To simplify the expression \( 37 \cdot 3^{-9} \) using the properties of integer exponents, we first recognize that \( 3^{-9} \) can be rewritten with a positive exponent.
The property states:
\[ a^{-n} = \frac{1}{a^n} \]
So, we can rewrite \( 3^{-9} \) as follows:
\[ 3^{-9} = \frac{1}{3^9} \]
Now, substituting this back into the original expression:
\[ 37 \cdot 3^{-9} = 37 \cdot \frac{1}{3^9} \]
This can be simplified to:
\[ \frac{37}{3^9} \]
Next, we need to calculate \( 3^9 \):
\[ 3^9 = 19683 \]
Putting it all together, we have:
\[ \frac{37}{19683} \]
Thus, the expression \( 37 \cdot 3^{-9} \) simplifies to \( \frac{37}{19683} \) with all positive exponents.
Final answer:
\[ \frac{37}{19683} \]