If I understand what you're asking correctly... these are notes you took while your instructor worked a problem. You wrote down his/her steps but don't quite understand them. If this is not what you're having difficulty with, please correct me.
Assuming it is, then you're having trouble after the notes hit this step:
\(\displaystyle S=\frac{\sqrt{130464-112896}}{36\left(35\right)}\)
Well, let's try a few intermediate steps to hopefully clarify. If we subtract the two terms under the square root, we get:
\(\displaystyle S=\frac{\sqrt{17568}}{36\left(35\right)}\)
Now, you know that you can simply a square root by pulling out a perfect square. So does 17568 divide by any perfect squares? It does. And the largest of those is 144. That means we can pull out a 12 from the square root.
\(\displaystyle S=\frac{12\sqrt{122}}{36\left(35\right)}\)
\(\displaystyle S=\frac{\sqrt{122}}{105}\)
Now if I were solving this problem, I'd stop here and use an exact form. But if you want to approximate, we can do that. And then we realize that... whoops. This problem doesn't work out the way your notes say it should. Hence why even with a calculator you were getting a different answer. So, either you copied the notes down wrong, or your instructor messed up.