To simplify the expression \( 14^{-7} - 9^{-13} \) using the property of negative integer exponents, we can rewrite it in terms of positive exponents.
The property states that
\[ a^{-n} = \frac{1}{a^n}. \]
Applying this property to our expression:
- Rewrite \( 14^{-7} \) as \( \frac{1}{14^7} \).
- Rewrite \( 9^{-13} \) as \( \frac{1}{9^{13}} \).
Thus, we have:
\[ 14^{-7} - 9^{-13} = \frac{1}{14^7} - \frac{1}{9^{13}}. \]
However, the question specifically asks for an equivalent expression that uses only positive exponents.
Given the options:
- \( \frac{9}{13} \cdot 1^{47} \)
- \( \frac{14^7}{9^{13}} \)
- \( 14^{-7} \cdot 9^{13} \cdot 14 \)
- \( \frac{1}{9^{13}} \cdot 14^{-7} \)
The second option, \( \frac{14^7}{9^{13}} \), correctly shows \( 14^{-7} \) rewritten as \( \frac{1}{14^7} \) and does not include any negative exponents anymore. Hence, this is our equivalent expression.
So the equivalent expression to \( 14^{-7} - 9^{-13} \) with only positive exponents would be:
Answer: \( \frac{14^7}{9^{13}} \)