In the year 628 Brahmagupta wrote, "He, who distinctly and severally knows addition and the rest of the twenty logistics, and the eight determinations including measurement by shadow, is a mathematician."
The eight determinations are: addition, subtraction, multiplication, division, square, square root, cube and cube root.
As you know, Brahmagupta was the first to write about zero as a number and he gave the arithmetic rules for handling positives and negatives more than 1000 years before the western finally accepted negative numbers.
He also wrote about 'pulverizing' so maybe this word applies to your way of determining roots.
So if you haven't already, you may enjoy reading Chapters 12 and 18 from his book Brahmasphutasiddhanta. This book changed the way the world does math from Roman to Hindu Arabic numerals.
Brahmagupta at 628 CE may have been one of the first to create a way to solve* roots mentally. Until proven otherwise, you may be the most recent!
You can read and download Brahmagupta's method (and much more) via this link.