Question

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

100

4^2 + 60

54

76

Answer 1

To simplify the expression \((16−12)^2 + 5(8 + 2^2) / 5^3 − (1 + 4)^2\), let's break it down step by step.

1. Calculate \( (16 - 12)^2 \):
\[
16 - 12 = 4 \implies (4)^2 = 16
\]

2. Calculate \( 2^2 \):
\[
2^2 = 4
\]

3. Calculate \( 8 + 2^2 \):
\[
8 + 4 = 12
\]

4. Calculate \( 5(8 + 2^2) \):
\[
5 \cdot 12 = 60
\]

5. Calculate \( 5^3 \):
\[
5^3 = 125
\]

6. Now compute \( \frac{5(8 + 2^2)}{5^3} \):
\[
\frac{60}{125}
\]
To simplify \( \frac{60}{125} \), divide the numerator and denominator by 5:
\[
\frac{60 \div 5}{125 \div 5} = \frac{12}{25}
\]

7. Calculate \( (1 + 4)^2 \):
\[
1 + 4 = 5 \implies (5)^2 = 25
\]

8. Now, combine everything in the original expression:
\[
(16 - 12)^2 + \frac{5(8 + 2^2)}{5^3} - (1 + 4)^2
\]
becomes:
\[
16 + \frac{12}{25} - 25
\]

9. Combine the whole numbers:
\[
16 - 25 = -9
\]

10. The final expression is:
\[
-9 + \frac{12}{25}
\]
To combine these, convert \(-9\) to a fraction with the same denominator:
\[
-9 = -\frac{225}{25}
\]
Thus, we have:
\[
-\frac{225}{25} + \frac{12}{25} = \frac{-225 + 12}{25} = \frac{-213}{25}
\]

The numerator of the simplified expression is \(-213\).

Therefore, none of the answer choices (100, \(4^2 + 60\), 54, 76) match. The final answer for the numerator of the entire expression is:

\(\text{Numerator: } -213\)

If you specifically meant only to derive the intermediate assertions or match these calculations against provided choices, kindly specify, as the concluding evaluation contradicts available selections.