1. **State the problem:** Simplify the expression $a^4 \cdot a^3$. 2. **Recall the rule for multiplying powers with the same base:** When multiplying powers with the same base, add the exponents: $$a^m \cdot a^n = a^{m+n}$$ 3. **Apply the rule:** $$a^4 \cdot a^3 = a^{4+3}$$ 4. **Simplify the exponent:** $$a^{7}$$ 5. **Final answer:** The simplified form of $a^4 \cdot a^3$ is $$a^7$$.