  In this work, we use reconstruction methods to obtain cosmological solutions
in the recently developed scalar-tensor representation of $f(R,T)$ gravity.
Assuming that matter is described by an isotropic perfect fluid and the
spacetime is homogeneous and isotropic, i.e., the
Friedmann-Lema\^itre-Robsertson-Walker (FLRW) universe, the energy density, the
pressure, and the scalar field associated with the arbitrary dependency of the
action in $T$ can be written generally as functions of the scale factor. We
then select three particular forms of the scale factor: an exponential
expansion with ${a(t)\propto e^t}$ (motivated by the de Sitter solution); and
two types of power-law expansion with ${a(t)\propto t^{1/2}}$ and ${a(t)\propto
t^{2/3}}$ (motivated by the behaviors of radiation- and matter-dominated
universes in general relativity, respectively). A complete analysis for
different curvature parameters ${k=\{-1,0,1\}}$ and equation of state
parameters ${w=\{-1,0,1/3\}}$ is provided. Finally, the explicit forms of the
functions $f\left(R,T\right)$ associated with the scalar-field potentials of
the representation used are deduced.
