prooving irrational n rational numbers.?

Question

let a=(sqr root 2) ^ (sqr root 2). simplify a ^(sqr root 2). Conclude that there are examples of irrational numbers x and y such that x^y is rational.

Answer 1

Either sqrt(2)^(sqrt(2)) is rational or it is irrational.

If sqrt(2)^(sqrt(2)) is rational, then setting x = sqrt(2) and y = sqrt(2) gives an example of irrational numbers x and y such that x^y is rational.

Otherwise, if sqrt(2)^(sqrt(2)) is irrational, then look at (sqrt(2)^(sqrt(2)))^(sqrt 2). By a property of exponents, this equals (sqrt(2))^(sqrt(2) * sqrt(2)), which equals (sqrt(2))^2, which equals 2, which is certainly rational. So if sqrt(2)^(sqrt(2)) is irrational, then x = sqrt(2)^(sqrt(2)) and y = sqrt(2) gives an example of irrational numbers x and y such that x^y is rational.

Thus, whether sqrt(2)^(sqrt(2)) is rational or irrational, we can find irrational x and y such that x^y is rational. (The proof doesn't give a definite example of x and y, but instead shows that either one pair or the other above works.)

Answer 2

Let x=e and y=log 5 (here log is base e) which are both irrational.
Then x^y=5, a rational number

Answer 3

A, comes first and the x and y is it rational to forget about Z You do not know that Z is a very valued letter.Do you want us to say petsa and pussle without our beloved Z tho Z came last but is much appreciated.Can you tell me why you omitted Z ?

Answer 4

this whole question seems IRRATIONAL, and if I were RATIONAL, I probably would not even have answered it.

Answer 5

psssssh wat the heck??