Other mistake: The very last scene of the movie shows Supergirl flying underwater holding the Omegadron. Oddly enough, though her body is realistic, her arms are now very thin and measure two meters long.
Supergirl (1984)
1 other mistake
Directed by: Jeannot Szwarc
Starring: Faye Dunaway, Peter O'Toole, Mia Farrow, Hart Bochner, Helen Slater
Continuity mistake: When Supergirl crushes the rock, her hair is curly and messy. In the following shot it's brushed and straight.
Lucy Lane: All you need is a couple of streaks and your ears pierced. I could do it for you, it's easy. You just get a needle, heat it up, dab it with some alcohol, and zap! The guys go crazy.
Linda Lee: My ears what?
Lucy Lane: Pierced. Like and you know, I take a needle, and then I heat it up, dab it with some alcohol and, zap, all the guys go crazy.
Linda Lee: Because I have holes in my ears?
Lucy Lane: What, are you putting me on? Sometimes I can't figure you out, Linda.
More questions & answers from Supergirl
Answer: The Omegaahedron is the power source for the whole city. It supplies the air, the dome that protects them, the lights, and the machines they need to survive.
The question was why would Zoltar need permission to use the Omegahedron, not what it could do.