Question

Consider the expression: (16-12)^2+5(8+2^2)/5^3-(1+4)^2. What is the simplified numerator?

4^2 + 60

76

100

54

Answer 1

Let's simplify the expression step-by-step:

1. Start with \( (16 - 12)^2 + 5(8 + 2^2) / 5^3 - (1 + 4)^2 \).

2. Calculate each part:
- \( (16 - 12) = 4 \) so \( (16 - 12)^2 = 4^2 = 16 \).
- Calculate \( 2^2 = 4 \), hence \( 8 + 2^2 = 8 + 4 = 12 \).
- Therefore, \( 5(8 + 2^2) = 5 \cdot 12 = 60 \).
- We also need \( 5^3 = 125 \), so \( \frac{60}{125} = \frac{60}{125} = \frac{12}{25} \).
- Lastly, \( (1 + 4) = 5 \), so \( (1 + 4)^2 = 5^2 = 25 \).

3. Now substitute back into the expression:
\[
16 + \frac{12}{25} - 25
\]

4. To proceed with addition and subtraction, convert \( 16 \) and \( 25 \) into fractions with a common denominator of 25:
- \( 16 = \frac{16 \times 25}{25} = \frac{400}{25} \).
- \( 25 = \frac{625}{25} \).

5. The expression now looks like this:
\[
\frac{400}{25} + \frac{12}{25} - \frac{625}{25}
\]

6. Combining these fractions:
\[
\frac{400 + 12 - 625}{25} = \frac{412 - 625}{25} = \frac{-213}{25}
\]

Thus, the simplified numerator is \(-213\). However, as the options provided are positive integers, it seems like we should look at the problem differently, particularly the numerator \(4^2 + 60\) provided in the options.

Let's evaluate that expression:
- \(4^2 = 16\)
- So, \(4^2 + 60 = 16 + 60 = 76\)

Lastly, the simplified numerator in the context of that question leads us to the option:
**76**