Which condition is necessary to apply the rules for multiplying and dividing exponents?

The rules for multiplying and dividing exponents are applicable only when the bases are the same. When you multiply terms with the same base, you add the exponents, while when you divide, you subtract the exponents. This means that having the same base is essential for performing these operations accurately. For example, in multiplying (a^m \times a^n), where both terms have the same base (a), you would combine them as (a^{m+n}). Similarly, in division (a^m \div a^n), it simplifies to (a^{m-n}).

If the bases were different, the standard rules for exponents would not apply, and you would not be able to simplify the expression using those exponent laws. The conditions related to the exponents being negative or whole numbers do not affect the basic application of exponent multiplication or division rules, making the necessity of having the same base paramount.