To simplify the expression \((2 - 3 \cdot 93) - 429 \cdot 9^{-10}\) using the properties of exponents, we start by evaluating it step by step.
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The expression to simplify is \(2^{-3} \cdot 9^3 - 4 \cdot 29 \cdot 9^{-10}\).
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For \(2^{-3}\): \[ 2^{-3} = \frac{1}{2^3} = \frac{1}{8} \]
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For \(9^3\): \[ 9 = 3^2, , \text{so} , 9^3 = (3^2)^3 = 3^{2 \cdot 3} = 3^6 \]
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For \(9^{-10}\): \[ 9^{-10} = \frac{1}{9^{10}} = \frac{1}{(3^2)^{10}} = \frac{1}{3^{20}} \]
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The powers of both bases can be dealt with as normal numeric operations, but directly working with \(2^{-3} \cdot 9^3 - 4 \cdot 29 \cdot 9^{-10}\) yields: \[ 2^{-3} \cdot 9^3 - 4 \cdot 29 \cdot 9^{-10} = \frac{1}{8} \cdot 3^6 - 4 \cdot 29 \cdot \frac{1}{3^{20}} \]
Now, let's explore the multiple choice options provided:
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Option 1: \(2392\)
This option does not appear to relate to the expression derived from exponent manipulation.
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Option 2: \( \frac{2^{12} \cdot 9^{-12}}{2^{9} \cdot 9^{-10}} \)
This needs further simplification. This is equivalent to: \[ 2^{12 - 9} \cdot 9^{-12 + 10} = 2^{3} \cdot 9^{-2} \] Which is equivalent to \(\frac{2^3}{9^2}\). This looks promising.
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Option 3: \(2^{3} \cdot 9^{2}\)
This option actually does not hold true since we are in fact aiming to limit using negative exponents but they multiply different bases which may create complexity.
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Option 4: \(2^{-7} \cdot 9^{-12} \cdot 9^{-10}\)
This also doesn’t follow or even follow through with the structure as there will be direct multiplication merging negative fraction exponents that lead into improper subtraction.
In conclusion, the simplified expression should be equivalent to Option 2: \[ \frac{2^{12} \cdot 9^{-12}}{2^9 \cdot 9^{-10}} = \frac{2^{3}}{9^{2}} \] Which is the correct choice when analyzed regarding positive exponents generation.