To simplify the expression \( (16 - 12)^2 + 5(8 + 2^2) / 5^3 - (1 + 4)^2 \), we will break it down step by step.
1. Calculate \( 16 - 12 \):
\[
16 - 12 = 4
\]
2. Calculate \( (16 - 12)^2 \):
\[
(16 - 12)^2 = 4^2 = 16
\]
3. Calculate \( 2^2 \):
\[
2^2 = 4
\]
4. Calculate \( 8 + 2^2 \):
\[
8 + 2^2 = 8 + 4 = 12
\]
5. Now calculate \( 5(8 + 2^2) \):
\[
5(8 + 2^2) = 5 \cdot 12 = 60
\]
6. Next, calculate \( 5^3 \):
\[
5^3 = 125
\]
7. Now we compute \( \frac{5(8 + 2^2)}{5^3} \):
\[
\frac{60}{125} = \frac{12}{25}
\]
8. Calculate \( 1 + 4 \):
\[
1 + 4 = 5
\]
9. Now calculate \( (1 + 4)^2 \):
\[
(1 + 4)^2 = 5^2 = 25
\]
Now we will combine everything:
\[
(16 - 12)^2 + \frac{5(8 + 2^2)}{5^3} - (1 + 4)^2 = 16 + \frac{12}{25} - 25
\]
To combine these numbers, let's express 16 and 25 with a common denominator of 25:
- Convert 16:
\[
16 = \frac{400}{25}
\]
- Convert 25:
\[
25 = \frac{625}{25}
\]
So 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 \(-213\). However, since the question asks for the simplified numerator directly before we rearranged the entire expression, let's review.
The simplified numerator as derived before is:
\[
4^2 + 60 = 16 + 60 = 76
\]
Thus, the simplified numerator of this expression is:
\[
\boxed{76}
\]