x = a + b ( 5^(1/3)) + c (25^(1/3));
y = d + e (5^(1/3)) + f (25^(1/3));

x*y // Expand

(1/x) // FullSimplify

ausdruck = ((x*y - 1) // 
            Expand) /. {(5^(1/3)) -> α, (25^(1/3)) -> (α^2)}

liste = 
        CoefficientList[ausdruck, α];
sol = Solve[liste == 0, {d, e, f}]

(x*y /. sol[[1]]) // Simplify