[lainga]

What privileges the square root over any other fractional power?

[xyzzyz]

Finding intersection points of a circle with a line is equivalent to solving a system of equations, where one equation is that of a circle, (x-a)^2 + (y-b)^2 = r^2, and the second is that of a line, Ax + By = C. To solve it, you’ll be taking square roots, and not other roots. Similarly, to find intersection of two circles, you’ll be taking square roots, and not other roots.

[xrisk]

Presumably the hypotenuse of a right angled triangle.

[gilleain]

Indeed. Related is the Ammann-Beenker tiling and it's connections to the 'Silver Ratio' of sqrt(2) + 1.

* https://en.wikipedia.org/wiki/Silver_ratio

* https://en.wikipedia.org/wiki/Ammann%E2%80%93Beenker_tiling

[hinkley]

Hypotenuse of a 1x2 unit right triangle, to be precise. By Pythagoras, the square root of any sum of squares can be drawn trivially with a compass and a straight edge. So 2, 5, 7, 10, 13, 17, etc