To generate an expression equivalent to \( \frac{5^{-9}}{4^{-12}} \) using positive exponents only, we can apply the property of negative integer exponents which states that \( a^{-n} = \frac{1}{a^n} \).
Starting with the original expression:
\[ \frac{5^{-9}}{4^{-12}} \]
We can rewrite the negative exponents as follows:
\[ = \frac{1}{5^{9}} \div \frac{1}{4^{12}} \]
Dividing by a fraction is equivalent to multiplying by its reciprocal:
\[ = \frac{1}{5^{9}} \times 4^{12} \]
This can be written as:
\[ = \frac{4^{12}}{5^{9}} \]
Now, both exponents are positive. Thus, the equivalent expression using only positive exponents is:
\[ \frac{4^{12}}{5^{9}} \]