To rationalize a fraction, you have to multiply the numerator and denominator by the same expression such that the denominators are only integers (have no more radicals)
When the denominator is a cubic binomial, you would multiply the numerator and denominator both by the radical's conjugate. For the denominator cuberoot(3) - 1, the conjugate would be cuberoot(9) + cuberoot(3) +1.
Distributing the conjugate to the denominator would leaves you with
cuberoot(27)+cuberoot(9)+cuberoot(3)-cuberoot(9)-cuberoot(3) - 1.
The middle terms cancel leaving cuberoot(27) - 1 which equals 3 -1 = 2.
Your final rationalized fraction would be
(cuberoot(9) + cuberoot(3) + 1) / 2
