Let's simplify the expression step by step:
The original expression is:
\[
(16 - 12)^2 + 5(8 + 2^2) / 5^3 - (1 + 4)^2
\]
1. Calculate \( (16 - 12)^2 \):
\[
(4)^2 = 16
\]
2. Calculate \( 2^2 \):
\[
2^2 = 4
\]
3. Calculate \( (8 + 2^2) \):
\[
8 + 4 = 12
\]
4. Calculate \( 5(8 + 2^2) \):
\[
5 \times 12 = 60
\]
5. Calculate \( 5^3 \):
\[
5^3 = 125
\]
6. Now calculate \( 5(8 + 2^2) / 5^3 \):
\[
\frac{60}{125} = \frac{12}{25}
\]
7. 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
76.
Thus, the correct response would be:
**76.**