AMC 8 2000 Problem 17

AMC8 2000 年第 17 题

表达式与运算★★★☆☆
The operation ⊗\otimes is defined for all nonzero numbers by a⊗b=a2b\displaystyle a\otimes b =\frac{a^{2}}{b} . Determine [(1⊗2)⊗3]−[1⊗(2⊗3)][(1\otimes 2)\otimes 3]-[1\otimes (2\otimes 3)] .
运算 ⊗\otimes 对所有非零数定义为 a⊗b=a2b\displaystyle a\otimes b = \frac{a^{2}}{b}。求 [(1⊗2)⊗3]−[1⊗(2⊗3)][(1\otimes 2)\otimes 3]-[1\otimes (2\otimes 3)] 的值。
  1. A.−23\displaystyle -\frac{2}{3}
  2. B.−14\displaystyle -\frac{1}{4}
  3. C.0
  4. D.14\displaystyle \frac{1}{4}
  5. E.23\displaystyle \frac{2}{3}

正确答案 · Correct Answer

A

解析 · Solution

Solution 1

Follow PE(MD)(AS), doing the innermost parentheses first. [(1⊗2)⊗3]−[1⊗(2⊗3)][(1\otimes 2)\otimes 3]-[1\otimes (2\otimes 3)], [122⊗3]−[1⊗223]\displaystyle [\frac{1^2}{2} \otimes 3]-[1\otimes \frac{2^2}{3}], [12⊗3]−[1⊗43]\displaystyle [\frac{1}{2} \otimes 3]-[1\otimes \frac{4}{3}], [(12)23]−[12(43)]\displaystyle [\frac{(\frac{1}{2})^{2}}{3}]-[\frac{1^2}{(\frac{4}{3})}], [14⋅13]−[34]\displaystyle [\frac{1}{4} \cdot \frac{1}{3}]-[\frac{3}{4}], 112−34\displaystyle \frac{1}{12} - \frac{3}{4}, 112−912\displaystyle \frac{1}{12} - \frac{9}{12}, −812\displaystyle \frac{-8}{12}, −23\displaystyle -\frac{2}{3}, which is answer AA

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