You can find counterexamples to disprove this claim. We have positive integers that are perfect square numbers; when we take the square root of those numbers, we get an integer.
For example, the square root of 1 is 1, which is an integer. So if y = 1, then the denominator becomes an integer and thus we get a quotient of two integers (since x is also defined to be an integer), the definition of a rational number.
Example: x = 2, y = 1 ends up with [tex] \frac{2}{\sqrt{1}} = \frac{2}{1} [/tex] which is rational. This goes against the claim that [tex] x/\sqrt{y} [/tex] is always irrational for positive integers x and y.
Any integer y that is a perfect square will work to disprove this claim, e.g. y = 1, y = 4, y= 9, y = 16. So it is not always irrational.
