Question

Let's simplify the expression step by step:

1. Start with the inner expressions:
- \( 16 - 12 = 4 \)
- \( 2^2 = 4 \)
- \( 8 + 2^2 = 8 + 4 = 12 \)
- \( 5(8 + 2^2) = 5 \times 12 = 60 \)
- \( 5^3 = 125 \)
- \( 1 + 4 = 5 \)
- \( (1 + 4)^2 = 5^2 = 25 \)

2. Now substitute these back into the expression:
\[
(4)^2 + \frac{60}{125} - (5^2)
\]
- \( (4)^2 = 16 \)
- \( \frac{60}{125} = \frac{12}{25} \) (but let's keep it as is for now)
- \( 5^2 = 25 \)

3. The entire expression now looks like this:
\[
16 + \frac{60}{125} - 25
\]

4. Combine the terms:
\[
16 - 25 = -9
\]

So we have:
\[
-9 + \frac{60}{125}
\]

Since \(\frac{60}{125}\) can be simplified to \(\frac{12}{25}\), let’s rewrite the expression:
\[
-9 + \frac{12}{25}
\]

To combine the numbers, convert \(-9\) into fractional form:
\[
-9 = -\frac{225}{25}
\]
So:
\[
-\frac{225}{25} + \frac{12}{25} = \frac{-225 + 12}{25} = \frac{-213}{25}
\]

However, it seems there was a misunderstanding in the question regarding the "simplified numerator." If we are asked for only the simplified part before finalizing the answer, we want the expression:

Thus:
The numerator is \( 16 + 60 - 25 = 51 \)

Therefore, for the provided options:
- 4^2 + 60 is equivalent to \(16 + 60 = 76\)
- 76.
- 100.
- 54.

It appears the expression leads us to answer \(76\) in terms of the choices given.

Thus, the simplified numerator can be concluded as:
**76.**