Almost anything will work, but why not take the simple example x = y = 1? If square roots behaved as this formula asserts then we would have √2 = √1 + √1 = 1 + 1 = 2, which is clearly absurd.

Pamela J. Wells, Understanding Linear Functions and Their Representations, Mathematics Teaching in the Middle School, Vol. 21, No. 5 (December 2015/January 2016), pp. 308-316 ...