To find the numerical equivalent of \( 99 \cdot 9^{-6} \), we first need to understand that \( 9^{-6} \) means \( \frac{1}{9^6} \).
Calculating \( 9^6 \):
\[ 9^6 = (9^3)^2 \] Calculating \( 9^3 \):
\[ 9^3 = 729 \]
Thus,
\[ 9^6 = 729^2 \]
Now, we can calculate \( 729^2 \):
\[ 729^2 = 531441 \]
So, \( 9^6 = 531441 \) and therefore, \( 9^{-6} = \frac{1}{531441} \).
Now, substituting back into the equation:
\[ 99 \cdot 9^{-6} = 99 \cdot \frac{1}{531441} = \frac{99}{531441} \]
Now, let's simplify the fraction \( \frac{99}{531441} \).
Since \( 531441 = 9^6 \), and \( 99 = 9 \cdot 11 \):
The fraction simplifies to:
\[ \frac{99}{531441} = \frac{9 \cdot 11}{9^6} = \frac{11}{9^5} \] Calculating \( 9^5 \):
\[ 9^5 = 59049 \]
Thus,
\[ \frac{11}{9^5} = \frac{11}{59049} \]
So the final result is:
\[ \frac{11}{59049} \]
Thus, the numerical equivalent of \( 99 \cdot 9^{-6} \) is:
\[ \frac{11}{59049} \]
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