I graduated college a year ago and I'm not sure if I've ever in my life been asked to manually compute a square root (except for a perfect square integer).
Same here. And I distinctly remember asking a math teacher in middle school if a number was a perfect square and we manually ended up checking and incrementing numbers until we decided that the number was not. It makes me think that this was possibly not such a common task to assign in school?
Depends on your course load. My EE courses forced you to do this (and then forced you to do it in hardware... on a crappy chip that I swear lacked an overflow bit, but my memory is fuzzy).