To simplify the expression \((16−12)^2 + 5(8 + 2^2) / 5^3 − (1 + 4)^2\), let's break it down step by step.
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Calculate \( (16 - 12)^2 \): \[ 16 - 12 = 4 \implies (4)^2 = 16 \]
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Calculate \( 2^2 \): \[ 2^2 = 4 \]
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Calculate \( 8 + 2^2 \): \[ 8 + 4 = 12 \]
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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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Now compute \( \frac{5(8 + 2^2)}{5^3} \): \[ \frac{60}{125} \] To simplify \( \frac{60}{125} \), divide the numerator and denominator by 5: \[ \frac{60 \div 5}{125 \div 5} = \frac{12}{25} \]
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Calculate \( (1 + 4)^2 \): \[ 1 + 4 = 5 \implies (5)^2 = 25 \]
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Now, combine everything in the original expression: \[ (16 - 12)^2 + \frac{5(8 + 2^2)}{5^3} - (1 + 4)^2 \] becomes: \[ 16 + \frac{12}{25} - 25 \]
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Combine the whole numbers: \[ 16 - 25 = -9 \]
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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.
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