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.

I think the kid has misunderstood something his teacher said about cancelling. A math student of any age is bound to misunderstand his teacher at some point or other. When I was in high school, long, long ago, Mr. Jones was fond of saying that such and such equation has no real solutions, equations like, say, $x^2 + 9 = 0$. I thought he meant such an equation has no solutions at all. It wasn't until long after college that I learned about imaginary numbers.
The equation $x^4 - 81 = 0$ has four solutions, anyone who has studied the fundamental theorem of algebra can tell you. If we want to limit ourselves to real solutions, there are still two solutions left. But when we punch up $\root 4 \of {81}$ on a calculator, we want just one answer, and we want that answer to be the same each time, e.g., if it says the answer is $3$ one time and $-3$ another time, we'd think the calculator has a malfunction of some sort.
And so it is with the square root. We want the calculator to say $\sqrt 9 = 3$ every time, no matter how it is that we got the $9$ in the first place, whether by $(-3)^2$ or $3^2$ or any other operation that could possibly give $9$, such as $56 - 47$.
