Question 37838: Prove that if x^y=y^x then (x/y)^(x/y)=x^((x/y)-1)
Answer by ilana(307)   (Show Source):
You can put this solution on YOUR website! 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:).
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