The quotient rule of exponents states that when you divide two exponential expressions with the same base, you subtract the exponents. Mathematically, it's expressed as:
\[ \frac{a^m}{a^n} = a^{m-n} \]
where \( a \) is the base (and \( a \neq 0 \)), \( m \) and \( n \) are the exponents.
Quick Check:
- Same Base: Ensure that both the numerator and denominator have the same base.
- Subtract Exponents: Subtract the exponent in the denominator from the exponent in the numerator.
Example:
\[ \frac{x^5}{x^2} = x^{5-2} = x^3 \]
Special Cases:
- If \( n = m \), then \(\frac{a^m}{a^n} = a^{m-n} = a^0 = 1\).
- If \( m < n \), then the result will be a fraction: \(\frac{a^m}{a^n} = \frac{1}{a^{n-m}}\).
This rule allows for simplification of expressions involving exponents.