Two irrational numbers, such that their sum is rational and their product is irrational??

Question

Give Example of two irrational numbers, such that their sum is rational and their product is irrational:

Answer 1

Fourth root of two and negative fourth root of two.

Answer 2

Dumb example: $\pi$ and $-\pi$

Answer 3

Try $x=1+\pi$ and $y=1-\pi$ then $$x+y=2$$ and $$xy=1-\pi^2$$