To find the numerical equivalent of \( 9^9 \cdot 9^{-6} \), we can use the properties of exponents. Specifically, we can combine the exponents since they have the same base:
\[ 9^9 \cdot 9^{-6} = 9^{9 + (-6)} = 9^{9 - 6} = 9^3 \]
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:
\[ 81 \cdot 9 = 729 \]
Thus, the numerical equivalent of \( 9^9 \cdot 9^{-6} \) is
\[ \boxed{729} \]
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