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A discrete-time predator-prey system with Michaelis-Menten functional response and Gompertz growth of prey is examined to reveal its chaotic dynamics. We prove algebraically that when one of the model parameter passes its critical value, the system passes through a flip bifurcation (FB) and Neimark-Sacker bifurcation (NSB) in the interior of R2+. We apply the center manifold theorem and bifurcation theorems to determine the existence conditions and direction of bifurcations. Numerical simulations are employed which include the diagram of bifurcations, phase portraits, periodic orbits, invariant cycle, abrupt emergence of chaos, and attracting chaotic sets. In addition, maximum Lyapunov exponents (MLEs) and fractal dimension (FD) are computed numerically to justify the existence of chaos in the system. Finally, we apply feedback control method to control chaotic trajectories.