<s>[INST] Find two real numbers x and y such that x, y, x + y, x^2, y^2, x^2 - y^2, x^3, y^3, x^4, y^4 are all irrational but x^2 + y^2 is rational. Find two real numbers $x$ and $y$ such that $x, y, x + y, x^2 , y^2 ,  x^2 - y^2, x^3, y^3, x^4, y^4$ are all irrational but $x^2 + y^2$ is rational.
What is the approach to solve this problem? It cannot be guess and check, right?
(Shout-outs to CI students. D is for diploma!)
 [/INST] If we want to avoid transcendentals, use $x=\sqrt{3}+\sqrt{2}$ and $y=\sqrt{3}-\sqrt{2}$. 
The price is that we need to work a little harder to prove the irrationality of $x$, $y$, $x+y$, $x^2$, $y^2$, $x^2-y^2$, $x^3$, $y^3$, $x^4$ and $y^4$. But not much harder, there is a lot of symmetry. 
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