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Suppose that in the problem you describe, instead of a and b, we talk about actual numbers, such as 4 and 9. The square root of 4 is 2, and the square root of 9 is 3; so if we add the square root of 4 and the square root of 9, that is 2 + 3 = 5. Now, suppose we add 4 + 9 and get 13. The square root of 13 is just over 3.6, which you will note is considerably less than 5. This is because multiplication is not a linear process; as you multiply larger numbers, the total increases more rapidly than if you were merely adding numbers. A square root is the number which, multiplied by itself produces the number of which it is the square root, so it is basically connected to multiplication, not addition. Hence, you can't get the same result through addition.

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Q: Why is square root of a plus square root of b not equal square root of a plus b?
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What is the square root of a negative number plus the square root of a negative number?

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