Say you have an equation which looks like this:

y^2 = x + 4

Assuming you want to rearrange the equation to solve for y, you would do this:

y = square root(x + 4)

I understand what is being done here, however I don't fully understand the reasoning behind it. I know that you are taking the square root of both sides, but I wondered why you couldn't do this another way, the way I thought of, is as follows:

y = square root(x) + square root(4)

Now, I know that these two ways are not the same, but I wondered why they aren't the same, and what I wondered about even more, is how you know to choose one way over the other. If the equation was set up like this:

2y = x + 4

Then you could do this:

y = (x + 4)/2

Or you could do this, and still get the same result:

y = x/2 + 4/2

Dividing each term individually and then adding the results has the same effect as adding the terms, and then dividing the result. If you can use two different methods in this case, and get the same result, then how do you know which way to use in the other case(with squares/square roots)? This is my main question, but I'm also wondering why the answers end up different in the first place.