In this paper, the modified exponential function method is applied to find the exact solutions of the Radhakrishnan-Kundu-Lakshmanan equation with Atangana's conformable beta-derivative. For this, the definition of the conformable beta derivative proposed by Atangana and the properties of this derivative are firstly given. Then, the exact solutions of the nonlinear Radhakrishnan-Kundu-Lakshmanan equation which can be stated with the conformable beta-derivative of Atangana are obtained by using of the presented method. For the related problem in this paper, two solution cases are obtained, in each case five different solution families. The exact solutions found as a result of the application of the method seem to be 1-soliton solutions, dark soliton solutions, periodic soliton solutions and rational function solutions. According to the obtained results, it can be said that the Radhakrishnan-Kundu-Lakshmanan equation with Atangana's conformable beta-derivative has different kinds of soliton solutions. Also, three-dimensional contour and density graphs and two-dimensional graphs drawn with different parameters are given of these new exact solutions. These graphs give detailed informations about the physical behavior of the real and imaginary parts of the exact solutions obtained.