How to explain to a 14-year-old that $\sqrt{(-3)^2}$ isn't $-3$? I had this problem yesterday. I tried to explain to the kid this: $$\sqrt{(-3)^2} = 3,$$ and he immediately said: "My teacher told us that we can cancel the square with the square root, so it's $$\sqrt{(-3)^2} = -3."$$
He has a lot of problems with maths, and I don't know how I can explain to him this as easily as possible. He still thinks that I lied him.
Thank you.

By definition, the square root of $x$ (if it exists) is the (unique) non-negative number $y \geq 0$ such that $x = y^2$. The square root only exists if $x \geq 0$, and in this case it always exists and is unique.
If $y \geq 0$ then it is true that $\sqrt{y^2} = y$. More generally,
$$ \sqrt{y^2} = |y|. $$
For a more complete understanding, your high-schooler will have to study some complex analysis. The square root function is multi-valued, and the usual positive square root function is a specific branch of the function. Another possibility is using the non-positive square root instead of the non-negative square root. These are the only two possibilities which are continuous.
