To solve the expression \( 4^{2 - 3 \cdot 4 + 7} \), follow the order of operations (PEMDAS/BODMAS: Parentheses/Brackets, Exponents/Orders, Multiplication and Division, Addition and Subtraction).
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Start with the subtraction and multiplication inside the exponent: \[ 2 - 3 \cdot 4 + 7 \] First, calculate \( 3 \cdot 4 \): \[ 3 \cdot 4 = 12 \]
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Now, substitute this back into the expression: \[ 2 - 12 + 7 \]
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Calculate \( 2 - 12 \): \[ 2 - 12 = -10 \]
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Now substitute -10 back into the expression: \[ -10 + 7 = -3 \]
Now, we have: \[ 4^{-3} \]
- Finally, evaluate \( 4^{-3} \): \[ 4^{-3} = \frac{1}{4^3} = \frac{1}{64} \]
Thus, the final result is: \[ \frac{1}{64} \]