To solve the expression \( 3^7 \cdot 3^{-9} \) using the properties of integer exponents, we can apply the rule that states \( a^m \cdot a^n = a^{m+n} \).
In this case, we have:
\[ 3^7 \cdot 3^{-9} = 3^{7 + (-9)} = 3^{7 - 9} = 3^{-2} \]
Next, to express \( 3^{-2} \) with only positive exponents, we use the property that states \( a^{-n} = \frac{1}{a^n} \):
\[ 3^{-2} = \frac{1}{3^2} \]
Now we can calculate \( 3^2 \):
\[ 3^2 = 9 \]
So,
\[ 3^{-2} = \frac{1}{9} \]
Therefore, the equivalent expression with only positive exponents is \( \frac{1}{9} \).
The final answer is:
\[ \frac{1}{9} \]