I was exploring tetration and came across the following identities:
$${^0}(ab) = 1$$ $${^2}(ab) = ({^2}a)^b * ({^2}b)^a$$ $${^3}(ab) = (ab)^({^2}(ab)) = (ab)^{(({^2}a)^b * ({^2}b)^a)}$$
That third identity I would like to close up into a nicer form:
But I don't know how:
what I mean by nicer form is that I want to have:
${^3}a$ and ${^3}b$ in the expression so I can find some sort of binomial theorem generalization for tetration. (Here we work with products as opposed to sums)