Think about it. There is a rule that the square root of a * the square root of b = the square root of ab. This applies to cube roots as well because what is really happening is that a cube root is the same thing as x1/3. When you multiply several of those together, x^1/3*y1/3 = (xy)1/3. You can combine terms like that.
You also might find it easier if you factor out common multiples at the beginning. You can factor out 6 for the first bracket and 6^(1/3) from the second bracket. Doing this makes your job easier, because you are multiplying to smaller numbers. Also keep track of when you can take things out of roots. Cube root 8 is 2, cube root r^3 is r and so on.
Sounds like you're on the right path. If you're having a hard time with these complex terms, you can try replacing the roots with variables and then plugging them back in when you're done. Easier on the brain.