This paper investigates cosmological solutions in the context of the modified $f(R,T)$ theory of gravity, where $R$ is the Ricci scalar and $T$ is the trace of the energy-momentum tensor. The authors consider the Friedmann-Robertson-Walker (FRW) space-time to find solutions of the field equations. They explore the nature of the universe by examining acceleration expansion, ultra relativistic, sub-relativistic, dust, radiation, and stiff universes. Additionally, they apply the power law technique to two different $f(R,T)$ gravity models to observe the expanding nature of the universe. The bouncing scenario is discussed by choosing specific values of the model parameters, and the energy conditions are observed to be satisfied for a successful bouncing model. The study concludes that some solutions in the $f(R,T)$ theory support the concept of exotic matter and accelerated expansion of the universe due to significant negative pressure.This paper investigates cosmological solutions in the context of the modified $f(R,T)$ theory of gravity, where $R$ is the Ricci scalar and $T$ is the trace of the energy-momentum tensor. The authors consider the Friedmann-Robertson-Walker (FRW) space-time to find solutions of the field equations. They explore the nature of the universe by examining acceleration expansion, ultra relativistic, sub-relativistic, dust, radiation, and stiff universes. Additionally, they apply the power law technique to two different $f(R,T)$ gravity models to observe the expanding nature of the universe. The bouncing scenario is discussed by choosing specific values of the model parameters, and the energy conditions are observed to be satisfied for a successful bouncing model. The study concludes that some solutions in the $f(R,T)$ theory support the concept of exotic matter and accelerated expansion of the universe due to significant negative pressure.