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

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:).
RELATED QUESTIONS
if x^y=y^x then prove that... (answered by venugopalramana)

Prove that if |x + y| = |x| + |y| then xy >... (answered by ikleyn)

If x=e^y-e^(-y)/e^y+e^(-y) then prove that... (answered by jsmallt9)

if x+1/y=1 & y+1/z=1 then prove that... (answered by Fombitz)

Prove that |x + y| ≦ |x| +... (answered by AnlytcPhil)

Prove that |x+y| ≤... (answered by Edwin McCravy)

Prove... (answered by Fombitz)

Please help prove "if x < y, then... (answered by jim_thompson5910)

If log (x+y)/2 = 1/2 (log x + log y) prove that x =... (answered by ikleyn)