To find an equivalent expression to \( 14^{-7} - 9^{-13} \) with only positive exponents, we can apply the property of negative integer exponents, which states that \( a^{-n} = \frac{1}{a^n} \).
Let's rewrite the original expression:
- Rewrite \( 14^{-7} \) and \( 9^{-13} \) using positive exponents:
\[ 14^{-7} = \frac{1}{14^7} \] \[ 9^{-13} = \frac{1}{9^{13}} \]
- Now substituting these into the expression gives:
\[ \frac{1}{14^7} - \frac{1}{9^{13}} \]
Next, let's analyze the provided response options:
- \( 14^{-7} \cdot 9^{13} \) (not correct, as it does not represent the original expression's form)
- \( \frac{1}{9^{13} \cdot 14^{-7}} \) (this simplifies to \( \frac{1 \cdot 14^7}{9^{13}} \) which is not equivalent to the original expression)
- \( \frac{9^{13}}{14^{7}} \) (this is not how the expression started, as it's a difference, not a fraction of summands)
- \( \frac{14^7}{9^{13}} \) (similarly, incorrect as it does not retain the subtraction aspect)
Thus, we see that none of the answer choices reflect the equivalent desired format or maintain the original subtraction nature of \( 14^{-7} - 9^{-13} \).
If instead, we consider potential answers that were not provided in the choices, we could write:
- To combine the fractions into a single expression, you would find a common denominator (e.g., \( 14^7 \cdot 9^{13} \)), but that's not part of the original prompt.
Hence, based on the options you provided, the correct form in terms of equivalent expressions with positive exponents does not appear to be present among those choices. If you wish for clarification on rewriting or seeking alternatives, please let me know!