Calculation in Spin-Orbital Coupling for Hydrogen atom

I was calculating the spin-orbital correction in the fine structure of Hydrogen, but then bumped into an irritating problem.

In the process of calculation, I must calculate $langle n,l,j,m_j|frac{overrightarrow{S}cdotoverrightarrow{L}}{r^3}|n,l,j,m_jrangle $. In all of the textbooks and posts I read, they simply calculate the term as $langleoverrightarrow{S}cdotoverrightarrow{L}ranglelanglefrac{1}{r^3}rangle$.

Since I consider $frac{1}{r^3}$ an operator, I am wondering how the expectation value of the product of operators can be treated as the product of the expectation values.

Thanks in advance 🙂