When is it possible to add different powers of a dimensioned quantity?

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When is it possible to add different powers of a dimensioned (i.e. non-dimensionless) quantity, e.g. consider power series for dimensioned quantities. For example, when can you exponentiate a dimensioned quantity? I have heard that this is done in symplectic geometry (e.g. $e^\omega$) but I don't know any more beyond this.