SOLUTION: if x^y=y^x then prove that (x/y)^(x/y)=x^((x/y)-1)

Question 37586: if x^y=y^x then prove that (x/y)^(x/y)=x^((x/y)-1)
Answer by venugopalramana(3286) (Show Source): You can put this solution on YOUR website!
if x^y=y^x then prove that (x/y)^(x/y)=x^((x/y)-1)
LET X^Y=Y^X=K
SO...X=K^(1/Y)
Y=K^(1/X)
LET X/Y=Z={K^(1/Y)}/{K^(1/X)}=K^{(1/Y)-(1/X)}=K^{(X-Y)/XY}
SO T.P.T......Z^Z=X^(Z-1)
T.P.T..........[K^{(X-Y)/XY}]^Z=[K^(1/Y)]^(Z-1)
T.P.T.........Z(X-Y)/XY=(Z-1)/Y
HENCE
LHS =Z(X-Y)/XY=X(X-Y)/XY*Y=(X-Y)/Y^2
RHS=(Z-1)/Y={(X/Y)-1}/Y=(X-Y)/Y^2=LHS...PROVED
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