Why do even the simplest calculators feature a square root key?

[EdS2]

It's an interesting question, and I think there will be several answers
- the same thing happened with mechanical calculators: they evolved from add/sub to four-function to square root, and presumably for the same reason(s)
- it's easy to implement square root, much easier than x^y or log or trig
- an extra function makes for a more attractive product - it's a marketing arms race, and the electronic calculators might have been competing still with the mechanical ones
- buyers will remember being taught to calculate square roots at school, as being the next algorithm you learn after long division
- if buyers remember any formulae from school, the length of the hypotenuse and the solution of a quadratic are very likely to be in the list

Of course, schooling has changed, and numeracy has changed, so it's harder to see now what would have been understood, what the competition would be, and what would have seemed attractive.

It would be good to have a killer app for square root - it would be good to know why it was taught in schools, but it might not have been because it was useful.

First mechanical calculator to offer square root, the Friden SRW in 1952:

http://www.hpmuseum.org/srw.htm "The Friden SRW
By the 1950's, four function motor driven mechanical calculators had become common on the desks of engineers and scientists. In 1952 Friden shook up the market with the SRW, with the amazing ability to extract a square root at the touch of a single key."

http://www.hpmuseum.org/root.htm "How The Friden Extracted Square Roots - It's Basically What You Learned In School

There are many ways to calculate a square root, and I'd be rather surprised if these early five-bangers used Newton's method. Square root is only about as expensive as division, and it wouldn't be normal to use lots of divisions to do the job. Here are some ways:
http://user.mendelu.cz/marik/mechmat/sqrt/