Question

this is my question!!

Answer 1

Let's derive the answer. We have ...

A^(1/A) = B^(1/B) = C^(1/C) .......(1)
=> A^(BC) = B^(AC) = C^(AB) = X, some variable. [ By raising to power ABC ]

Now, let's take the second equation..

A^(BC) + B^(AC) + C^(AB) = 729
=> 3X = 729
=> X = 243 ....... (2)

Now, think about equation (1). Obviously, for such a condition to satisfy, the following should hold...
A = B = C = K (say)

Now from (2), we can say...
K^(K*K) = 243

Think of this now. Assume, K is even, then there is no way we can have a number 243. If K is odd, what you have is some odd number raised to an even number. But this is in contrast with what we should be having, because we have a number with 3 in the unit place. Thus, K wont belong to the  set of whole numbers.
 
Lets have a different approach, we have  ....
K^(K*K) = K^(K^2) = 243
=> K^(K^2) = 3^5

Now its clear from this that there could be no possible solution to this.

:) Hope this Helps!!!