Ugh, that's a messy one. Let's try to control the mess by writing #a=\sqrt{x}# so #x=a^2# and #b=sqrt{y}# so #y=b^2#.
# ( frac{ x sqrt{x} - y sqrt{y} }{x - sqrt{x} sqrt{y} } - frac { x- y} {sqrt x + sqrt y} )( frac{sqrt{x} sqrt{y} + 2x }{3x})^{-1} #
= # ( frac{ a^3 - b^3}{a^2 -ab } - frac { a^2-b^2} {a+b} )( frac{a b + 2a^2 }{3a^2})^{-1} #
Well that's less messy already, isn't it? It's easier to see the factors this way:
= # ( frac{ (a-b)(a^2 + ab + b^2) }{a(a-b)} - frac { (a+b)(a-b)} {a+b} )( frac{3a^2}{a( 2a + b) }) #
We got some cancelling we can do,
# = ( frac{ a^2 + ab + b^2 }{a} - (a-b) )( frac{3a}{ 2a + b }) #
# = ( frac{ a^2 + ab + b^2 - a^2 + ab}{a} )( frac{3a}{ 2a + b }) #
# = ( frac{ b(2a + b) }{a} )( frac{3a}{ 2a + b }) #
# = 3 b #
# = 3 sqrt y #
