(In reply to Not sure if rigorous but...
by Kenny M)
"take the 0.4th root of both sides:
x^0.4^(x/0.4) = 0.4^x"
The given equation is X^(0.4^X) = 0.4^(X^0.4)
Taking the 0.4th root on both sides, we have:
X^((0.4^X)/0.4) = 0.4^((X^0.4)/0.4)
Or, X^((0.4^(X-1)) = 0.4^((X^0.4)/0.4)
*** Your error lies in assuming that X^(0.4^X) = (X^0.4)^X, which is not true since:
(X^0.4)^X = X^(0.4*X), and we cannot have:
X^(0.4^X) = X^(0.4*X)
*** The value of a^(b^c) as defined in the problem is similar to the wikipedia article on integer exponentiation in the following location.
Edited on March 11, 2008, 2:58 am