For example, any number tetrated to a positive integer can be calculated by just doing repeated exponentiation from top to bottom. Similarly, it is possible to calculate values of $^{0.5}2$, $^{3}3$, $^{1.2}4$, $^{1.5}2$, etc, i.e., when the fractional part of the tetration exponent is of the form 1/n where n is a positive integer. I mean, it is possible to get a unique value of these. But for expressions like $^{0.3}2$, it is not possible to solve it without using the extensions which have been proposed for tetration, and these extensions usually give different answers. So, is there some other category of fractional tetration exponents such that the expressions involving them can be uniquely determined without using the extensions of the tetration function?


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