To solve the expression \((4^3 \cdot 7^{-2})^{-5} \div (7^9 \cdot 4^{-13})\) with positive exponents, we will simplify step by step.
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Start with the expression: \[ (4^3 \cdot 7^{-2})^{-5} \div (7^9 \cdot 4^{-13}) \]
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Apply the negative exponent to the first part of the expression: \[ (4^3)^{-5} \cdot (7^{-2})^{-5} = 4^{-15} \cdot 7^{10} \]
Thus, we rewrite the expression as: \[ 4^{-15} \cdot 7^{10} \div (7^9 \cdot 4^{-13}) \]
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Rewrite the division as multiplication by the reciprocal: \[ = 4^{-15} \cdot 7^{10} \cdot \frac{1}{7^9 \cdot 4^{-13}} = 4^{-15} \cdot 7^{10} \cdot 7^{-9} \cdot 4^{13} \]
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Combine the exponents:
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For the base \(4\): \[ 4^{-15} \cdot 4^{13} = 4^{-15 + 13} = 4^{-2} \]
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For the base \(7\): \[ 7^{10} \cdot 7^{-9} = 7^{10 - 9} = 7^{1} \]
Combining these, we have: \[ 4^{-2} \cdot 7^{1} \]
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Convert to positive exponents: \[ = \frac{7}{4^{2}} = \frac{7}{16} \]
Thus, the final answer, simplified without any remaining exponents, is: \[ \frac{7}{16} \]