To solve the expression \((4^3 \cdot 7^{-2})^{-5} / (7^9 \cdot 4^{-13})\), we'll first apply the properties of integer exponents to rewrite everything with positive exponents before simplifying.
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Apply the negative exponent on the numerator: \[ (4^3 \cdot 7^{-2})^{-5} = 4^{-15} \cdot 7^{10} \] Here, we use the property \((a^m \cdot b^n)^p = a^{m \cdot p} \cdot b^{n \cdot p}\).
- \(4^3\) raised to \(-5\) becomes \(4^{-15}\) (since \(3 \cdot -5 = -15\)).
- \(7^{-2}\) raised to \(-5\) becomes \(7^{10}\) (since \(-2 \cdot -5 = 10\)).
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Rewrite the overall expression: Now substituting back, we get: \[ \frac{4^{-15} \cdot 7^{10}}{7^9 \cdot 4^{-13}} \]
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Combine the terms in the fraction: Recall that when dividing exponents with the same base, we subtract the exponents: \[ = 4^{-15 - (-13)} \cdot 7^{10 - 9} = 4^{-15 + 13} \cdot 7^{1} = 4^{-2} \cdot 7^{1} \]
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Simplify the expression: Knowing that \(4^{-2} = \frac{1}{4^2}\), we have: \[ = \frac{7}{4^2} = \frac{7}{16} \]
Therefore, the final simplified answer is: \[ \boxed{\frac{7}{16}} \]