If x and y are irrational and positive, then is x*y irrational?

Sometimes, but not always. For example, if x equals the square root of 2 and y equals the square root of 32, then x*y=8, which is rational. However, if y were instead to equal the square root of 3, then x*y=sqrt6, which is irrational.

Most of the time the answer will be irrational, though.

No, not necessarily. For example sqrt(2) and sqrt(8) are both irrational and positive, but: sqrt(2)*sqrt(8) = 4.

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