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.
After having given him the square root function definition, it may be helpful to show the following:
$$
\sqrt{(-3)^2}=\sqrt{9}=\sqrt{3^2}=3\neq -3.
$$
Although well-definedness may be a bit of a heavy topic for a 14-year old, it should be very easy to explain why you cannot have ambiguous definitions in mathematics.
I think perhaps the boy is right and You've not told him the whole truth. $\sqrt{(-3)^2}$ does actually equal -3 but it also equals +3. In the same way as $\sqrt{9}$ = 3 or -3.


I wonder if he has a grasp of negative numbers yet? If not then I think that would be the place to start. If he does then start by showing him that $3^2 = 9$ and $(-3)^2 = 9$, then you can tell him that $\sqrt{9}$ takes us back from the square to either of the original numbers.