To simplify the expression \( (15 - 3 \cdot 47) \cdot 04^{-3} \), let's first evaluate the terms inside the parentheses:
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Calculate \( 3 \cdot 47 \): \[ 3 \cdot 47 = 141 \]
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Then, calculate \( 15 - 141 \): \[ 15 - 141 = -126 \]
Now we need to substitute this back into the expression: \[ (-126) \cdot 04^{-3} \] Since \( 04^{-3} = 0^{-3} \), which is undefined, the expression can't be simplified normally. However, if we consider the multiplication with \( 0^{-3} \), any multiplication involving zero yields zero.
Thus, the expression simplifies to: \[ 0 \]
However, if we interpret \( 4^{-3} \) correctly (as there's a possibility of misinterpretation), we can evaluate \( 4^{-3} \) separately: \[ 4^{-3} = \frac{1}{4^3} = \frac{1}{64} \]
With this, the expression thus transforms to: \[ (-126) \cdot \frac{1}{64} = \frac{-126}{64} \]
Now we compare with the options given:
- \( 1 \cdot 4^{-3} \) is a positive expression.
- \( \frac{1}{64} \) doesn't match.
- \( 64 \) doesn't match either.
- \( \frac{60}{4^{-3}} \) doesn't match.
None of these options appear to fully capture the essence of the expression if following the calculations and given a potential typo.
So from the context of properties of exponents through simplification, if any sort of manipulation happened where \( 64 \) could emerge from terms of \( 4^{-3} \) (perhaps wrong transformation was done with the exponents), perhaps indicating through the 'properties of integers', or misinterpretation of the exponent. Assuming intended operations led to revisions of those terms, we might find related fractions appearing, yet no direct equivalent from the provided response could be categorized.
If the emphasis lies solely on calculation through integers leading to 64 but it seems unanchored. Kindly inspect if the response cues toward such mistakes or define transformations that result on exponents level within equivalents since the responses listed do not line up with fundamental evaluation of the expression stepwise both positive and negative constructs influencing negatives as defined upon simplifications.
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