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By the extended <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo stretchy="false">(</mo><mfrac><msup><mi>G</mi><mo>′</mo></msup><mi>G</mi></mfrac><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula> method and the improved tanh function method, the exact solutions of the (2+1) dimensional Boussinesq equation are studied. Firstly, with the help of the solutions of the nonlinear ordinary differential equation, we obtain the new traveling wave exact solutions of the equation by the homogeneous equilibrium principle and the extended <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo stretchy="false">(</mo><mfrac><msup><mi>G</mi><mo>′</mo></msup><mi>G</mi></mfrac><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula> method. Secondly, by constructing the new ansatz solutions and applying the improved tanh function method, many non-traveling wave exact solutions of the equation are given. The solutions mainly include hyperbolic, trigonometric and rational functions, which reflect different types of solutions for nonlinear waves. Finally, we discuss the effects of these solutions on the formation of rogue waves according to the numerical simulation.