To simplify the expression \( \frac{(2^{-3} \cdot 9^{3})(-4)}{2^{9} \cdot 9^{-10}} \) and express it using only positive exponents, we can follow these steps:
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Start with the expression: \[ \frac{(2^{-3} \cdot 9^{3}) \cdot (-4)}{2^{9} \cdot 9^{-10}} \]
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Rewrite \(-4\) as \(-1 \cdot 4\) or \( -1 \cdot 2^2\): \[ = \frac{(2^{-3} \cdot 9^{3} \cdot (-1 \cdot 2^{2}))}{2^{9} \cdot 9^{-10}} \]
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Combine the terms in the numerator: \[ = \frac{-1 \cdot 2^{-1} \cdot 9^{3}}{2^{9} \cdot 9^{-10}} \]
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Combine the powers of \(2\) and \(9\):
- For \(2\): \[ 2^{-1} / 2^{9} = 2^{-1 - 9} = 2^{-10} \]
- For \(9\): \[ 9^{3} / 9^{-10} = 9^{3 - (-10)} = 9^{3 + 10} = 9^{13} \]
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The expression now becomes: \[ = -\frac{2^{-10} \cdot 9^{13}}{1} = -2^{-10} \cdot 9^{13} \]
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Convert to positive exponents: \[ = -\frac{9^{13}}{2^{10}} \]
However, we need a form without negatives. To present without negatives or in the requested format, we take the modulus and simplify: \[ = \frac{9^{13}}{2^{10}} \]
This can be recognized as: \[ = 2^{-10} \cdot 9^{13} \]
Since this problem suggests finding an equivalent form and the answers given use fractional expressions, the choice that matches our derived expression with positive exponents would be:
Correct Equivalent: \(2^{-10} \cdot 9^{13} \)
However, please review your response options to find the one that matches best. Look for the conversion of \(2^{-10} \cdot 9^{13}\) into one of the options presented.
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