To rewrite the expression \( 14^{-7} \cdot 9^{-13} \) with only positive exponents, we apply the Property of Negative Integer Exponents, which states that \( a^{-n} = \frac{1}{a^n} \).
Applying this property to our expression:
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Rewrite \( 14^{-7} \): \[ 14^{-7} = \frac{1}{14^7} \]
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Rewrite \( 9^{-13} \): \[ 9^{-13} = \frac{1}{9^{13}} \]
Thus, combining these gives you: \[ 14^{-7} \cdot 9^{-13} = \frac{1}{14^7} \cdot \frac{1}{9^{13}} = \frac{1}{14^7 \cdot 9^{13}} \] This can also be expressed as: \[ \frac{1}{9^{13} \cdot 14^7} \]
Now, let's match this to the responses provided:
- \( \frac{9^{13}}{14^7} \) - This has positive exponents but does not match as it inverts the fraction.
- \( \frac{1}{9^{13} \cdot 14^{-7}} \) - This has a negative exponent for \( 14 \).
- \( 14^{-7} \cdot 9^{13} \) - This has a negative exponent for \( 14 \).
- \( \frac{14^7}{9^{13}} \) - This is just correcting the base order without addressing the negatives.
None of the responses provided an exact solution to document \( \frac{1}{9^{13} \cdot 14^7} \) which is what was derived, but it is worth rechecking. However, if \( \frac{1}{9^{13} \cdot 14^7} \) is effectively represented as \( \frac{1}{9^{13}} \cdot \frac{1}{14^{7}} \), it is the equivalent of using 14 and 9 with correct exponent force.
So, the response that closely represents an equivalent expression is: \[ \frac{1}{9^{13} \cdot 14^7} \] If you are allowed to specify or reconfigure representations that are equivalent.
Based on strict form where only positive outcomes manifest and interpret correctly would consider to mark if rearranged albeit not perfectly matched.