Originally posted by Fat Ladysee-simple
a and b are rational. Therefore a can be written a1/a2, b as b1/b2 (a1,a2,b1,b2 all integers).
Assume that c is rational too. So c can be written as c1/c2 (c1 and c2 integers).
(a1/a2)^n + (b1/b2)^n = (c1/c2)^n
Multiply both sides by (a2*b2*c2)^n
This gives (a1*b2*c2)^n + (b1*a2*c2)^n = (c1*a2*b2)^n
Clearly a1*b2*c2, b1*a2*c2 and c1*a2*b2 are ...[text shortened]... we know has no integer solutions.
Therefore our assumption that c is rational must be false.