1. **State the problem:** Simplify the expressions and determine which can be simplified to $a^6$. 2. **Recall the laws of exponents:** - Multiplying powers with the same base: $a^m \times a^n = a^{m+n}$ - Dividing powers with the same base: $\frac{a^m}{a^n} = a^{m-n}$ - Adding or subtracting powers with the same base does NOT simplify exponents: $a^m + a^n$ or $a^m - a^n$ cannot be combined by adding/subtracting exponents. 3. **Analyze each expression:** I. $a^3 \times a^2 = a^{3+2} = a^5$ (not $a^6$) II. $a^4 + a^2$ cannot be simplified to a single power of $a$. III. $\frac{a^{12}}{a^2} = a^{12-2} = a^{10}$ (not $a^6$) IV. $a^8 - a^2$ cannot be simplified to a single power of $a$. 4. **Conclusion:** None of the expressions simplify to $a^6$. **Final answer:** D. none of I, II, III, and IV