To solve the expression \(6 - 4(3 - 1)^2 + 16\), we will follow the order of operations, which is often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right)).
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Evaluate the expression inside the parentheses: \[ 3 - 1 = 2 \]
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Substitute back into the expression: \[ 6 - 4(2)^2 + 16 \]
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Evaluate the exponent: \[ (2)^2 = 4 \]
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Substitute back into the expression: \[ 6 - 4 \cdot 4 + 16 \]
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Perform the multiplication: \[ 4 \cdot 4 = 16 \]
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Substitute back into the expression: \[ 6 - 16 + 16 \]
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Perform the addition and subtraction from left to right:
- First, calculate \(6 - 16\): \[ 6 - 16 = -10 \]
- Then, add \(16\): \[ -10 + 16 = 6 \]
Thus, the final answer is: \[ \boxed{6} \]
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