[en] This work presents simultaneous branching fraction measurements of the decay modes τ^-→K^-nπ⁰ν_τ with n=0,1,2,3 and τ^-→π^-nπ⁰ν_τ with n=3,4. The analysis is based on a data sample of 427 x 10⁶τ⁺τ^- pairs recorded with the BABAR detector, which corresponds to an integrated luminosity of 464.4 fb^-1. The measured values are B(τ^-→K^-ν_τ)=(6.57±0.03±0.11) x 10^-3, B(τ^-→K^-π⁰ν_τ)=(4.61±0.03±0.11) x 10^-3, B(τ^-→K^-π⁰π⁰ν_τ)=(5.05±0.17±0.44) x 10^-4, B(τ^-→K^-π⁰π⁰π⁰ν_τ)=(1.31±0.43±0.40) x 10^-4, B(τ^-→π^-π⁰π⁰π⁰ν_τ)=(1.263±0.008±0.078) x 10^-2 and B(τ^-→π^-π⁰π⁰π⁰π⁰ν_τ)=(9.6±0.5±1.2) x 10^-4, where the uncertainties are statistical and systematic, respectively. All measurements are compatible with the current world averages whereas the uncertainties are significantly smaller by a factor of up to five. The determination of B(τ^-→π^-π⁰π⁰π⁰π⁰ν_τ) is the first measurement of this branching fraction. The measured branching fractions are combined with the current world averages. Using the new averages, an updated determination of vertical stroke V_us vertical stroke from hadronic τ decays yields vertical stroke V_us vertical stroke =0.2146±0.0025, which improves previous measurements by 19%. Its uncertainty is comparable to the one of the current world average from semileptonic kaon decays. (orig.)