  We consider the theory of algebraically closed fields of characteristic zero
with multivalued operations $x\mapsto x^r$ (raising to powers). It is in fact
the theory of equations in exponential sums.
  In an earlier paper we have described complete first-order theories of such
structures, conditional on a Diophantine conjecture that generalises the
Mordell-Lang conjecture (CIT). Here we get this result unconditionally.
  The field of complex numbers with raising to real powers satisfies the
corresponding theory if Schanuel's conjecture holds. In particular, we have
proved that a (weaker) version of Schanuel's conjecture implies that every
well-defined system of exponential sums with real exponents has a solution in
the complex numbers. Recent result by Bays, Kirby and Wilkie states that the
required version of Schanuel's conjecture holds for almost every choice of
exponents. It follows that for the corresponding choice of real exponents we
have an unconditional description of the first order theory of the complex
numbers with raising to these powers.
