Do not fret, your answer is correct.
Don't second-guess your work: you did very well, (and no, the power rule applied to terms with negative exponents does not make them $0$!
As a rule, solution manuals/solutions in the appendices of books are less carefully edited than are the problems or text themselves. (Solutions are often not authored by the author of the text/exercises themselves, and proof-readers/publishers, etc. don't usually have the expertise to check for "correctness".) This can be terribly aggravating, because such errors are typically not discovered until students/instructors encounter discrepancies, like you yourself encountered.
Check to see of your text/solution manual has an "Errata" page on line (Google "text title": Errata). Those are more easily compiled and published these days with the availability of doing so on the web, on-going-as-discovered.
ADDED: Since you encountered this through Khan Adademy, I suspect there's a "contact" link somewhere on site where users can offer feedback and/or report errors. I'm sure there are folks "in charge" of creating/compiling/maintaining exercises and (correct, when possible!) solutions.