Let's simplify the expression step by step:
The original expression is:
\[ (16 - 12)^2 + 5(8 + 2^2) / 5^3 - (1 + 4)^2 \]
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Calculate \( (16 - 12)^2 \): \[ (4)^2 = 16 \]
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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 \( 5(8 + 2^2) \): \[ 5 \times 12 = 60 \]
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Calculate \( 5^3 \): \[ 5^3 = 125 \]
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Now calculate \( 5(8 + 2^2) / 5^3 \): \[ \frac{60}{125} = \frac{12}{25} \]
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Calculate \( (1 + 4)^2 \): \[ (5)^2 = 25 \]
Now, we can plug everything back into the expression:
\[ 16 + \frac{12}{25} - 25 \]
Convert \( 16 \) and \( -25 \) to have a common denominator of 25:
\[ 16 = \frac{400}{25} \] \[ -25 = \frac{-625}{25} \]
Thus, we rewrite the expression:
\[ \frac{400}{25} + \frac{12}{25} - \frac{625}{25} \]
Combine the fractions:
\[ \frac{400 + 12 - 625}{25} = \frac{-213}{25} \]
Now, if the question is asking for the simplified numerator before dividing by 25, the numerator would be:
\[ 400 + 12 - 625 = -213 \]
However, since the question asks specifically for the simplified numerator based on what was done in the prior steps we can consider the part \( 4^2 + 60 \) as a potential candidate.
So, evaluating \( 4^2 + 60 \): \[ 4^2 = 16 \] \[ 16 + 60 = 76 \]
Thus if we consider \( 4^2 + 60 \) as a standalone component of the larger expression that combines numerator components before outputs:
The simplified numerator could therefore be
Thus, the correct response would be:
76.