To simplify the expression \((16 - 12)^2 + 5(8 + 2^2)/5^3 - (1 + 4)^2\), we will evaluate each component step-by-step.
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Calculate \((16 - 12)^2\): \[ 16 - 12 = 4 \implies (16 - 12)^2 = 4^2 = 16 \]
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Calculate \(2^2\): \[ 2^2 = 4 \]
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Calculate \(8 + 2^2\): \[ 8 + 2^2 = 8 + 4 = 12 \]
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Calculate \(5(8 + 2^2)\): \[ 5 \cdot 12 = 60 \]
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Calculate \(5^3\): \[ 5^3 = 125 \]
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Calculate \(\frac{5(8 + 2^2)}{5^3}\): \[ \frac{60}{125} = \frac{12}{25} \]
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Calculate \((1 + 4)^2\): \[ 1 + 4 = 5 \implies (1 + 4)^2 = 5^2 = 25 \]
Now, combine all calculated parts back into the expression: \[ (16) + \left(\frac{12}{25}\right) - (25) \]
Next, we'll convert everything to a common denominator:
- The common denominator of \(25\) will be used.
Convert \(16\) and \(25\) to have a denominator of \(25\): \[ 16 = \frac{16 \times 25}{25} = \frac{400}{25}, \quad 25 = \frac{25 \times 25}{25} = \frac{625}{25} \]
Now, substitute back into the expression: \[ \frac{400}{25} + \frac{12}{25} - \frac{625}{25} \]
Combine these fractions: \[ \frac{400 + 12 - 625}{25} = \frac{412 - 625}{25} = \frac{-213}{25} \]
Hence, we have the simplified numerator \( -213 \).
However, returning to the original task of finding only the numerator, the correct answer stated in the options is:
- 54
- 76
- 42 + 60
- 100
None of these appear correct based on my calculations. Please verify the initial expression and the response options again. If needed, I can reassess the expression.