To simplify the expression \((16 - 12)^2 + 5(8 + 22)53 - (1 + 4)^2\), let's break it down step by step:
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Simplify \( (16 - 12)^2 \): \[ 16 - 12 = 4 \implies (4)^2 = 16 \]
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Simplify \( 5(8 + 22)53 \): \[ 8 + 22 = 30 \implies 5 \cdot 30 \cdot 53 \] We need to calculate \( 5 \cdot 30 \): \[ 5 \cdot 30 = 150 \] Now, multiply \( 150 \cdot 53 \): \[ 150 \cdot 53 = 7950 , (\text{calculated as } 150 \cdot (50 + 3) = 7500 + 450) \]
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Simplify \( (1 + 4)^2 \): \[ 1 + 4 = 5 \implies (5)^2 = 25 \]
Now, putting it all together in the original expression: \[ (16) + (7950) - (25) = 16 + 7950 - 25 \]
Calculating this step by step: \[ 16 + 7950 = 7966 \] Then, \[ 7966 - 25 = 7941 \]
So the simplified numerator is \( 7941 \).
None of the response options given match \( 7941 \), indicating there may have been an error in how the responses were structured.