To do something scrupulously is to do it very carefully. It's the opposite of doing something sloppily.
To be scrupulous is to be careful, exact, and conscientious, so to do something scrupulously is to give it close attention and to strive for perfection. Diligent workers do their jobs scrupulously. A teacher who provides students with extra tutoring is behaving scrupulously. An accountant who does your taxes scrupulously is not likely to make a mistake. The Latin root word is scrupulus, literally "a small, sharp stone," and figuratively meaning "uneasiness or pricking of conscience."