a square root of an irrational number

Question

I wonder if a square root of an irrational number is always irrational?

I would tend to think that yes, but I can´t think of any justification. Also there are cases which are rather hard to decide like sqrt(Pi).

Answer 1

Yes. The square of a rational is rational, so the identity $x = (\sqrt{x})^2$ tells us that if the square root is rational, the original number must be too.