To apply the property of negative integer exponents, we can rewrite the expression \( \frac{5^{-9}}{4^{-12}} \) using the rule that states \( a^{-n} = \frac{1}{a^n} \).
Starting with the expression:
\[ \frac{5^{-9}}{4^{-12}} \]
We can rewrite this as:
\[ \frac{1}{5^9} \div \frac{1}{4^{12}} \]
Dividing by a fraction is equivalent to multiplying by its reciprocal, so we can rewrite the expression as:
\[ \frac{1}{5^9} \times 4^{12} \]
This simplifies to:
\[ \frac{4^{12}}{5^{9}} \]
Now, since the expression contains only positive exponents, the final equivalent expression is:
\[ \frac{4^{12}}{5^{9}} \]
So the expression with positive exponents only is:
\[ \frac{4^{12}}{5^{9}} \]