Precalculus algebra problem about rational and irrational numbers.

Question

Let $ a, b $ be irrational numbers. We know that $ a + b $, $ a^3 + b^3 $ and $ a^2 + b $ are rational.

I have proved that $ ab $, $ a + b^2 $ are also rational. I tried to find some examples: $ (1 - \sqrt{x}, 1 + \sqrt{x}) $, $ (1 - \sqrt[3]{x}, 1 + \sqrt[3]{x}) $, $ (1 - \sqrt[6]{x}, 1 + \sqrt[6]{x}) $, even trigonometric functions.

Answer 1

Take $$(a,b)=\left(\frac{1+\sqrt{r}}{2},\frac{1-\sqrt{r}}{2}\right),$$ where $r\in\mathbb Q$, $r>0$ and $\sqrt{r}\in\mathbb R\setminus\mathbb Q$.