# Newton's method, assuming a > 0 and n > 0
create or replace function nth_root(a,n) as (
  with recursive cte as (
    select 0 as ix, a::DOUBLE as x, (a-1.0)::DOUBLE as xprime
    union all
    select ix+1 as ix,
           ((n - 1) * x / n ) +  (a / (n * x ** (n-1))) as x,
           x as xprime
    from cte
    where @(x - xprime) > 1e-15
    and ix<10
  )
  select last(x order by ix)
  from cte
);

select n,
       5.0 ^ (1 / cast(n as double)) as builtin,
       nth_root(5.0, n) as nth_root
from unnest( [1,3,5,10,1000,10000] )  _(n);