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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).

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The square of a rational number is rational, so the square root of an irrational number must be irrational. – Daniel Fischer Sep 20 '13 at 18:29
Great answer. How come I didn´t think of this? :-D – Adam Sep 20 '13 at 18:31
up vote 3 down vote accepted

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.

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