The field of reals with a predicate for the real algebraic numbers and a predicate for the integer powers of two

Given a theory T̃ of a polynomially bounded expansion R̃ of R̄ with the field of exponents Q, we introduce a theory T whose models are comprised of a dense pair of models of T̃ and a discrete multiplicative group. We will then prove that T is complete and admits quantifier elimination when predicates are added for certain existential formulas. In particular if T̃ = RCF then T axiomatises 〈R̄,Ralg, 2Z〉. In the final section we prove that T is dependent.