Question

对所有非零数，运算 $\otimes$ 定义为 $a\otimes b =\dfrac{a^{2}}{b}\text{。}$

求 $[(1\otimes 2)\otimes 3]-[1\otimes (2\otimes 3)]\text{。}$

The operation $\otimes$ is defined for all nonzero numbers by $a\otimes b =\dfrac{a^{2}}{b}.$

Determine $[(1\otimes 2)\otimes 3]-[1\otimes (2\otimes 3)].$

$-\dfrac{2}{3}$

$-\dfrac{1}{4}$

$0$

$\dfrac{1}{4}$

$\dfrac{2}{3}$

答案：A

知识点：自定义运算分数

难度评级：1410

小提示：

从内到外计算每个 $\otimes$ 表达式。

Evaluate each $\otimes$ expression from the inside out

大提示：

保留括号；这个运算不满足结合律。

Keep the parentheses; the operation is not associative

解答：

计算如下： $\begin{gather*} [(1\otimes 2)\otimes 3]-[1\otimes (2\otimes 3)] \\ = [\dfrac{1^2}{2} \otimes 3] - [1 \otimes \dfrac{2^2}{3}] \\ = [\dfrac{1}{2} \otimes 3] - [1 \otimes \dfrac{4}{3}] \\ = \dfrac{(\frac{1}{2})^2}{3} - \dfrac{1^2}{(\frac{4}{3})} \\ = \dfrac{1}{4} \cdot \dfrac{1}{3} - \dfrac{3}{4} \\ = \dfrac{1}{12} - \dfrac{3}{4} \\= -\dfrac{2}{3} \end{gather*}\text{。}$

所以正确答案是 A。

We can calculate it as follows. $\begin{gather*} [(1\otimes 2)\otimes 3]-[1\otimes (2\otimes 3)] \\ = [\dfrac{1^2}{2} \otimes 3] - [1 \otimes \dfrac{2^2}{3}] \\ = [\dfrac{1}{2} \otimes 3] - [1 \otimes \dfrac{4}{3}] \\ = \dfrac{(\frac{1}{2})^2}{3} - \dfrac{1^2}{(\frac{4}{3})} \\ = \dfrac{1}{4} \cdot \dfrac{1}{3} - \dfrac{3}{4} \\ = \dfrac{1}{12} - \dfrac{3}{4} \\= -\dfrac{2}{3}. \end{gather*}$

Thus, A is the correct answer.

2000 AMC 8 第 17 题

答案：A

难度评级：1410

小提示：

大提示：

解答：

其他年份的第 17 题