To rewrite the expression \( (2 - 3 \cdot 9^3) - 429 \cdot 9^{-10} \) in an equivalent form with only positive exponents, we start by simplifying the expression step by step.
- Rewrite negative exponents as positive exponents:
- \( 9^{-10} = \frac{1}{9^{10}} \)
Now, we will go through each option provided and check if it can be derived from simplifying the given expression with positive exponents:
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Response 1: \( 2^{12} \cdot 9^{-12} \div (2^9 \cdot 9^{-10}) \)
- This expression involves a division of exponents and does simplify but doesn't correspond with the original expression as no exponent transformations match. Plus, it retains negative exponents.
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Response 2: \( \frac{2^3}{9^2} \)
- Simplifying this yields a straightforward expression but does not match the original expression.
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Response 3: \( 2^3 \cdot 9^2 \)
- This is a positive exponent form but also bears no relation to the original form.
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Response 4: \( 2^{-7} \cdot 9^{-12} \cdot 9^{-10} \)
- Similar to previous responses, contains negative exponents and does not provide positive equivalent.
From this breakdown, we need to check if any transformations yield a combination that strictly uses positive exponents. However, none of the choices accurately reframe the original into a simpler configuration with strictly positive exponents.
Please re-evaluate the main expression: If the need arises to address each of these further or seek assistance on computation specifics for accurately determining your equivalent expression based off what I have approached is needed, please let me know!