Powers of commutators in linear algebraic groups

Let G be a linear algebraic group over k , where k is an algebraically closed field, a pseudo-finite field or the valuation ring of a non-archimedean local field. Let G= G(k) . We prove that if γ∈ G such that γ is a commutator and δ∈ G such that δ= γ then δ is a commutator. This generalises a result of Honda for finite groups. Our proof uses the Lefschetz principle from first-order model theory.