Combine like bases (indices law): Evaluate a^5 × a^7 and express the result as a single power of a.

Introduction / Context:This is a direct application of the index law a^m * a^n = a^(m+n). We combine exponents with the same base by simple addition.

Given Data / Assumptions:

• a^5 × a^7
• a ≠ 0 (usual nonzero base assumption)

Concept / Approach:When multiplying powers with the same base, add exponents: m + n. No further simplification is needed beyond adding 5 and 7.

Step-by-Step Solution:a^5 × a^7 = a^(5 + 7) = a^12

Verification / Alternative check:For a = 2: 2^5 × 2^7 = 32 × 128 = 4096 = 2^12, as expected.

Why Other Options Are Wrong:a^35 multiplies exponents incorrectly; a^2 subtracts instead of adds; a^(5/7) is unrelated; a^10 is adding wrongly.

Common Pitfalls:Multiplying exponents (5*7) instead of adding—only (a^m)^n multiplies exponents.

Final Answer:a^12