Question

To simplify the expression \ ((16−12) ^2+5(8+2^2)/5^3−(1+4) ^2\), we will break it down step by step and first focus on the numerator.
76
54
42+60
10

Answer 1

Let's simplify the expression step by step, focusing first on the numerator:

\[
(16 - 12)^2 + 5(8 + 2^2)
\]

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

2. Now square that result:
\[
(16 - 12)^2 = 4^2 = 16
\]

3. Now calculate \(2^2\):
\[
2^2 = 4
\]

4. Next, calculate the expression inside the parentheses \(8 + 2^2\):
\[
8 + 2^2 = 8 + 4 = 12
\]

5. Now multiply that by 5:
\[
5(8 + 2^2) = 5 \times 12 = 60
\]

6. Now combine the results of steps 2 and 5:
\[
(16 - 12)^2 + 5(8 + 2^2) = 16 + 60 = 76
\]

Now, we can rewrite the entire expression:

\[
\frac{76}{5^3} - (1 + 4)^2
\]

Next, let's simplify the denominator and the final term:

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

2. Calculate \(1 + 4\):
\[
1 + 4 = 5
\]

3. Now square that result:
\[
(1 + 4)^2 = 5^2 = 25
\]

Now we can substitute these values into our expression:

\[
\frac{76}{125} - 25
\]

To combine these, we need a common denominator. Rewrite 25 as a fraction:

\[
25 = \frac{25 \times 125}{125} = \frac{3125}{125}
\]

Thus, we rewrite our expression:

\[
\frac{76}{125} - \frac{3125}{125} = \frac{76 - 3125}{125} = \frac{-3049}{125}
\]

This results in our final simplified expression:

\[
\frac{-3049}{125}
\]

Since your query had options like 54 or 10, it appears we've included many calculations, and the mistakenly derived expression doesn't seem to align with any straightforward conclusion. If you require corrections based on intermediate steps or further simplifications, please let me know!