Let's simplify the given expression step by step:
The expression is:
\[ (16−12)^2 + \frac{5(8+2^2)}{5^3} − (1+4)^2 \]
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Calculate \( (16 - 12)^2 \): \[ (16 - 12) = 4 \implies (4)^2 = 16 \]
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Calculate \( 2^2 \): \[ 2^2 = 4 \]
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Calculate \( 8 + 2^2 \): \[ 8 + 4 = 12 \]
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Calculate \( 5(8 + 2^2) \): \[ 5 \cdot 12 = 60 \]
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Calculate \( 5^3 \): \[ 5^3 = 125 \]
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Now compute \( \frac{5(8 + 2^2)}{5^3} \): \[ \frac{60}{125} = \frac{12}{25} \]
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Calculate \( (1 + 4)^2 \): \[ (1 + 4) = 5 \implies (5)^2 = 25 \]
Now combine all the parts calculated:
\[ 16 + \frac{12}{25} - 25 \]
Convert \( 16 \) and \( 25 \) to have a common denominator of \( 25 \): \[ 16 = \frac{400}{25}, \quad 25 = \frac{625}{25} \]
So now we have: \[ \frac{400}{25} + \frac{12}{25} - \frac{625}{25} \]
Combine the fractions: \[ \frac{400 + 12 - 625}{25} = \frac{-213}{25} \]
The numerator is \(-213\).
None of the responses provided match this, indicating a possible error in interpretation or calculations. If we only need the simplified numerator without considering the entire expression at this moment, it would have been \(-213\).
Since there is no option that represents this, please double-check any context given for the numerator, or the expression itself.
However, if you meant to evaluate the expression differently or needed further clarification, please let me know!