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.
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Start with the first term: \[ (16 - 12)^2 = 4^2 = 16 \]
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Simplify \(5(8 + 2^2)\): \[ 2^2 = 4 \quad \text{so} \quad 8 + 4 = 12 \] Then, \[ 5(12) = 60 \]
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The full second part before the division is: \[ 60 / 5^3 \] Calculate \(5^3\): \[ 5^3 = 125 \] So the second part now is: \[ 60 / 125 = \frac{60}{125} \]
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Now, simplify \(\frac{60}{125}\). The greatest common divisor (GCD) of 60 and 125 is 5: \[ \frac{60 \div 5}{125 \div 5} = \frac{12}{25} \]
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Simplify the last term: \[ (1 + 4)^2 = 5^2 = 25 \]
Now, combine these results into the overall numerator: \[ 16 + \frac{12}{25} - 25 \]
To combine these, convert \(16\) and \(-25\) to fractions with a common denominator, which will be \(25\): \[ 16 = \frac{16 \times 25}{25} = \frac{400}{25} \] \[ -25 = \frac{-25 \times 25}{25} = \frac{-625}{25} \]
Now, add these fractions: \[ \frac{400}{25} + \frac{12}{25} - \frac{625}{25} = \frac{400 + 12 - 625}{25} = \frac{412 - 625}{25} = \frac{-213}{25} \]
Thus, the simplified numerator is: \[ \boxed{-213} \]