Operator ordering is often fallen back on due to its convenience in quantum optics and quantum statistics, thus it is an important task to derive the various ordered forms of operators as directly as possible. In this paper we arrange quantum mechanical operators $ {\left(a{a}^\dagger \right)}^{\pm n} $ and $ {\left({a}^\dagger a\right)}^{\pm n} $ in their normally and anti-normally ordered product forms by using special functions and general mutual transformation rules between normal and anti-normal orderings of operators. Furthermore, the Q- and P-ordered forms of power operators $ {\left(XP\right)}^{\pm n} $ and $ {\left(PX\right)}^{\pm n} $ are also obtained by the analogy method. Finally, some applications are discussed, such as the Glauber-Sudarshan $ P $ -representation of chaotic light field and the generating functions of even and odd bivariate Hermite polynomials.