Advanced Algebra Challenge

Note: Many of the ideas below came from the excellent note in the Reader Reflections of the March 2007 Mathematics Teacher, contributed by Warren Groskreutz. One of his students derived and proved his own 'theorem' about certain radical expressions. I decided to develop these ideas into an activity. The last part is a bit different from 'Nathan's Theorem.' Mr. Groskreutz is to be commended for creating an environment in which such 'discoveries' can be made.

[Click on the small image below to enlarge.]