Question 1113751
By inspection, there is a trivial answer  x = y = 1.

But there is also a nontrivial answer.
——

{{{ x^2 = y^3 }}} —>  {{{ y = matrix(3,3,"","","","",x^((2/3)),"", "", "" ,"") }}}

{{{ x^y = y^x }}} and the above —> {{{ matrix(3,3,"", "", "", "", "", "", "", x^((x^(2/3))),"") }}}  = {{{ matrix(3,3,"","","","",(x^(2/3))^x,"", "", "" ,"") }}} <br>

Taking logs:

   {{{ matrix(3,3, "", "", "", "", x^(2/3) *ln(x), "", "", "", "") =  x*(2/3)*ln(x) }}}

   {{{ matrix(3,3, "", "", "", "", x^(-1/3) , "", "", "", "") = 2/3 }}}  —>  {{{ x = (3/2)^3  = 27/8 }}}  —>  {{{ y = 9/4 }}}
—

 Ans:  {{{  highlight( x = 27/8 ) }}}  —>   {{{ highlight(  y = 9/4 ) }}}
—

<br>
Check:
                {{{ x^2 = (27/8)^2 = 729/64 }}}
               {{{ y^3 = (9/4)^3 = 729/64   }}} (ok) <br>

               {{{ x^y  =  (27/8)^(9/4) =  ( 3^27 / 2^27 ) ^(1/4)  }}}
               {{{ y^x =   (9/4)^(27/8)  = ( 3^54 / 2^54 ) ^(1/8) =  ( 3^27 / 2^27 ) ^(1/4) }}}  (ok)


———
EDITED:  removed statement that x=y=0 is a trivial answer, it is not (0/0 form).