In this paper, we show that the expansion history of the Universe in power-law cosmology essentially depends on two crucial parameters, namely the Hubble constant H0 and deceleration parameter q. We find the constraints on these parameters from the latest H(z) and SNe Ia data. At 1σ level the constraints from H(z) data are obtained as and km s−1 Mpc−1, while the constraints from the Type Ia supernovae (SNe Ia) data are and km s−1 Mpc−1. We also perform the joint test using H(z) and SNe Ia data, which yields the constraints and km s−1 Mpc−1. The estimates of H0 are found to be in close agreement with some recent probes carried out in the literature. The analysis reveals that the observational data successfully describe the cosmic acceleration within the framework of power-law cosmology. We find that the power-law cosmology accommodates well the H(z) and SNe Ia data. We also test the power-law cosmology using the primordial nucleosynthesis, which yields the constraints q≳ 0.72 and H0≲ 41.49 km s−1 Mpc−1. These constraints are found to be inconsistent with the ones derived from the H(z) and SNe Ia data. We carry out the statefinder analysis, and find that the power-law cosmological models approach the standard Λ cold dark matter (ΛCDM) model as q→−1. Finally, we conclude that despite having several good features power-law cosmology is not a complete package for the cosmological purposes.