Factor this expression completely. x^3 1/8 - y^3 1/27

This expression is the difference of two cubes:

x^3     y^3
-
2^3     3^3

The rule here is  a^3 - b^3 = (a - b)(a^2 + ab + b^2), so

x^3     y^3                          x^2       x^3y^3        y^2
- = (x/2 - y/3)*( + + )
2^3     3^3                          2^2          6^3          3^2