  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 $H_{0}$ and deceleration parameter $q$. We find the constraints
on these parameters from the latest $H(z)$ and SNe Ia data. At 1$\sigma$ level
the constraints from $H(z)$ data are obtained as $q=-0.18_{-0.12}^{+0.12}$ and
$H_{0}=68.43_{-2.80}^{+2.84}$ km s$^{-1}$ Mpc$^{-1}$ while the constraints from
the SNe Ia data read as $q=-0.38_{-0.05}^{+0.05}$ and
$H_{0}=69.18_{-0.54}^{+0.55}$ km s$^{-1}$ Mpc$^{-1}$. We also perform the joint
test using $H(z)$ and SNe Ia data, which yields the constraints
$q=-0.34_{-0.05}^{+0.05}$ and $H_{0}=68.93_{-0.52}^{+0.53}$ km s$^{-1}$
Mpc$^{-1}$. The estimates of $H_{0}$ 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
$q\gtrsim 0.72$ and $H_{0}\lesssim 41.49$ 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 $\Lambda$CDM model as
$q\rightarrow -1$. Finally, we conclude that despite having several good
features power-law cosmology is not a complete package for the cosmological
purposes.
