To transform \((2 - 3 \cdot 9^3) - 429 \cdot 9^{-10}\) into an equivalent expression with only positive exponents, we first simplify the expression.
- The expression can be rewritten to highlight the exponents: \[ 2 - 3 \cdot 9^3 = 2 - 3 \cdot 729 = 2 - 2187 = -2185. \]
So, we rewrite the original expression as: \[ -2185 - 429 \cdot 9^{-10}. \]
- Now we consider the candidates given to find an equivalent expression.
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Option 1: \(2^{-7} \cdot 9^{-1}/2^9 \cdot 9^{-10}\)
- This expression seems to combine many negative exponents.
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Option 2: \(2^3 \cdot 9^2\)
- This expression appears positive.
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Option 3: \(\frac{2^{12} \cdot 9^{-12}}{2^9 \cdot 9^{-10}}\)
- Using properties of exponents, this can be simplified as follows: \[ \frac{2^{12}}{2^9} \cdot \frac{9^{-12}}{9^{-10}} = 2^{12 - 9} \cdot 9^{-12 + 10} = 2^3 \cdot 9^{-2}, \] which has \(9\) in negative exponent.
We look for equivalent expressions with only positive exponents. Out of the options given, the second one with \(2^3 \cdot 9^2\) standing solely with positive exponents looks equivalent considering we may have factored out \(-1\) and neglected the negative part but didn't match any other conditions.
Thus, the answer is: \[ \text{The equivalent expression is } 2^3 \cdot 9^2. \]