You need to change the mid term such that 6`root(3)(a^3)* sqrt b` to become `6 root(3)(a)*root(3)(b)`

The denominator is now changed into `4 root(3)(a^2) - 6 root(3)(a)*root(3)(b) + 9 root(3)(b^2)` .

Multiplying the denominator by `2 root(3)(a) + 3 root(3) (b)` yields the difference of cubes 8a + 27b.

1/`(4 root(3)(a^2) - 6 root(3)(a)*root(3)(b) + 9 root(3)(b^2))` =`(2 root(3)(a) + 3 root(3) (b))/((4 root(3)(a^2) - 6 root(3)(a)*root(3)(b) + 9 root(3)(b^2))*(2 root(3)(a) + 3 root(3) (b)))`

`1/(4 root(3)(a^2) - 6 root(3)(a)*root(3)(b) + 9 root(3)(b^2))` =`(2 root(3)(a) + 3 root(3) (b))/(8a+27b)`

**Rationalizing the given expression yields `(2 root(3)(a) + 3 root(3) (b))/(8a+27b)` .**