  We calculate the pi+ K+ scattering length in fully-dynamical lattice QCD with
domain-wall valence quarks on MILC lattices with rooted staggered sea-quarks at
a lattice spacing of b=0.125 fm, lattice spatial size of L =2.5 fm and at pion
masses of m_pi=290, 350, 490 and 600 MeV. The lattice data, analyzed at
next-to-leading order in chiral perturbation theory, allows an extraction of
the full pi K scattering amplitude at threshold. Extrapolating to the physical
point gives m_pi a_3/2 = -0.0574 (+- 0.0016)(+0.0024 -0.0058) and m_pi a_1/2 =
0.1725 (+- 0.0017)(+0.0023 -0.0156) for the I=3/2 and I=1/2 scattering lengths,
respectively, where the first error is statistical and the second error is an
estimate of the systematic due to truncation of the chiral expansion.
