Working with surds recently, and found some cool new identities that I don't think were known before now.

if n = x*y, and z = n.sqrt(), assuming n is known but x and y are not..

q = (surd(n, (1/(1/((n+z)-1))))*(n**2))
r = (surd(n, (surd(n, x)-surd(n, y)))*surd(n, n))

s = abs(surd(abs((surd(n, q)-q)), n)/(surd(n, q)-q))
t = (abs(surd(abs((surd(n, q)-q)), n)/(surd(n, q)-q)) - abs(surd(n, abs(surd(n, q)+r)))+1)

(surd(n, (1/(1/((n+z)-1))))*(n**2)) ~=
(surd(n, (surd(n, x)-surd(n, y)))*surd(n, n))

for every n I checked.

likewise.

s/t == r.sqrt() / q.sqrt()

and

(surd(n, q) - surd(s, q)) ==
(surd(n, t) - surd(s, t))

Even without knowing x, y, r, or t.

Not sure if its useful, but its cool.

surd() is just..

surd(j, k ) = return (j+k.sqrt())*(j-k.sqrt())

and d() is just the python decimal module for ease of use.