In the movie "Gifted" a girl calculates the approximate square root of a 4-digit number in her head. Here the scene on YouTube: https://www.youtube.com/watch?v=37meAwQqPsE

Later, it is said she is using the Trachtenberg system: https://en.wikipedia.org/wiki/Trachtenberg_system

The square root method seems to be described in the book "The Trachtenberg speed system of basic mathematics", the text is available on archive.org: https://archive.org/stream/TheTrachtenbergSpeedSystemOfBasic... (search for "square root").