Other mistake: Spock lifts his crew mates with the rocket boots. He passes deck numbers 35 through 78 from bottom to top. First off, deck numbers go from top to bottom. The bridge is on deck 1. Second, the Enterprise of that class only had 23 decks.
Other mistake: If Sha ka ri is in the center of the galaxy then it should have taken about 70 years to reach if the Delta Quadrant is opposite the Alpha Quadrant. Even if this is not so, it has been stated that the nearest point in the Delta Quadrant is 70 years away from earth, therefore the center of the galaxy should not be within travelling distance of earth, or anywhere within the federation.
Other mistake: If Uhura was supposed to recall the rest of the officers over a priority seven situation why did it take so long to go get Kirk, Spock, and McCoy?
Other mistake: In the fight in Paradise City, the Enterprise people have shields that they didn't have in evidence when they ran out of the shuttle.
Other mistake: As the shuttle craft approaches the planet, there is only a view of the stars, rather than any of the mountains or other scenery.
