Powers in finitely generated groups

In this paper we study the set Γⁿ of n^th-powers in certain finitely generated groups Γ. We show that, if Γ is soluble or linear, and Γⁿ contains a finite index subgroup, then Γ is nilpotent-by-finite. We also show that, if Γ is linear and Γⁿ has finite index (i.e. Γ may be covered by finitely many translations of Γⁿ), then Γ is soluble-by-finite. The proof applies invariant measures on amenable groups, number-theoretic results concerning the S-unit equation, the theory of algebraic groups and strong approximation results for linear groups in arbitrary characteristic.