptions:

• Expression: 27^3 × 3^4 ÷ 3^10.
• Recall: 27 = 3^3.
• Exponent rules: a^m * a^n = a^(m+n), and a^m / a^n = a^(m−n).

Concept / Approach: Express all terms as powers of 3. Then use exponent laws to combine them into a single power of 3. Finally, evaluate that power or identify the corresponding integer.

Step-by-Step Solution:

Verification / Alternative check: You can also pair 3^4 ÷ 3^10 = 3^(−6) and then multiply 27^3 * 3^(−6) = 3^9 * 3^(−6) = 3^3 = 27.

Why Other Options Are Wrong: 9 and 81 are 3^2 and 3^4 respectively—off by exponent arithmetic; 1/27 corresponds to 3^(−3), which would arise from reversing the subtraction; 3 would require exponent 1.

Common Pitfalls: Forgetting to convert 27 into 3^3, or subtracting exponents in the wrong order when dividing.

Final Answer: 27