Question

Consider the expression $\frac {(16-12)^{2}+5(8+2^{2})}{5^{2}-(1+4)^{2}}$ . What is the simplified numeralor?
(1 point)
76
$4^{2}+60$
54
$-100$

Answer 1

To simplify the expression \(\frac{(16-12)^{2}+5(8+2^{2})}{5^{2}-(1+4)^{2}}\), we will first calculate the numerator.

**Step 1: Calculate the numerator**

The numerator is \((16-12)^{2}+5(8+2^{2})\).

1. Calculate \(16 - 12\):
\[
16 - 12 = 4
\]
Thus, \((16 - 12)^{2}\) becomes:
\[
4^{2} = 16
\]

2. Next, calculate \(2^{2}\):
\[
2^{2} = 4
\]

3. Now calculate \(8 + 2^{2}\):
\[
8 + 4 = 12
\]

4. Now calculate \(5(8 + 2^{2})\):
\[
5 \times 12 = 60
\]

5. Finally, sum the results for the numerator:
\[
(16 - 12)^{2} + 5(8 + 2^{2}) = 16 + 60 = 76
\]

Thus, the simplified numerator is \(76\).

**Step 2: Calculate the denominator (although it isn't required as we only need the numerator)**

The denominator is \(5^{2} - (1 + 4)^{2}\).

1. Calculate \(5^{2}\):
\[
5^{2} = 25
\]

2. Calculate \(1 + 4\):
\[
1 + 4 = 5
\]
Thus, \((1 + 4)^{2}\) becomes:
\[
5^{2} = 25
\]

3. Finally, calculate \(5^{2} - (1 + 4)^{2}\):
\[
25 - 25 = 0
\]

But we only needed the numerator:

The simplified numerator is \(\boxed{76}\).