Factor out common terms from a list of terms

Question

I was looking for an automated way to factor out common terms from a list.

Example (assuming n<1):

 list={(1 + r^2)^(-1/2 + n/2) (1 + w^2)^(-1/2 - n/2) (1 + r^(1 - n) w^(1 + n)) x12^-n, 
       r^(1 - n) (1 + r^2)^(-1/2 + n/2) w^(1 + n) (1 + w^2)^(-1/2 - n/2) x12^-n, 
       (1 + r^2)^(-1/2 + n/2) (1 + w^2)^(-1/2 - n/2) x12^-n}

Expected output:

(1 + r^2)^(-1/2 + n/2) (1 + w^2)^(-1/2 - n/2)   x12^-n  {  (1 + 
r^(1 - n) w^(1 + n)) , r^(1 - n)  w^(1 + n), 1  } 
or 
{ CommonFactor->(1 + r^2)^(-1/2 + n/2) (1 + w^2)^(-1/2 - n/2)   x12^-n , 
  { (1 + r^(1 - n) w^(1 + n)) , r^(1 - n)  w^(1 + n), 1  }}

There are some ways (Extract common factor from vector or matrix) to factor out common terms using PolynomialGCD, which works in most of the cases, however, in this case, it does not do it properly (probably due to the unknown n?) Using their approach I get

 {"CommonFactor" ->   r^(-2 n) (1 + r^2)^(-(1/2) + n/2) (1 + w^2)^(-1 - n) x12^(-2n), 
  {r^(2 n) (1 + r^2)^(1/2 + 1/2 (-1 + n) - n/2) (1 + w^2)^(1/2 + n/2) (1 + r^(1 - n) w^(1 + n)) x12^n, 
   r^(1 + n) w^(1 + n) (1 + w^2)^(1/2 + n/2) x12^n, 
   r^(2 n) (1 + w^2)^(1/2 + n/2) x12^n}}

which is not the wanted behavior. Assuming[0<n<1 also does not help.

Is there any better way?

Answer 1

{#, list/#} &@Fold[Intersection, list]

$$\left\{\left(r^2+1\right)^{\frac{n}{2}-\frac{1}{2}} \left(w^2+1\right)^{-\frac{n}{2}-\frac{1}{2}} \text{x12}^{-n},\left\{r^{1-n} w^{n+1}+1,r^{1-n} w^{n+1},1\right\}\right\}$$

---

EDIT a possible failsafe strategy where the Intersection will likely not work

Let's say the original list is now called alist:

alist = {(1 + r^2)^(-1/2 + n/2) (1 + w^2)^(-1/2 - n/2) (1 + 
     r^(1 - n) w^(1 + n)) x12^-n, 
  r^(1 - n) (1 + r^2)^(-1/2 + n/2) w^(1 + n) (1 + w^2)^(-1/2 - 
      n/2) x12^-n, (1 + r^2)^(-1/2 + n/2) (1 + w^2)^(-1/2 - 
      n/2) x12^-n}

and another list is called blist:

blist = {a^-n (1 + a^n b), b, a^-n}

Then

{Head /@ alist, Head /@ blist}

{{Times, Times, Times}, {Times, Symbol, Power}}

A possible strategy could be to check the Head in advance and choose 1 or another method instead of the Intersection. As an example:

If[! SameQ @@ (Head /@ #), 1, Fold[Intersection, #]] & /@ {alist, 
  blist}

{(1 + r^2)^(-(1/2) + n/2) (1 + w^2)^(-(1/2) - n/2) x12^-n, 1}