1. **Problem:** Simplify $x^{12} \cdot x^3$. 2. **Formula:** When multiplying powers with the same base, add the exponents: $$a^m \cdot a^n = a^{m+n}$$ 3. **Work:** $$x^{12} \cdot x^3 = x^{12+3} = x^{15}$$ 4. **Answer:** The simplified expression is $x^{15}$. This corresponds to option A.