  In this paper, we investigate the newly developed $f(R,\mathbf{T}^2)$ theory
($R$ is the Ricci scalar and $\mathbf{T}^2=T_{\alpha\beta}T^{\alpha\beta},~T
_{\alpha\beta}$ demonstrates the energy-momentum tensor) to explore some viable
cosmological models. For this purpose, we use the Noether symmetry approach in
the context of flat Friedmann-Robertson-Walker (FRW) universe. We solve the
Noether equations of this modified theory for two types of models and obtain
the symmetry generators as well as corresponding conserved quantities. We also
evaluate exact solutions and investigate their physical behavior via different
cosmological parameters. For the prospective models, the graphical behavior of
these parameters indicate consistency with recent observations representing
accelerated expansion of the universe. In the first case, we take a special
model of this theory and obtain new class of exact solutions with the help of
conserved quantities. Secondly, we consider minimal and non-minimal coupling
models of $f(R,\mathbf{T} ^{2})$ gravity. We conclude that conserved quantities
are very useful to derive the exact solutions that are used to study the cosmic
accelerated expansion.
