How to represent fraction $\frac{1}{a+b\cdot \sqrt[3]{2} + c\cdot (\sqrt[3]{2})^2}$ as $a_1 + b_1 \cdot \sqrt[3]{2} + c_1 \cdot (\sqrt[3]{2})^2$?

Question

What do I have to multiply both numerator and denominator with to get the representation is asked? $a, b, c, a_1, b_1, c_1$ are rational numbers.

Answer 1

I think, the easiest if you solve this: $$(a_1+b_1\cdot \sqrt[3]2 + c_1\sqrt[3]{2^2})(a+b\cdot \sqrt[3]2 + c\sqrt[3]{2^2}) =1 $$