Simplify $w^{3n}, w^{6n}, w^{3 + 3n}, w^{4+3n}$, etc. using $w^3=1$

Question

I having an expression containing terms that are powers of $w$. How can I simplify these terms using the assumption $w^3=1$?

Simplify[expr, w^3==1 && n ∈ PositiveIntegers]

doesn't work for terms that have symbolic variable, e.g. $n$, in the exponent.

EDIT: Example expression:

( -a*w^3 + b*(n+1)*w^6 + 9*w^(3+3*n)*Sum[d[i], {i,1,n}]^2 - w^(3+6*n)*Sum[c[i], {i,1,n}] ) / (s^2*w^(4 + 3*n) )

Answer 1

Since $w^3=1$, one can use $w^\alpha=w^{\alpha \text{ mod } 3}$ to simplify arbitrary powers of $w$. This can be implemented as a simple rule:

wrule = w^(x_) :> w^Refine[Mod[x, 3], Assumptions->Element[n, Integers]];

Examples:

{w^1, w^2, w^3, w^4, w^(3n), w^(6n), w^(3+3n), w^(4+3 n), w^(-4-3n)} /. wrule

{w, w^2, 1, w, 1, 1, 1, w, w^2}

Answer 2

A good strategy when attempting to replace portions of an expression is to look at the FullForm.

Below is an example expression:

expr = w^4 + w^(3 n) + w^(6 n) + w^(3 + 3 n) + w^(4 + 3 n)

and the associated FullForm

FullForm[expr]

By examining the FullForm of expr we see three forms where w is raised to a power involving integers: directly, with a times or with a plus and times.

Use patterns with ReplaceAll to accomplish the job.

expr //. {
  Power[w, Plus[integer_Integer /; integer > 2, rest_]] -> 
   Power[w, Plus[integer - 3, rest]],
  Power[w, integer_Integer] -> Power[w, integer - 3],
  Power[w, Times[integer_Integer, n]] -> 
   Power[w, Times[integer - 3, n]]
  }

Update

Upon request, the following expression was given as an example.

expr = (-a*w^3 + b*(n + 1)*w^6 + 
    9*w^(3 + 3*n)*Sum[d[i], {i, 1, n}]^2 - 
    w^(3 + 6*n)*Sum[c[i], {i, 1, n}])/(s^2*w^(4 + 3*n))

Now one additional rule to handle negative integers is required.

expr //. {
  Power[w, Plus[integer_Integer /; integer > 2, rest_]] -> 
   Power[w, Plus[integer - 3, rest]],
  Power[w, Plus[integer_Integer /; integer < -2, rest_]] -> 
   Power[w, Plus[integer + 3, rest]],
  Power[w, integer_Integer] -> Power[w, integer - 3],
  Power[w, Times[integer_Integer, n]] -> 
   Power[w, Times[integer - 3, n]]
  }

One might want to go further and assume that n is an integer. I leave that as an exercise for you.