Question 37838
Oh, this is a fun 1!!
x^y=y^x, so x^y/y^x=1.  We want it to look like x^(x/y)/y^(x/y), so first we can raise each side of our equation to 1/y.  So x^y/y^x=1 becomes (x^y)^(1/y)/(y^x)^(1/y)=1^(1/y), so x/(y^(x/y))=1.  Now we just need x^(x/y) in the numerator.  If we multiply (a^1)*(a^x), we get a^(x+1).  So to get from x^1 to x^(x/y), we use the equation 1+m(or any variable)=x/y, so m=x/y-1.
So finally, we multiply our equation by x^((x/y)-1), and:
x/(y^(x/y))=1 goes to x*x^((x/y)-1)/(y^(x/y))=1*x^((x/y)-1), so:
(x*(x/y))/(y^(x/y)) = x^((x/y)-1), which completes our proof:).