To simplify the expression \((16 - 12)^2 + \frac{5(8 + 2^2)}{5^3} - (1 + 4)^2\):
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Calculate \((16 - 12)^2\): \[ (16 - 12) = 4 \quad \text{so} \quad (4)^2 = 16 \]
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Calculate \(2^2\): \[ 2^2 = 4 \]
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Then calculate \(8 + 2^2\): \[ 8 + 4 = 12 \]
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Now calculate \(5(8 + 2^2)\): \[ 5(12) = 60 \]
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Calculate \(5^3\): \[ 5^3 = 125 \]
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Now, calculate \(\frac{5(8 + 2^2)}{5^3}\): \[ \frac{60}{125} = \frac{12}{25} \]
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Calculate \((1 + 4)^2\): \[ (5)^2 = 25 \]
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Putting it all together: \[ 16 + \frac{12}{25} - 25 \]
To simplify \(16 - 25\): \[ 16 - 25 = -9 \]
Now combine with \(\frac{12}{25}\): \[ -9 + \frac{12}{25} = -\frac{225}{25} + \frac{12}{25} = -\frac{213}{25} \]
The entire expression simplifies to \[ -\frac{213}{25} \]
Now assessing the list of given responses for the numerator (assuming integers):
- 100
- \(42 + 60\) (which evaluates to \(102\))
- 54
- 76
None of these seem to match \(-213\) or any equivalent form.
Given that we're examining for simplification in the numerator:
- From parts of the expression computed: \(16\) and \(-25\),
- There isn't a clear match.
Therefore, it seems none of the given responses match the simplified values obtained from the expression as examined here.
If this is potentially a flaw in interpreting the task or if any alternative simplification directly points to a choice made earlier, please clarify. But with straightforward calculations, results deviate from presented options.