Let's break down the expression step by step to find the simplified numerator of \( (16−12)^2 + 5(8+22)/53−(1+4)^2 \).
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Simplify \( (16 - 12)^2 \): \[ (4)^2 = 16 \]
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Simplify \( 5(8 + 22) \): \[ 8 + 22 = 30 \quad \Rightarrow \quad 5(30) = 150 \]
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The denominator is \( 53 \), thus we have: \[ \frac{150}{53} \]
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Simplify \( (1 + 4)^2 \): \[ (5)^2 = 25 \]
Now we can plug these values back into the expression: \[ (16) + \frac{150}{53} - (25) \]
Combine the values: \[ 16 - 25 = -9 \] So the expression now looks like: \[ -9 + \frac{150}{53} \]
To have a common denominator for these two parts:
- Convert \(-9\) to a fraction with denominator \(53\): \[ -9 = -\frac{477}{53} \]
Now we combine: \[ -\frac{477}{53} + \frac{150}{53} = \frac{-477 + 150}{53} = \frac{-327}{53} \]
The entire numerator we simplified is \(-327\).
However, the question is asking for the simplified numerator of the expression: \[ (16 - 12)^2 + 5(8 + 22) - (1 + 4)^2. \]
Calculating this: \[ 16 + 150 - 25 = 141 \]
So the simplified numerator of the given expression is:
141
Review the options again: None of the choices listed before (100, 76, 54, or 42+60) are correct, as the simplified numerator is 141.
There appears to be a discrepancy in the response options.