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We show that, generically, finding the $k$-th root of a braid is very fast.More precisely, we provide an algorithm which, given a braid $x$ on $n$ strandsand canonical length $l$, and an integer $k>1$, computes a $k$-th root of $x$,if it exists, or guarantees that such a root does not exist. The generic-casecomplexity of this algorithm is $O(l(l+n)n^3\log n)$. The non-generic cases aretreated using a previously known algorithm by Sang-Jin Lee.

The root extraction problem for generic braids