Well it would equal 3xyz/(a-b)^3 + (b-c)^3 + (c-a)^3 wouldnt it? I dont know how to simplify the denomiator.
$\displaystyle (\sqrt{a}- \sqrt{b})^2>0$
$\displaystyle \Rightarrow a+b-2 \sqrt{ab}>0$
$\displaystyle \Rightarrow a+b>2 \sqrt{ab}$ .......(1)
similarly $\displaystyle b+c> 2 \sqrt{bc}$ ......(2)
and $\displaystyle c+a > 2 \sqrt{ca}$ ...... (3)
multiply the above three inequalities.