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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.

Shaun
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Rida
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  • How can you "split" square-root over addition? In general, $\sqrt{a + b} \neq \sqrt{a} + \sqrt{b}$. That is the problem with what you have written. If you write $\sqrt{10}$, you mean a real number whose square is $10$. $\sqrt{10}$ is only a symbol and it is to be treated that way! – Aniruddha Deshmukh Apr 07 '21 at 08:03
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    It is not true for general functions that $f(a+b)=f(a)+f(b)$ (This is Cauchy's functional equation https://en.wikipedia.org/wiki/Cauchy%27s_functional_equation) – Mark Bennet Apr 07 '21 at 08:07
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    what is true is that $\sqrt{a+b}\leq\sqrt{a}+\sqrt{b}$. – C Squared Apr 07 '21 at 08:15
  • Have you thought about what you get if you square both sides of $$\sqrt a+\sqrt b = \sqrt{a+b}?$$ – saulspatz Apr 07 '21 at 08:19
  • If $a,b≥0 $ then $$\sqrt{ab}=\sqrt a×\sqrt b$$ not $a+b$ – lone student Apr 07 '21 at 08:57
  • If you imagine you can equate the square root of a partition of some number to the sum of square roots of the partition elements, then why stop at $\sqrt{10}=\sqrt{9}+\sqrt{1}$? How about $\sqrt{10}=\sqrt{4}+\sqrt{4}+\sqrt{1}+\sqrt{1}=6$, or even $\sqrt{k}=k\sqrt{1}=k$. Assert that every number its own square root, and from that argue that the only possible numbers are $0$ and $1$. – Keith Backman Apr 07 '21 at 15:25

2 Answers2

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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.

5xum
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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 enter image description here

Khosrotash
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