Is it Orthopae dics or Orthope dics?

The word orthopaedics comes from two greek words:
ortho meaning straight
and paedia which means children.

That is because orthopaedic surgeons were physicians who spent their time taking care of what used to be called "crippled children" and straightening out crooked spines in these poor children. In fact the emblem of orthopaedics is that of a crooked tree being straightened by making it grow next to a straight splint (see above). Like many words originating from the greek, the correct spelling includes the combined letters ae.

The alternative spelling of orthopedics has come into semi-accepted use in the US because so many of the lay public would assume it was a typographic error when they saw it spelled with ae and this would distract from the content of the text.

While we are being semantically correct, there is no such thing as "orthopaedics" or "orthopaedist" even though these words have come into common usage. The correct title of the specialty is "orthopaedic surgery", which itself is descriptively incorrect because statistically orthopaedic surgeons take care of 20-25 patients non surgically in their offices for every patient that ends up needing surgery. So they really should be called orthopaedic physicians.

As mentioned above, the specialty of orthopaedic surgery was one that took care of children with deformities. In those early days orthopaedic surgeons did not take care of fractures at all. Fractures were taken care of by the general surgeons. In time though, it became obvious that the expertise gained in working with the bones and muscles of the "crippled children" made orthopaedic surgeons far more knowledgeable in dealing with bones, muscles, spines, etc than the general surgeons and orthopaedic surgeons reluctantly took over the care of fractures and other bony problems in adults. Now there are orthopaedic surgeons who concentrate in taking care of the orthopaedic problems of adults and never treat children.