To find the numerical equivalent of \( \frac{3}{9} \cdot 3^{-5} \), we can simplify the expression step by step.
First, simplify \( \frac{3}{9} \):
\[ \frac{3}{9} = \frac{1}{3} \]
Now, substituting this back into the original expression:
\[ \frac{1}{3} \cdot 3^{-5} \]
Next, we can use the property of exponents which states that \( a^{-n} = \frac{1}{a^n} \):
\[ 3^{-5} = \frac{1}{3^5} \]
Thus, our expression becomes:
\[ \frac{1}{3} \cdot \frac{1}{3^5} = \frac{1}{3 \cdot 3^5} \]
Using the property of exponents \( a^m \cdot a^n = a^{m+n} \), we simplify the denominator:
\[ 3 \cdot 3^5 = 3^{1+5} = 3^6 \]
So we have:
\[ \frac{1}{3^6} \]
To find the numerical value of \( 3^6 \):
\[ 3^6 = 729 \]
Therefore,
\[ \frac{1}{3^6} = \frac{1}{729} \]
The numerical equivalent of the original expression \( \frac{3}{9} \cdot 3^{-5} \) is
\[ \boxed{\frac{1}{729}} \]
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