To rewrite the expression \(14^{-7} \cdot 9^{-13}\) using only positive exponents, we can apply the property of negative exponents, which states that \(a^{-n} = \frac{1}{a^n}\).
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Rewrite \(14^{-7}\): \[ 14^{-7} = \frac{1}{14^7} \]
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Rewrite \(9^{-13}\): \[ 9^{-13} = \frac{1}{9^{13}} \]
Putting these together: \[ 14^{-7} \cdot 9^{-13} = \frac{1}{14^7} \cdot \frac{1}{9^{13}} = \frac{1}{14^7 \cdot 9^{13}} = \frac{1}{14^7 \cdot 9^{13}} \]
Now, this can also be written as: \[ \frac{1}{9^{13}} \cdot \frac{1}{14^7} = 9^{-13} \cdot 14^{-7} \]
Based on the provided options, the expression with only positive exponents equivalent to \(14^{-7} \cdot 9^{-13}\) would be: \[ \frac{1}{9^{13} \cdot 14^{7}} \quad \text{or} \quad 9^{13} \cdot 14^{-7} \]
To choose from the options given:
- \( \frac{14^{7}}{9^{13}} \): This is not correct because it inverts the entire fraction.
- \( 14^{-7} \cdot 9^{-13} \): This is the original expression; not rewritten as positive exponents.
- \( \frac{1}{9^{13} \cdot 14^{7}} \): This is the rewritten expression with positive exponents and is correct.
- \( 9^{13} \cdot 14^{-7} \): This is not entirely correct since it includes a negative exponent.
- \( \frac{9^{13}}{14^{7}} \): This would be incorrect; the exponents must be positive but doesn't match with what we had derived.
Therefore, the correct choice is: \[ \frac{1}{9^{13} \cdot 14^{7}} \]
That means the equivalent expression to \(14^{-7} \cdot 9^{-13}\) with only positive exponents is: \(\text{Final Answer:} \frac{1}{9^{13} \cdot 14^{7}}\)
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