To find the numerical equivalent of \( 99 \cdot 9^{-6} \), we begin by calculating \( 9^{-6} \).
The expression \( 9^{-6} \) can be rewritten as: \[ 9^{-6} = \frac{1}{9^6} \]
Next, we calculate \( 9^6 \): \[ 9^6 = (3^2)^6 = 3^{2 \cdot 6} = 3^{12} \]
Now we can evaluate \( 3^{12} \): \[ 3^{12} = 531441 \]
So, \[ 9^6 = 531441 \] and therefore, \[ 9^{-6} = \frac{1}{531441} \]
Now we substitute back into the original expression: \[ 99 \cdot 9^{-6} = 99 \cdot \frac{1}{531441} = \frac{99}{531441} \]
Now we need to simplify \( \frac{99}{531441} \). We can check if \( 99 \) and \( 531441 \) have a common factor. Since \( 99 = 3^2 \cdot 11 \) and \( 531441 = 3^{12} \), the common factor we can reduce is \( 3^2 \).
So we can simplify: \[ \frac{99}{531441} = \frac{3^2 \cdot 11}{3^{12}} = \frac{11}{3^{10}} \]
Calculating \( 3^{10} \): \[ 3^{10} = 59049 \]
Thus, \[ \frac{11}{59049} \]
Therefore, the numerical equivalent of \( 99 \cdot 9^{-6} \) written without exponents is: \[ \frac{11}{59049} \]