Simplifying expressions - factoring or expanding? The term "simplify" has always confused me. What does "simplifying" mean? More specifically, which is more "simple" - a fully factored expression or a fully expanded one?

Simple is in the eye of the beholder.  For some purposes a fully factored expression is valuable-finding roots, for example.  For some purposes a fully expanded one is valuable-especially for behavior at infinity.  There is a similar tension regarding radicals in the denominator.  You can clear them, but do you want to?  
I am not in math education, but if somebody wants you to submit answers in the "simplest form" they should define it clearly.  It makes grading easier and you can follow the rules.  Otherwise, you use the form that suits the next thing you will do with it.
