To simplify the expression \( 3^7 \cdot 3^{-9} \), you can use the property of exponents that states \( a^m \cdot a^n = a^{m+n} \).
Applying this property:
\[ 3^7 \cdot 3^{-9} = 3^{7 + (-9)} = 3^{-2} \]
Now, to express \( 3^{-2} \) as a positive exponent, you can use the negative exponent rule, which states that \( a^{-n} = \frac{1}{a^n} \):
\[ 3^{-2} = \frac{1}{3^2} \]
Calculating \( 3^2 \):
\[ 3^2 = 9 \]
Thus, we have:
\[ 3^{-2} = \frac{1}{9} \]
So, the simplified form of \( 3^7 \cdot 3^{-9} \) is:
\[ \frac{1}{9} \]