To simplify the expression \( 14^{-79} / 9^{-13} \) and convert it to an equivalent expression with only positive exponents, we will apply the property of negative exponents, which states that \( a^{-n} = \frac{1}{a^n} \).
Start by rewriting the given expression:
\[ \frac{14^{-79}}{9^{-13}} = 14^{-79} \cdot 9^{13} \]
Now, using the property of negative exponents, we convert \( 14^{-79} \):
\[ 14^{-79} = \frac{1}{14^{79}} \]
So, the expression becomes:
\[ \frac{1}{14^{79}} \cdot 9^{13} = \frac{9^{13}}{14^{79}} \]
Next, let's analyze the options provided:
- \( \frac{9^{13}}{14^{7}} \)
- \( \frac{1}{9^{13} \cdot 14^{-7}} \)
- \( 14^{-7} \cdot 9^{13} \)
- \( \frac{14^{7}}{9^{13}} \)
Among these, none of the responses match \( \frac{9^{13}}{14^{79}} \) directly. However, we notice that by interpreting the terms involved, if we consider \( \frac{9^{13}}{14^{-7}} \) to be incorrectly simplified.
Thus, the expression does not have a direct match in the given responses. If generating a proper equivalent from the original expression simply converting to positive powers, we find:
\[ \frac{9^{13}}{14^{79}} \]
The response with the closest structure focusing on formatted negatives could align if properly factored.
Final derived equivalency would resonate best as:
\[ \text{None from listed correctly simplify structurally,} ; \frac{9^{13}}{14^{79}} \text{ persists within direct equivalence.} \]
Providing the original requires full reform in mathematical rigor to presented structure, none validly apply straightforwardly without simplification. Proper structure reiterates fundamentals underpinning proposed numerals.