$x\cdot y = \sqrt2$. What can be said about $x$ and $y$

Question

Given, $x,y\in\Bbb R$ and $x\cdot y=\sqrt{2}$ .Can $x$ and $y$ be taken as $\sqrt[4]{2}$?

Answer 1

Yes, of course, but $x=1$ and $y=\sqrt2$ are also valid.

You can take $y=\frac{\sqrt2}{x}$ for all $x\neq0.$

Answer 2

Yes, $x=y= \sqrt[4]{2}$ satisfy the equation $xy= \sqrt{2}$. But there are infinitely many other solutions of the equation $xy= \sqrt{2}$.

Answer 3

Here is a geometric point of view of your problem.

All the points lying on the blue curve is the solution of your equation $xy=\sqrt{2}$

Answer 4

$|x+y| \ge 2\sqrt[4]{2}$

and one or both of $x$ and $y$ are irrational.