You can approximate the real $n$th root of $k$ as the real solution to $x^n-k=0$. Since this is a monotone function on $[0,k]$, you can use a bisection method to approximate it to an arbitrary degree. So finding a root amounts to iterated computation of $x^n$. So if divide-and-conquer makes computing $x^n$ more efficient so it should make computing $k^{1/n}$ more efficient.
You can approximate the real $n$th root of $k$ as the real solution to $x^n-k=0$. Since this is a function you can use a bisection method to approximate it to an arbitrary degree. So finding a root amounts to iterated computation of $x^n$. So if divide-and-conquer makes computing $x^n$ more efficient so it should make computing $k^{1/n}$ more efficient.