To simplify \( 9^9 \cdot 9^{-6} \), we can use the property of exponents that states \( a^m \cdot a^n = a^{m+n} \).
Applying this property:
\[ 9^9 \cdot 9^{-6} = 9^{9 + (-6)} = 9^{9 - 6} = 9^3 \]
Next, we need to calculate \( 9^3 \):
\[ 9^3 = 9 \cdot 9 \cdot 9 \]
Calculating step-by-step:
\[ 9 \cdot 9 = 81 \]
Now multiplying by 9 again:
\[ 81 \cdot 9 = 729 \]
Therefore, the numerical equivalent of \( 9^9 \cdot 9^{-6} \) is
\[ \boxed{729} \]