Move blue T-bar to enter positive number x to be squared. Orange T-bar shows z = x^2
Move orange T-bar to enter positive number z to be rooted. Blue T-bar shows x = sqrt(z)
At point C there are 3 pink sliders (sliding in slots of violet, orange and blue bars respectively) connected together by revolution joints. The screw lead must be large enough to avoid self-locking.
The video shows operation 40^2 = 1600 or sqrt(1600) = 40 and then 70^2 = 4900 or sqrt(4900) = 70.
The mechanism works on similar triangles rule.
From triangles OBC and OAD: OB/OD = BC/AD
OB = x ; AD = y ; BC = z ; OD = k = 100 mm (constant) then z = (xy)/k
Because pink bevel gears have the same tooth number and their screws have the same lead, so x = y hence z = (x^2)/k or x = sqrt(z*k)
For the X and Y scale, 1 mm corresponds 1 unit.
For the Z scale, 1 mm corresponds 100 units.