How is your problem solving skills?

Question

Find all positive rational number pairs, not equal, which satisfy the following equation (write a formula which gives all solutions):

x^y=y^x

Hint: Let k = y/x, then y = kx. Substitute for y and take the xth root of both sides.

Answer 1

OK Jerry, I like this one, though the testing is getting tiresome.

If x and y are both rational, then there must be some a,b such that x=ay/b. In fact, a and b can be relatively prime if x and y are different.

x^y=y^x=> (ay/b)^y=y^(ay/b)=>((ay/b)^b/a)^ay/b=y^(ay/b)
=>(ay/b)^(b/a)=y
Therefore ay/b=y^(a/b)
y=b/a*y^(a/b)
bth root of y ath power of y>ay
and for a not equal to b this can't be a linear deviation, so the two errors won't offset.

This suggests to me the only positive answers are a=b so x=y.

Answer 2

1. I analyze every statement of the problem because these statements are your scenarios.

2. I always list all the given parameters.

3. Identify what is "required" in the problem

4. List the formula

5. Substitution

6. Solution

7. Final Answer

Answer 3

either x is 1 or y is 1

Answer 4

I've been working on this for a long time and here's all I've got so far:

y=2x

for any even integer x, and of course

x=2y
for any even integer y.

Is that all?

Answer 5

x=y

That would be the only formulaic solution to the formula you listed.

Answer 6

Excellent!

And you should have asked, "How ARE your problem solving skills" ... see problem solved.

Answer 7

ylnx = xlny
y/x = lny / lnx

Answer 8

amkind of bad at maths don't even ask!!!!!!!!!!1