# How to eliminate a constant factor from an expression, ideally without Rules

Consider this expression as rendered by default StandardForm in MMA (my installation, 11.0.1.0, Windows)

expr = ((3*2^Sum[Subscript[q, j], {j, 1, -1 + P}] + 3^P)*(2^Sum[Subscript[q, j], {j, 1, P}] + 3^P) + 3^(1 + 2*P)*n +
3*(2^Sum[Subscript[q, j], {j, 1, P}] + 3^P)*Sum[2^Sum[Subscript[q, j], {j, 1, i}]*3^(-i + P), {i, 2, -1 + P}])/
(3*2^Sum[Subscript[q, j], {j, 1, P}])


When written out (here via TeXForm, which is not quite the same as StandardForm) it is readily apparent that the factor of $$1/3$$ cancels numerous explicit factors of $$3$$ and can be absorbed in factors of $$3^m$$ by the replacement $$m \rightarrow m-1$$

$$\frac{1}{3} 2^{-\sum _{j=1}^P q_j} \left(3 \left(2^{\sum _{j=1}^P q_j}+3^P\right) \left(\sum _{i=2}^{P-1} 3^{P-i} 2^{\sum _{j=1}^i q_j}\right)+\left(3\ 2^{\sum _{j=1}^{P-1} q_j}+3^P\right) \left(2^{\sum _{j=1}^P q_j}+3^P\right)+n 3^{2 P+1}\right)$$

The question is, how can I force MMA to make this simplification? Simplify does not do it, FullSimplify times-out.

If I use Expand[expr] I obtain

$$3^P 2^{-\sum _{j=1}^P q_j} \sum _{i=2}^{P-1} 3^{P-i} 2^{\sum _{j=1}^i q_j}+\sum _{i=2}^{P-1} 3^{P-i} 2^{\sum _{j=1}^i q_j}+n 3^{2 P} 2^{-\sum _{j=1}^P q_j}+3^P 2^{\sum _{j=1}^{P-1} q_j-\sum _{j=1}^P q_j}+3^{2 P-1} 2^{-\sum _{j=1}^P q_j}+2^{\sum _{j=1}^{P-1} q_j}+3^{P-1}$$

and then have factors such as $$3^P$$ that can/should be factored out again (where they occur in isolation; I would strongly prefer not to have $$3^{p-1} \rightarrow 3^P /3$$, for example} (Is there a pattern I could supply to Expand to avoid this?)

I suspect this could be done with replacement rules, but I fear the required rules would be rather ad hoc and would much prefer a (simpler) generic solution.

• Related to getting 3^p out of the sum: (16969), (58091), (226003) -- Another point: 3^p/3 won't stay in that form. Oct 18, 2022 at 11:24
• @MichaelE2 Indeed! $3^P/3 \rightarrow 3^{-1+P}$, with leading -1, yet another - but separate - peeve! Oct 18, 2022 at 17:11