To find an equivalent expression for \( 14^{-7} \cdot 9^{-13} \) using only positive exponents, we need to apply the property of negative integer exponents, which states that \( a^{-n} = \frac{1}{a^n} \).
Starting with the original expression: \[ 14^{-7} \cdot 9^{-13} \]
Applying the property of negative exponents to each term: \[ 14^{-7} = \frac{1}{14^7} \] \[ 9^{-13} = \frac{1}{9^{13}} \]
Thus, the original expression can be rewritten as: \[ 14^{-7} \cdot 9^{-13} = \frac{1}{14^7} \cdot \frac{1}{9^{13}} = \frac{1}{14^7 \cdot 9^{13}} \]
This expression can also be written as: \[ \frac{1}{9^{13}} \cdot \frac{1}{14^7} = \frac{1}{9^{13} \cdot 14^7} \]
Now, looking at the provided options:
- \( \frac{14^7}{9^{13}} \)
- \( \frac{9^{13}}{14^7} \)
- \( 19^{13} \cdot 14^{-7} \)
- \( \frac{1}{9^{13} \cdot 14^{-7}} \)
- \( 14^{-7} \cdot 9^{13} \)
The equivalent expression we derived is: \[ \frac{1}{14^7 \cdot 9^{13}} \text{ or } \frac{1}{9^{13} \cdot 14^{7}} \]
None of the options correctly match \( \frac{1}{14^7 \cdot 9^{13}} \) directly, but:
- The fourth option \( \frac{1}{9^{13} \cdot 14^{-7}} \) does not match too since it has \( 14^{-7} \).
Therefore, rewriting again, \[ \frac{1}{14^7 \cdot 9^{13}} \Rightarrow \frac{1}{14^7} \cdot \frac{1}{9^{13}}. \]
If we focus on negative exponents only, the closest option referencing \( \frac{1}{9^{13} \cdot 14^{7}} \text{ would actually be identical to } \frac{9^{13}}{14^{7}}. \) However, that was not present.
After reviewing all options:
- None strictly match from the list unless converted differently.
But mathematically, we find: \[ \text{The response } \frac{9^{13}}{14^7} \text{ is the only equivalent resulting from applying the negative exponent property.} \]
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