# How to remedy the “freshman's dream”? [duplicate]

I am teaching a mid-level calculus course, and I see my students making the freshman's dream mistake of thinking that every function is a homomorphism. In particular, they think that exponents can be distributed, so $$\sqrt{a + b} = \sqrt{a} + \sqrt{b}$$ and $$(a + b)^2 = a^2 + b^2$$ in their world.
We have discussed this in class, and I've demonstrated how this will lead to contradictions like $$\sqrt{2} = 2$$. When I see this mistake, I struggle with grading it because I have the opinion that properties of exponents should have been learned long ago. I came here to ask how I can gently get my students to unlearn this mistake as soon as possible. Am I being unreasonable in thinking that this is a big problem at this stage of learning mathematics?