To simplify the expression \( 9^9 \cdot 9^{-6} \), we can use the properties of exponents.
According to the exponent rule \( a^m \cdot a^n = a^{m+n} \), we can combine the exponents:
\[ 9^9 \cdot 9^{-6} = 9^{9 + (-6)} = 9^{9 - 6} = 9^3 \]
Now we need to calculate \( 9^3 \):
\[ 9^3 = 9 \times 9 \times 9 \]
Calculating step-by-step:
\[ 9 \times 9 = 81 \] \[ 81 \times 9 = 729 \]
Thus, the numerical equivalent of \( 9^9 \cdot 9^{-6} \) is \( 729 \).
The answer is \( \boxed{729} \).