We can simplify the expression \( 9^9 \cdot 9^{-6} \) using the properties of exponents.
According to the property of exponents, when we multiply two expressions with the same base, we add their exponents:
\[ 9^9 \cdot 9^{-6} = 9^{9 + (-6)} = 9^{9 - 6} = 9^3 \]
Now, we need to calculate \( 9^3 \):
\[ 9^3 = 9 \cdot 9 \cdot 9 \]
Calculating step-by-step:
\[ 9 \cdot 9 = 81 \] \[ 81 \cdot 9 = 729 \]
Thus, the numerical equivalent of \( 9^9 \cdot 9^{-6} \) is
\[ \boxed{729}. \]