To simplify the expression given in your question, let's rewrite it clearly. The expression appears to be:
\[ 2^{-3} \cdot (9^3)^{-4} / (2^{9} \cdot 9^{-10}) \]
Let's break this down step by step.
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Simplify the individual terms:
- \( 2^{-3} \) remains as is.
- For \( (9^3)^{-4} \), we have: \[ (9^3)^{-4} = 9^{-12} \] because \( (a^m)^n = a^{m \cdot n} \).
- The term \( 2^9 \) remains as is, and \( 9^{-10} \) also remains as is.
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Combine the terms: Now the expression can be rewritten as: \[ \frac{2^{-3} \cdot 9^{-12}}{2^9 \cdot 9^{-10}} \]
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Apply the property of exponents: For the base 2: \[ \frac{2^{-3}}{2^9} = 2^{-3 - 9} = 2^{-12} \] For the base 9: \[ \frac{9^{-12}}{9^{-10}} = 9^{-12 + 10} = 9^{-2} \]
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Combine the results: Now we have: \[ 2^{-12} \cdot 9^{-2} \]
This would be the simplified expression. Now, let's match it with the answer choices provided:
- 1. \( 2^{3} \cdot 9^2 \)
- 2. \( 2^{12} \cdot 9^{-12} \)
- 3. \( 2^{3/9} \cdot 9^{2} \)
- 4. \( 2^{-7} \cdot 9^{-1} \) over \( 2^{9} \cdot 9^{-10} \)
None of the options directly match our simplified expression, but if we adjust \( 2^{-12} \cdot 9^{-2} \) it is equivalent to:
\[ \frac{1}{2^{12} \cdot 9^2} \]
If any option embodies this form in a complex fraction, then, based on definitions of negatives, we could equate that way, but currently, none of the given choices reflect the exact simplified expression.
If you need further assistance or a different question analysis, feel free to ask!
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