Is there a method for finding two irrational numbers that multiply together to form a rational number

Question

I know multiplying $x*\frac{1}{x}$ gives us the rational number 1 as long as x is nonzero. I also know the square of rational numbers multiplied by itself gives us a rational number. Are these the only way of finding irrational numbers that multiply to a rational one? What about numbers $a$ and $b$ where $b\ne\frac{1}{a}$? Do such numbers exist?

Answer 1

If you want $wv = r$ where $w,v$ are irrational and $r$ is rational, that must mean that $w =\frac rv$ (since $v$ is irrational it isn't $0$ so we don't have to worry about that)

So to find any pair you want just pick any arbitrary rational number (so long as it is not zero; as $w \ne 0$ and $v\ne 0$ then $wv \ne 0$; but that is the only restriction; we can pick any other rational at all), and pick any arbitrary irrational $v$. Then let $w = \frac rv$. As $v$ is irrational and $r$ is non-zero rational we will have $w = \frac rv$ be irrational.

And we have, voila $wv = \frac rv \cdot v = r$.