It is possible to write \frac 2 4 or \frac \pi 4 to get 2/4 and \pi/4, respectively.

However, it is not possible to write \frac \sqrt{2} 2 to get \sqrt{2}/2.

Are there consistent rules on how to write shorthand \frac?

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Note that this is nothing to do with \frac per se: you will see the same with almost any commands where you provide single tokens as mandatory arguments. –  Joseph Wright Aug 8 '12 at 18:45
Enclose the \sqrt{2} with {} to make it a single token. –  cyanide-based food Aug 8 '12 at 18:45