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A common example is that students think that $\sqrt{ab}=\sqrt a \sqrt b$ for all $a,b$; try tricking them by saying that $1=\sqrt{1}=\sqrt{-1\cdot -1} = \sqrt{-1}\sqrt{-1} = i\cdot i = -1$. Similarly students will not often realize that $f(x) = \sqrt{x^2}$ is NOT $x$, but rather $|x|$. Extraneous solutions (i.e. solutions that pop out when solving but are ...