document.write( "Question 37586: if x^y=y^x then prove that (x/y)^(x/y)=x^((x/y)-1) \n" ); document.write( "

Algebra.Com's Answer #25692 by venugopalramana(3286) $\"\"$ $\"About$

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if x^y=y^x then prove that (x/y)^(x/y)=x^((x/y)-1)
\n" ); document.write( "LET X^Y=Y^X=K
\n" ); document.write( "SO...X=K^(1/Y)
\n" ); document.write( "Y=K^(1/X)
\n" ); document.write( "LET X/Y=Z={K^(1/Y)}/{K^(1/X)}=K^{(1/Y)-(1/X)}=K^{(X-Y)/XY}
\n" ); document.write( "SO T.P.T......Z^Z=X^(Z-1)
\n" ); document.write( "T.P.T..........[K^{(X-Y)/XY}]^Z=[K^(1/Y)]^(Z-1)
\n" ); document.write( "T.P.T.........Z(X-Y)/XY=(Z-1)/Y
\n" ); document.write( "HENCE
\n" ); document.write( "LHS =Z(X-Y)/XY=X(X-Y)/XY*Y=(X-Y)/Y^2
\n" ); document.write( "RHS=(Z-1)/Y={(X/Y)-1}/Y=(X-Y)/Y^2=LHS...PROVED \n" ); document.write( "

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