What are the implications of binomials with tetration?

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The discussion explores the implications of tetration in relation to binomials and its failure to distribute over multiplication, similar to how exponentiation does not distribute over addition. Participants question how tetration interacts with multiplication, specifically asking about the expression (xy) tetration squared. There is curiosity about whether there are elegant patterns to study within this framework or if the concepts are trivial or unsolvable. The conversation references Knuth's up-arrow notation as a potential tool for understanding these operations. Overall, the implications of tetration on binomials remain an open area for exploration.
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"Binomials" with tetration

With the hyperoperations of addition, iterated addition (multiplcation), which distributes over addition, and iterated multiplication (exponentiation), which distributes over multiplcation, we can study how exponentiation fails to distribute over the operation which is two iterative operationsbelow it, addition...Now in extending to tetration, how would tetration fail to distribute over multiplication ? What is (xy) tetra squared? Is there a system of elegant patterns to study, or is it trivial, unsolvable, etc.?
 
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