Determine all possible positive real numbers X that satisfy this equation:

X^0.4X = 0.4^X0.4

Note: For the purposes of the problem, a^bc is equal to a^(b^c)

(In reply to Not sure if rigorous but... by Kenny M)

You say:

"take the 0.4th root of both sides:

x^0.4^(x/0.4) = 0.4^x"

But:

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.
http://en.wikipedia.org/wiki/Exponentiation#Identities_and_properties

Regards,

K Sengupta

 

Edited on March 11, 2008, 2:58 am

Posted by K Sengupta on 2008-03-11 02:52:02