Remove square roots in an expression

Question

I have an expression of the following form (yet way longer)

(u^2 - Sqrt[z1])^2 (u^3 - Sqrt[z1])^3 (u^4 - Sqrt[z1])^2
(u^2 + Sqrt[z1])^2 (u^3 + Sqrt[z1])^3 (u^4 + Sqrt[z1])^2
(-1 + u^18 z1^(5/2)) (1 + u^18 z1^(5/2))

As you can see, terms combine by pairs to produce

(u^4 - z1)^2 (u^6 - z1)^2 (u^8-z1)^2 (-1 + u^36 z1^5)

Does there exist a command in Mathematica to do this simplification?

The full expression is this one:

(1 + u^4)^3 (1 - u^3 + u^6)^2 (1 + u^3 + u^6)^2 (1 - u^4 + u^8)^3
(-1 + u^12)^7 (1 - u^6 + u^12)^2 (u^2 - Sqrt[z1])^2 (u^3 - Sqrt[z1])^3
(u^4 - Sqrt[z1])^2 (u^9 - Sqrt[z1])^2 (u^12 - Sqrt[z1])^3
(u^18 - Sqrt[z1])^2 (u^2 + Sqrt[z1])^2 (u^3 + Sqrt[z1])^3
(u^4 + Sqrt[z1])^2 (u^9 + Sqrt[z1])^2 (u^12 + Sqrt[z1])^3
(u^18 + Sqrt[z1])^2 (-1 + u^2 Sqrt[z1])^2 (1 + u^2 Sqrt[z1])^2
(-1 + u^3 Sqrt[z1])^3 (1 + u^3 Sqrt[z1])^3 (-1 + u^4 Sqrt[z1])^2
(1 + u^4 Sqrt[z1])^2 (-1 + u^9 Sqrt[z1])^2 (1 + u^9 Sqrt[z1])^2
(-1 + u^12 Sqrt[z1])^3 (1 + u^12 Sqrt[z1])^3 (-1 + u^18 Sqrt[z1])^2
(1 + u^18 Sqrt[z1])^2 (u^3 - z1)^2 (u^9 - z1)^2 (-1 + z1)^5 (1 + z1)^2
(u^3 + z1)^2 (u^6 + z1)^3 (u^9 + z1)^2 (u^12 + z1)^3 (u^18 + z1)^2
(1 - Sqrt[z1] + z1)^2 (1 + Sqrt[z1] + z1)^2 (u^4 - u^2 Sqrt[z1] + z1)^2
(u^4 + u^2 Sqrt[z1] + z1)^2 (u^8 - u^4 Sqrt[z1] + z1)^2
(u^8 + u^4 Sqrt[z1] + z1)^2 (u^12 - u^6 Sqrt[z1] + z1)^2
(u^12 + u^6 Sqrt[z1] + z1)^2 (-1 + u^3 z1)^2 (1 + u^3 z1)^2
(1 - u^2 Sqrt[z1] + u^4 z1)^2 (1 + u^2 Sqrt[z1] + u^4 z1)^2
(1 + u^6 z1)^3 (1 - u^4 Sqrt[z1] + u^8 z1)^2
(1 + u^4 Sqrt[z1] + u^8 z1)^2 (-1 + u^9 z1)^2 (1 + u^9 z1)^2
(1 + u^12 z1)^3 (1 - u^6 Sqrt[z1] + u^12 z1)^2 (1 + u^6 Sqrt[z1] + u^12 z1)^2
(1 + u^18 z1)^2 (u^6 + z1^2)^2 (u^12 + z1^2) (u^18 + z1^2)^2
(1 + u^6 z1^2)^2 (1 + u^12 z1^2) (1 + u^18 z1^2)^2 (u^18 - z1^(5/2))
(u^18 + z1^(5/2)) (-1 + u^18 z1^(5/2)) (1 + u^18 z1^(5/2))
(z1 + u^24 z1 - u^12 (1 + z1^2))^6

and FullSimplify does not modify it.

Answer 1

The simplest way is to expand the expression. If it's rational, the square roots will cancel. For an excessively large expression, a cleverer approach may be faster:

expr1 // Expand // Factor   (* OP's first expression *)
(*  (u^4 - z1)^2 (u^6 - z1)^3 (u^8 - z1)^2 (-1 + u^36 z1^5)  *)

expr2 // Expand // Factor   (* OP's second expression *)
(* ((-1+u)^7 (1+u)^7 (1+u^2)^7 (1-u+u^2)^7 (1+u+u^2)^7 (1+u^4)^3 (1-u^2+u^4)^7 (1-u^3+u^6)^2 (1+u^3+u^6)^2 (1-u^4+u^8)^3 (1-u^6+u^12)^2 (u^3-z1)^2 (u^4-z1)^2 (u^6-z1)^3 (u^8-z1)^2 (u^9-z1)^2 (u^12-z1)^6 (u^18-z1)^2 (u^24-z1)^3 (u^36-z1)^2 (-1+z1)^5 (1+z1)^2 (u^3+z1)^2 (u^6+z1)^3 (u^9+z1)^2 (u^12+z1)^3 (u^18+z1)^2 (-1+u^3 z1)^2 (1+u^3 z1)^2 (-1+u^4 z1)^2 (-1+u^6 z1)^3 (1+u^6 z1)^3 (-1+u^8 z1)^2 (-1+u^9 z1)^2 (1+u^9 z1)^2 (-1+u^12 z1)^6 (1+u^12 z1)^3 (-1+u^18 z1)^2 (1+u^18 z1)^2 (-1+u^24 z1)^3 (-1+u^36 z1)^2 (u^6+z1^2)^2 (u^12+z1^2) (u^18+z1^2)^2 (1+z1+z1^2)^2 (u^8+u^4 z1+z1^2)^2 (u^16+u^8 z1+z1^2)^2 (u^24+u^12 z1+z1^2)^2 (1+u^6 z1^2)^2 (1+u^4 z1+u^8 z1^2)^2 (1+u^12 z1^2) (1+u^8 z1+u^16 z1^2)^2 (1+u^18 z1^2)^2 (1+u^12 z1+u^24 z1^2)^2 (u^36-z1^5) (-1+u^36 z1^5)  *)

Update: Cleverer approach, perhaps (0.25 sec. vs. 25.6 sec.):

conjugateQ[e1_, e2_, v_] := NumberQ@Simplify[e1/(e2 /. v^p_ /; OddQ[2 p] :> -v^p)];
With[{conj = Gather[expr // FactorList,
       #1[[2]] == #2[[2]] && conjugateQ[#1[[1]], #2[[1]], z1] &]
 Times @@ Map[Expand[Times @@ #[[All, 1]]]^#[[1, 2]] &, conj]
 ]