Explanation

Put s=1/a+1/b=5/6 and t=1/(ab). Then 1/a^2+1/b^2=s^2-2t=13/36 gives 25/36-2t=13/36, so t=1/6. Now 1/a^3+1/b^3=s^3-3st=125/216-3(1/6)(5/6)=125/216-90/216=35/216. Forgetting the -3st cross term is the usual error.