(06/27/2010, 09:50 PM)moejoe Wrote: BUT, taking my 2nd example for why a^0=1 in exponentiation - this doesn't work at all with tetration because if you do the following:

"a^^k = y" and "a = y^^(1/k)"

...

So to me this all seems like some kind of paradox is happening but I hope that someone will be able to put this in a clear light.

I think your paradox emerges from the wrong assumption in your equations.

You dont get the inverse of f(x)=x^^k (i.e. the tetration root) by taking x^^(1/k).

This is the case for powers but not for tetration.

It works for x^k because we have for integer numbers.

If we extend this law to rational numbers we can set which means that is the inverse of , i.e. is the number such that .

All this consideration fails for tetration because we dont have the rule x^^(m*n)=(x^^m)^^n.