How to factorize this $\sqrt{8 - 2\sqrt{7}}$? When I was at high school, our teacher showed us a technique to simplify square roots like this  $\sqrt{8 - 2\sqrt{7}}$ that I forgot.
It was something like 8 = 7+1; 7 = 7*1; and using them we could represent $\sqrt{8 - 2\sqrt{7}}$ in simpler form. I would be happy if you can show how it works, and how this technique is called.

Such square roots can  be mechanically computed by a Simple Denesting Rule:
Here $\ 8-2\sqrt 7\ $ has norm $= 36.\:$ $\rm\ \color{blue}{Subtracting\ out}\,\ \sqrt{norm}\ = 6\,\ $ yields $\,\  2-2\sqrt 7\:$
which has $\, {\rm\ \sqrt{trace}}\, =\, \sqrt{4}\, =\, 2.\,\ \ \ \rm \color{brown}{Dividing\ it\ out}\,\ $ of the above  yields $\ \ \ 1-\sqrt 7$ 
Negate it to get the positive root.  Many more worked examples are in prior posts on this rule.
