thom, a side story about calc'ing 2nd roots.

once upon a time, way back when an electronic calculator cost more than 3 times
the price of a brand VW bug, and mechanical calcs cost about 140% of the price
of a VW, a group of us (National Science Foundation Summer Students) were shown
just how fantastic an expensive mechanical calc was.

(note, mechanical calc could add, subtract, multiply [slow] and divide [slow to
really slow, depending the numbers involved] out to about 18 digits.
Electronic calc's [made only by Friden as I recall] could do the same, plus
"hold" two additional numbers in a "stack". No calculators of the time had
printed output.)

The college instructor started working through the process involved to
calculate a square root using the process jeffies outlined. After he had been
working and calculating and working for maybe 5 minutes, I and another guy
started working the problem by hand. While the instructor had 20 digits to
work with in short order, his accuracy was only about two digits. In fact his
accuracy using the machine never did keep up on a digit by digit basis with us
working the problem by hand.

btw, most people who needed to do nth roots as part of their jobs all
personally owned a table of logs. If you will willing to pay the price, I
believe you could get tables out to 12 digits or more. Usually that kind of
accuracy was not needed.