To find an equivalent expression for \((2^{-3} \cdot 9^{3})(-4)^{2}(9) \cdot 9^{-10}\) using only positive exponents, let's simplify it step by step.
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Calculate \((-4)^{2}\) and express \(9\) in terms of \(9^{1}\): \[ (-4)^{2} = 16 \] Thus, we can rewrite the expression as: \[ (2^{-3} \cdot 9^{3}) \cdot 16 \cdot 9^{1} \cdot 9^{-10} \]
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Combine \(9^{3}\), \(9^{1}\), and \(9^{-10}\): When multiplying bases, we add the exponents: \[ 9^{3 + 1 - 10} = 9^{-6} \] Therefore, the expression can now be rewritten as: \[ 2^{-3} \cdot 16 \cdot 9^{-6} \]
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Next, express \(16\) as \(2^{4}\): \[ 16 = 2^{4} \] So the expression becomes: \[ 2^{-3} \cdot 2^{4} \cdot 9^{-6} \]
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Combine the \(2\) terms: \[ 2^{-3 + 4} = 2^{1} = 2 \] Thus, we have: \[ 2 \cdot 9^{-6} \]
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Express using only positive exponents: \(9^{-6}\) can instead be written as \(\frac{1}{9^{6}}\): \[ \frac{2}{9^{6}} \]
From the responses provided:
- \(2^{3} \cdot 9^{2}\) is not equivalent.
- \(2^{-7} \cdot 9^{-1} / (2^{9} \cdot 9^{-10})\) is too complex and does not match.
- \(2^{12} \cdot 9^{-12}/(2^{9} \cdot 9^{-10})\) also does not match.
- \(2^{3}/9^{2}\) does not match.
None of these responses directly reflect \(\frac{2}{9^{6}}\). Thus, re-evaluating this correctly implies that no options correspond correctly.
The simplified equivalent expression for the original is \( \frac{2}{9^{6}} \). However, based on the answers you provided, make sure to verify if other options exist or if they fit under transformable equivalences.
If you have a response equivalent to \( \frac{2}{9^{6}} \) in your options, that would be the correct selection.