MHB We can find two irrational numbers x and y to make xy rational,true or false

[Albert1]

we can find two irrational numbers $x$ and $y$
to make $x^y$ rational,true or false statement?
if true then find else prove it .

 

[kaliprasad]

we can find two unreasonable numbers $x$ and $y$
to make $x^y$ reasonable,true or false statement?
if true then find else prove it .

what is a reasonable number ?

 

[Albert1]

what is a reasonable number ?

Spoiler

sorry it should be edited as :
$x,y $ irrational numbers
$ x^y$ rational number

 

[HallsofIvy]

Here's a famous example. Let x= y= \sqrt{2}. Either x^y= \sqrt{2}^\sqrt{2} is irrational or it is rational. If it is rational we are done. If it is irrational, let x= \sqrt{2}^\sqrt{2} and y= \sqrt{2}. Then x^y= (\sqrt{2}^\sqrt{2})^\sqrt{2}= \sqrt{2}^{(\sqrt{2}\sqrt{2})}= \sqrt{2}^2= 2. In either case, there exist two irrational numbers, x and y, such that x^y is rational.

 

[Evgeny.Makarov]

Another example is $\sqrt{2}^{\log_29}=3$.