Exponent Quotient Rule

The exponent quotient rule (also known as the division property of exponents) states that when you divide two exponential expressions with the same nonzero base, you subtract the exponent in the denominator from the exponent in the numerator. In mathematical terms, for any nonzero number $a$ and any exponents $m$ and $n$:

\[ \frac{a^m}{a^n} = a^{m - n} \]

Where:

$a \neq 0$, $a$= real number

Example

For example, if you have:

\[ \frac{2^5}{2^3} = 2^{5-3} = 2^2 = 4 \]

Important Notes

Same Base: This rule only applies when the base $a$ is the same in both the numerator and the denominator.
Nonzero Base: The base $a$ must be nonzero because division by zero is undefined.
Negative Exponents: If $m < n$, the result will have a negative exponent. For example:

\[ \frac{5^2}{5^4} = 5^{2-4} = 5^{-2} = \frac{1}{5^2} = \frac{1}{25} \]

This rule is very useful for simplifying expressions in algebra that involve powers.

Latex

\frac{a^m}{a^n} = a^{m - n}