To simplify the expression \( 3^7 \cdot 3^{-9} \) using the properties of integer exponents, we can apply the product of powers property, which states that when multiplying two expressions with the same base, we add their exponents:
\[ 3^a \cdot 3^b = 3^{a+b} \]
Applying this to our expression:
\[ 3^7 \cdot 3^{-9} = 3^{7 + (-9)} = 3^{7 - 9} = 3^{-2} \]
Next, we can rewrite \( 3^{-2} \) using positive exponents. The property for negative exponents states that:
\[ x^{-n} = \frac{1}{x^n} \]
Therefore,
\[ 3^{-2} = \frac{1}{3^2} \]
Now, we calculate \( 3^2 \):
\[ 3^2 = 9 \]
Thus, we can write \( 3^{-2} \) as:
\[ 3^{-2} = \frac{1}{9} \]
In conclusion, the expression \( 3^7 \cdot 3^{-9} \) simplifies to:
\[ \frac{1}{9} \]
So, the final answer is:
\[ \frac{1}{9} \]