1. The problem: Understanding what to do when applying the product rule and the bases are not the same. 2. The product rule in algebra for exponents states: $$a^m \times a^n = a^{m+n}$$ where the bases $a$ are the same. 3. Important rule: You can only add the exponents when the bases are identical. 4. If the bases are different, for example, $a^m \times b^n$ where $a \neq b$, you cannot add the exponents. 5. In this case, the expression remains as a product: $$a^m \times b^n$$ and cannot be simplified further by combining exponents. 6. To simplify expressions with different bases, you may need to factor, use logarithms, or evaluate numerically if values are known. 7. Summary: When bases differ, do not add exponents; keep the product as is or use other algebraic methods depending on the problem context.