Taking the two expressions $\sqrt{2+x}$ and $\sqrt{2}+x$ it is clear that the horizontal bar on top is important in specifying what is included in the root operation.
But if there is only one term then is it correct to write the expression without any horizontal bar at all?
As in: $\surd 2$ instead of $\sqrt{2}$.
On
The rule is that the horizontal bar is always required. At least that's the rule I was taught (after all, it's just convention, so what is right is what we decide is right).
On
It's not strictly necessary, but it can sure help avoid a lot of ambiguity and confusion. Suppose I write $$\surd 2 + x$$ That could mean $\sqrt{2 + x}$ as easily as it could mean $\sqrt 2 + x$.
I'd like to mention a couple of related issues. There are still places on the Internet where mathematical typesetting is not available, and you may or may not have mathematical symbols available either, but you can almost always use parentheses. $(\surd 2) + \textrm{x}$ and $\surd(2 + \textrm{x})$ might be acceptable workarounds in those situations.
And then there's the issue of "source code." Both sqrt 2 and sqrt{2} will render the same here. Which one you use is entirely up to you. If you use the former rather than the latter, however, remember that you need to add braces if you later need to change it to a literal number with more digits, or an expression with more than one character.
It's not too unusual to see $\surd x$ without the bar in older books, probably because it was easier to typeset with older technology. But it's usually less clear. The modern convention is generally to prefer the bar: $\sqrt{x}$.