Our problem was that in the notes from class none of us could figure out how to get from √ 56 to 2√ 14 . He had several examples of this sort of simplification but none of us could figure out what was going on. Eventually, I found a web site with enough explanation for me to make sense of it and explain it to him.
Now this may seem elementary to some, but clearly it contained steps we all had forgotten or never really understood in the first place. What made sense was the use of factor trees. 56 = 4*14 = 22*14 All three are the same way of writing the same value. By removing a perfect square (4) from the root and reducing it, we can multiply it back against the remainder, 2√ 14 .
Now I'm not sure how that helps anyone, other than the opportunity to cancel out or move the doubling of the radical within a larger equation. (Is it doubled?) Clearly there is some sort of relationship between perfect squares, quadrants and curved lines represented by these complex formulas. And I understand that the concepts build on each other, but I wish textbooks did a better job of explaining those relationships with less emphasis on memorizing the steps. Even most web sites simply gave the step by step process instead of illustrating the rationale.
One final note, #33 in the picture has more problems than just determining Darnell's chances of catching the game. I really don't think the best use of math is to help friends circumvent their punishment. I don't appreciate textbooks circumventing my parental authority.