\text{This is a very common question type. So what we do is let} \\ x = \alpha ^{1/3} - \beta ^{1/3} \\ x^3 = (\alpha - \beta) -3\alpha ^{1/3} \beta ^{1/3}(\alpha ^{1/3} - \beta ^{1/3}) (Via binomial expansion).
x^3+3x\alpha ^{1/3} \beta ^{1/3} = (\alpha - \beta)
$ Now we simply...