Question

Consider the expression (16−12)exponent2+5(8+2exponent2)/5exponent3−(1+4)exponent2
. What is the simplified numerator?(1 point)
Responses

100

4exponent2 + 60

54

76

Answer 1

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

1. Start with the expression:
\((16 - 12)^{2} + 5(8 + 2^{2})/5^{3} - (1 + 4)^{2}\)

2. Calculate each part:

- \(16 - 12 = 4\)
Thus, \((16 - 12)^{2} = 4^{2} = 16\)

- \(2^{2} = 4\)
Thus, \(8 + 2^{2} = 8 + 4 = 12\)

- Now calculate \(5(12)/5^{3}\):
- First, \(5^{3} = 125\)
- Then \(5(12) = 60\)
- Now, \(5(8 + 2^{2})/5^{3} = 60/125 = \frac{60}{125} = \frac{12}{25}\)

- Lastly, \((1 + 4)^{2} = 5^{2} = 25\)

3. Now substitute these calculated values back into the expression:

\[
16 + \frac{12}{25} - 25
\]

4. Combine the terms:

- Convert \(16\) and \(-25\) into fractions with a common denominator of \(25\):
- \(16 = \frac{400}{25}\)
- \(-25 = -\frac{625}{25}\)

So we have:
\[
\frac{400}{25} + \frac{12}{25} - \frac{625}{25} = \frac{400 + 12 - 625}{25} = \frac{-213}{25}
\]

5. The numerator of the simplified expression is \(-213\), but since we are asked for the numerator from a selection of responses, it appears they might be focusing on a different part of the expression.

Now looking at the responses:
- 100
- \(4^{2} + 60\)
- 54
- 76

The part \(4^{2} + 60\) gives us:
\[
4^{2} = 16 \implies 16 + 60 = 76
\]

Thus, the simplified numerator is:
\[
\boxed{76}
\]