To solve the expression \( 3^7 \cdot 3^{-9} \) using the properties of integer exponents, we will first combine the exponents.
The property of exponents that applies here is:
\[
a^m \cdot a^n = a^{m+n}
\]
So, we can combine the exponents:
\[
3^7 \cdot 3^{-9} = 3^{7 + (-9)} = 3^{-2}
\]
Next, we convert \( 3^{-2} \) to a positive exponent. Using the property:
\[
a^{-n} = \frac{1}{a^n}
\]
We can rewrite \( 3^{-2} \) as:
\[
3^{-2} = \frac{1}{3^2}
\]
Now we calculate \( 3^2 \):
\[
3^2 = 9
\]
Therefore:
\[
3^{-2} = \frac{1}{9}
\]
So the equivalent expression with only positive exponents is \( \frac{1}{9} \).
In conclusion, the final answer is:
\[
\frac{1}{9}
\]