Problem:
The operation ⊗ is defined for all nonzero numbers by a⊗b=ba2​. Determine [(1⊗2)⊗3]−[1⊗(2⊗3)].
Answer Choices:
A. −32​
B. −41​
C. 0
D. 41​
E. 32​
Solution:
We have
(1⊗2)⊗3=212​⊗3=21​⊗3=3(21​)2​=341​​=121​
and
1⊗(2⊗3)=1⊗(322​)=1⊗34​=34​12​=43​
Therefore,
(1⊗2)⊗3−1⊗(2⊗3)=121​−43​=121​−129​=−128​=−32​.
Answer: A​.
The problems on this page are the property of the MAA's American Mathematics Competitions