A new method is proposed for evaluating electric multipole transition radial (EMTR) matrix elements of the hydrogen-like atoms using the expansion formulae suggested by one of the authors for the radial parts of 蠄伪-exponential type orbitals (蠄伪-ETO) in terms of radial parts of Slater type orbitals (STO), where 伪 is the frictional quantum number (鈭掆垶<伪鈮?). A comparative study is carried out between the proposed analytical computations and other numerical simulations. The effectiveness and accuracy of the obtained analytical expressions are demonstrated by calculation of concrete cases.