To simplify the expression \([(−4 + 2) − (7^1 + 5)] − (−5) − 5\), we'll use the order of operations (parentheses, exponents, multiplication and division (from left to right), addition and subtraction (from left to right)).
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Start with the expression: \[ [(−4 + 2) − (7^1 + 5)] − (−5) − 5 \]
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Calculate \(−4 + 2\): \[ −4 + 2 = −2 \]
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Calculate \(7^1 + 5\): \[ 7^1 = 7 \quad \text{so} \quad 7 + 5 = 12 \]
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Substitute back into the expression: \[ [−2 − 12] − (−5) − 5 \]
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Now simplify inside the brackets: \[ −2 − 12 = −14 \]
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Substitute back: \[ [−14] − (−5) − 5 \]
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Now handle the negative: \[ −14 + 5 − 5 \]
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Combine \(−14 + 5\): \[ −14 + 5 = −9 \]
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Finally, subtract \(5\): \[ −9 − 5 = −14 \]
Thus, the simplified expression is: \[ \boxed{−14} \]