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Algebra Difficulty 3.1 AMC 10/12 Find the answer

Calculate: (1)202412×[8(2)2](-1)^{2024}-\frac{1}{2} \times [8-(-2)^{2}].

A number or a short expression. Spacing and $ signs are ignored.

Solution

To calculate (1)202412×[8(2)2](-1)^{2024}-\frac{1}{2} \times [8-(-2)^{2}], we follow these steps:

1. Evaluate (1)2024(-1)^{2024}: Since any even power of 1-1 is 11, we have (1)2024=1(-1)^{2024} = 1.
2. Evaluate (2)2(-2)^{2}: Squaring 2-2 gives (2)2=4(-2)^{2} = 4.
3. Substitute and simplify inside the brackets: 8(2)2=84=48-(-2)^{2} = 8-4 = 4.
4. Multiply by 12\frac{1}{2}: 12×4=2\frac{1}{2} \times 4 = 2.
5. Subtract from the first part: 12=11 - 2 = -1.

Putting it all together in a detailed step-by-step calculation:

(1)202412×[8(2)2]=112×(84)=112×4=12=1. \begin{align*} (-1)^{2024}-\frac{1}{2} \times [8-(-2)^{2}] & = 1-\frac{1}{2} \times \left(8-4\right) \\ & = 1-\frac{1}{2} \times 4 \\ & = 1-2 \\ & = -1. \end{align*}

Therefore, the final answer is 1\boxed{-1}.

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Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.