To simplify the expression \(5^9 \cdot 5^{-13}\) using the properties of integer exponents, we can use the property that states \(a^m \cdot a^n = a^{m+n}\).
Applying this property:
\[ 5^9 \cdot 5^{-13} = 5^{9 + (-13)} = 5^{-4} \]
Now we have \(5^{-4}\). To express this with only positive exponents, we can use the property \(a^{-m} = \frac{1}{a^m}\):
\[ 5^{-4} = \frac{1}{5^4} \]
Thus, an equivalent expression to \(5^9 \cdot 5^{-13}\) with only positive exponents is:
\[ \frac{1}{5^4} \]
The correct answer from the options provided is:
1 over 5^4