To calculate \( 99 \cdot 9^{-6} \), we first need to rewrite \( 9^{-6} \).
Using the definition of negative exponents, we have:
\[ 9^{-6} = \frac{1}{9^6} \]
Calculating \( 9^6 \):
\[ 9^6 = (3^2)^6 = 3^{12} \]
Calculating \( 3^{12} \):
- First, calculate smaller powers:
- \( 3^1 = 3 \)
- \( 3^2 = 9 \)
- \( 3^3 = 27 \)
- \( 3^4 = 81 \)
- \( 3^5 = 243 \)
- \( 3^6 = 729 \)
- \( 3^7 = 2187 \)
- \( 3^8 = 6561 \)
- \( 3^9 = 19683 \)
- \( 3^{10} = 59049 \)
- \( 3^{11} = 177147 \)
- \( 3^{12} = 531441 \)
Thus,
\[ 9^6 = 531441 \]
Now plug this back into the expression for \( 9^{-6} \):
\[ 9^{-6} = \frac{1}{9^6} = \frac{1}{531441} \]
Now substitute this back into the original calculation:
\[ 99 \cdot 9^{-6} = 99 \cdot \frac{1}{531441} = \frac{99}{531441} \]
Thus, the numerical equivalent of \( 99 \cdot 9^{-6} \) is
\[ \frac{99}{531441} \]
In decimal form, this value will be very small. However, if we need to find the answer without exponents in simple fraction form, it is:
\[ \boxed{\frac{99}{531441}} \]