the probability of decay of a particle into two due to cubic in $\hat{x}$ interaction is given by $\langle f \mid \hat{x}^3\mid i\rangle$. The $\hat{x}^3$ term is written in the basis of ladder operators and takes the form: $$\sum_{k,k',q}\langle f \mid(a_k +a_k^\dagger) (a_{k'} + a_{k'}^\dagger) (a_q + a_q^\dagger)\mid i\rangle \delta (E_f-E_i)$$ where the states are number states of the form $\mid n_k \rangle \otimes \mid n_{k'} \rangle \otimes\mid n_q \rangle$. The are other factors of matrix elements and c-numbers. But let's get to the point. How do I simplify this expression and minimize the effort of considering all 8 terms? Properties like symmetry, cases where $k=k'=q$, etc., and other conditions might simplify the expression and make the calculation easier. Any words of wisdom or reference to sources like books, papers, and review articles will be a great help. Thank you.
Edit: @doublefelix pointed out that no commutator was prescribed. So let the commutator be $$[a_k,a_q^\dagger]=\delta_{k,q}$$.