Consider the equation q1αx1+…+qkαxk=q, with constants α∈Q‾∖{0,1}, q1,…,qk,q∈Q‾ and unknowns x1,…,xk, referred to in this paper as an algebraic equation with exponents. We prove that the problem to decide if a given equation has an integer solution is NP-complete, and that the same holds for systems of equations (whether α is fixed or given as part of the input). Furthermore, we describe the set of all solutions for a given system of algebraic equations with exponents and prove that it is semilinear.