I was arrested,booked and put to jail for 3-weeks for mistaken identity so the case was dismissed,my question is what should I do so as it won't show up on live scan when applying for work?

Others have been through hell over different departments not erasing dismissed cases from viewable files. I would pay for a background check on myself. If it shows, go to the DA's office and find out how to have all places where it was recorded made current.
As far as I know, the file is never totally erased from sight of the police, courts, and FBI. Doesn't seem fair, does it?