Question 9
(FRA 4) For all integers $n$, $n\ge0$, there exist uniquely determined integers $a_n,b_n,c_n$ such that
$$\left(1+4\sqrt[3]2-4\sqrt[3]4\right)^n=a_n+b_n\sqrt[3]2+c_n\sqrt[3]4.$$
Prove that $c_n=0$ implies $n=0$.
$$\left(1+4\sqrt[3]2-4\sqrt[3]4\right)^n=a_n+b_n\sqrt[3]2+c_n\sqrt[3]4.$$
Prove that $c_n=0$ implies $n=0$.