Multipying & Dividing Rules

How does an expression like simplify? Notice that both terms have the same base. When there is a quotient of two terms with the same base to a power, the powers can be subtracted.

Quotient of a Power Rule:

Why does this rule work? Using the expression above and writing out the expansion of each term creates the following steps:

In other words, when dividing expressions with the same base keep the base and subtract the exponent in the denominator from the exponent in the numerator. This rule is a nice "shortcut" for the division of terms, but it only applies to exponents with the same base. Therefore, it will not work for , because the bases are not the same.