How to approximate $z=a^{-1/4}$ using Newtons method?

Question

I need a function that isn't a polynomial and leads to a $f(x)$ that doesn't have a division by the iterate. The natural way of attempting this is by getting $x^4-\dfrac{1}{a}=0$ but this is using a polynomial. Is there any other way?

Answer 1

How about $x^{-4}-a=0$ ?

The Newton function is then $\dfrac{5x-ax^5}{4}$. Does division by $4$ count?

(Inspired in the division-free Newton's method for the reciprocal.)