Let Γ ⊂ Q ¯ × $\Gamma \subset \overline{\mathbb {Q}}^{\times }$ be a finitely generated multiplicative group of algebraic numbers. α 1 , … r ∈ $\alpha _1,\ldots ,\alpha _r\in {Q}}^\times$ numbers which are $\mathbb {Q}$ -linearly independent and let ε > 0 $\epsilon >0$ given real number. One the main results that we prove in this article is as follows: There exist only many tuples ( u q p ) Z +...