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To solve the expression \( 3^7 \cdot 3^{-9} \) using the properties of exponents, we can apply the rule that states:
\[
a^m \cdot a^n = a^{m+n}
\]
Applying this rule to our expression:
\[
3^7 \cdot 3^{-9} = 3^{7 + (-9)} = 3^{-2}
\]
Next, we can convert \( 3^{-2} \) to have a positive exponent by using the property of exponents that states:
\[
a^{-n} = \frac{1}{a^n}
\]
Thus:
\[
3^{-2} = \frac{1}{3^2}
\]
Now we can evaluate \( 3^2 \):
\[
3^2 = 9
\]
So we have:
\[
3^{-2} = \frac{1}{9}
\]
Therefore, the equivalent expression with only positive exponents is:
\[
3^7 \cdot 3^{-9} = \frac{1}{9}
\]
Final answer: \( \frac{1}{9} \)