Me: What does the square root of 5 mean?
Sophomore 1: A number times itself that equals 5.
M: So what’s \sqrt{5}\sqrt{5}?
S1: [Reaches for calculator, evaluates \sqrt{5}, then types 2.3*2.3] 5.29?

Me: What does the square root of 5 mean?
Sophomore 2: A number that if you times it by itself it equals 5.
M: So what’s this? [writing: \sqrt{5}\sqrt{5}]
S2: [Helpless look, grin] 2? 2.5?

And three more today that are similar. When I hold their hand through \sqrt{4}\sqrt{4} they seem to get it, but I think they’ve just memorized the visual pattern that \sqrt{x}\sqrt{x}=x with no connection to the definition they just gave me.

Cf. Substitute, then simplify.

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