Can √ 9 be √ (-3^2)?

Dear EBITDA,
I'm happy to respond. Many folks are confused on this particular point.

First of all, you may find this blog article helpful:
GMAT Quant: Roots

Here's the deal. We have to distinguish two cases that are easily confused.
1) Case #1: the √ appears printed on the page as part of the problem.
You see, technically, the √ symbol is the "find the positive square root" symbol. It always has a positive output (or a zero output), never a negative output. Thus, √9 equals +3 all the time.

2) Case #2: in the printed problem, a variable squared appears, and the student must take a square root in order to solve
In this suppose x^2 = 9 appears on the page: then we have to consider ALL roots, x = -3 and x = +3.

I'm not sure what the first book said, and in particular, I am not sure if you are aware of the subtleties of notation. Are you aware of the difference between these two:
(-3)^2 vs. -3^2
In the first, the squaring includes the negative, so the output is +9. In the second, by order of operations, we square the 3 first, then multiply by the negative, so the output is -9. We most certain cannot compute √ (-3^2) = √ (-9), because that is number not in the real number system. That's extremely different from
√ ((-3)^2) = √(9) = +3
I'm not sure whether the notational issue arose in the first book that you quoted or in your paraphrase of the first book.

It is definitely 100% true that
\(\sqrt{x^2} = |x|\)
That equation is true for every single number on the number line.

Does this answer your questions or not?
Mike