Yes, taking the nth root is an operator. Taking the SQUARE root is this operator with the operand 2, which is usually graphically omited. I totally agree with the top comment: it's burning one of the 2's.

The obvious counterargument is that the problem statement is

> Given exactly four instances of the digit 2 and some target natural number, use any mathematical operations to generate the target number with these 2s, using no other digits.

and the square root symbol is a mathematical operation and is not a digit. That it is equivalent to taking the 2nd root isn't relevant ... the whole puzzle is about finding and using equivalents.