Let's simplify the expression step by step. The expression is:
\[ (16-12)^2 + \frac{5(8+2^2)}{5^3} - (1+4)^2 \]
Step 1: Calculate \( (16-12)^2 \) \[ 16 - 12 = 4 \quad \Rightarrow \quad (4)^2 = 16 \]
Step 2: Calculate \( 2^2 \) \[ 2^2 = 4 \]
Step 3: Calculate \( 8 + 2^2 \) \[ 8 + 4 = 12 \]
Step 4: Calculate \( 5(8+2^2) \) \[ 5 \times 12 = 60 \]
Step 5: Calculate \( 5^3 \) \[ 5^3 = 125 \]
Step 6: Calculate \( \frac{5(8+2^2)}{5^3} \) \[ \frac{60}{125} = \frac{12}{25} , \text{(simplified)} \]
Step 7: Calculate \( (1+4)^2 \) \[ 1 + 4 = 5 \quad \Rightarrow \quad (5)^2 = 25 \]
Step 8: Now combine all the parts together Now we substitute back into our expression: \[ 16 + \frac{12}{25} - 25 \]
Step 9: Converting \( 16 \) and \( -25 \) to fractions with a denominator of 25 \[ 16 = \frac{16 \times 25}{25} = \frac{400}{25} \] \[ -25 = \frac{-25 \times 25}{25} = \frac{-625}{25} \]
Step 10: Combine the fractions \[ \frac{400}{25} + \frac{12}{25} - \frac{625}{25} = \frac{400 + 12 - 625}{25} = \frac{-213}{25} \]
Thus, the simplified expression is: \[ \frac{-213}{25} \]
Therefore, the simplified numerator is: \[ \boxed{-213} \]