To evaluate the expression \( 7 \cdot (6 + 2)^2 - 3^2 \), we will follow the order of operations (PEMDAS/BODMAS: Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), Addition and Subtraction (from left to right)).
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Calculate the expression inside the parentheses: \[ 6 + 2 = 8 \]
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Now, substitute back into the expression: \[ 7 \cdot 8^2 - 3^2 \]
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Calculate the exponents: \[ 8^2 = 64 \] \[ 3^2 = 9 \]
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Substituting these values back into the expression: \[ 7 \cdot 64 - 9 \]
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Now, perform the multiplication: \[ 7 \cdot 64 = 448 \]
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Finally, perform the subtraction: \[ 448 - 9 = 439 \]
Thus, the value of the expression \( 7 \cdot (6 + 2)^2 - 3^2 \) is \(\boxed{439}\).