Résumé : Constructive root bounds is the fundamental technique needed to achieve guaranteed accuracy, the critical capability in Exact Geometric Computation. Known bounds are overly pessimistic in the presence of general rational input numbers. In this paper, we introduce a method which greatly improves the known bounds for k-ary rational input numbers. Since majority of input numbers in scientific and engineering applications are such numbers, this could lead to a significant speedup for a large class of applications. We apply our method to the BFMSS Bound. Implementation and experimental results based on the CORE library are reported.