Question

which expression is equivalent to $9^{4} \cdot 9^{-6}$?
options: $\frac{1}{9^{-24}}$, $\frac{1}{9^{-2}}$, $\frac{1}{9^{2}}$, $\frac{1}{9^{24}}$

Explanation:

Step1: Apply exponent product rule

When multiplying terms with the same base, add exponents: $a^m \cdot a^n = a^{m+n}$.
$9^4 \cdot 9^{-6} = 9^{4 + (-6)} = 9^{-2}$

Step2: Rewrite negative exponent

A term with negative exponent becomes reciprocal with positive exponent: $a^{-n} = \frac{1}{a^n}$.
$9^{-2} = \frac{1}{9^2}$

Answer:

$\frac{1}{9^2}$ (third option)