What's wrong about $\sqrt{10} = \sqrt{9 + 1} = \sqrt{9} + \sqrt{1} = 3 + 1 = 4$?

Question

What's wrong about $\sqrt{10} = \sqrt{9 + 1} = \sqrt{9} + \sqrt{1} = 3 + 1 = 4$?

I know that it's logically wrong because $4 \times 4 = 16$, but the syntax to me seems to be healthy as long as I can see, well, of course, because I'm novice and I can't see much, but I think this silly question could have a very detailed and deep answer about what's wrong and what's right in math and how to reason while solving.

Thank you.

Answer 1

The problem is that $\sqrt{a+b} = \sqrt{a} + \sqrt{b}$ is not true for all values $a,b$.

In other words, the inequality $\sqrt{9+1}=\sqrt{9} + \sqrt{1}$ is not true. The other three equalities you wrote are true, but that one is not.

Answer 2

welcome to MSE. I think a visuality can make sense more about it. I made it by Desmos for you. https://www.desmos.com/calculator/3zpiiaqwtu
the blue one is $\sqrt{a}+\sqrt{b}$
red one is $\sqrt{a+b}$
We will observe it and scroll $a,b$ and see what happens!
hope it helps