#5707 ratsimp treats sqrt(x) and x^(1/4) as unrelated

[David Scherfgen]

By Claude:

ratsimp represents sqrt(x) and x^(1/4) by two independent variables. It knows that sqrt(x)^2 = x and (x^(1/4))^4 = x, but not that (x^(1/4))^2 = sqrt(x). So it misses simplifications, and calling it again gives a different result:

(%i1) ratsimp((7^(1/4)+1)*(7^(1/4)-1)*sqrt(7));
(%o1) (sqrt(7)-1)*sqrt(7)
(%i2) ratsimp(%);
(%o2) 7-sqrt(7)

fullratsimp repeats ratsimp until nothing changes, but it can stop too early. This expression equals sqrt(x):

(%i3) fullratsimp(sqrt(x)*(x^(1/4)*y+1) - x^(3/4)*y);
(%o3) sqrt(x)*(x^(1/4)*y+1)-x^(3/4)*y

expand and radcan simplify both examples fully.