The group \(\hbox{Sym }\mathbb{R}\) of permutations of the set \(\mathbb{R}\) of real numbers is considered in the paper. Let \(P_ 0\) be its subgroup of power functions \(x\mapsto x^{m/n}\) where \(m\), \(n\) are positive odd integers, \(T\) be the subgroup of translations \(x\mapsto x+a\) where \(a\) is a real algebraic number, and \(M\) be the subgroup of multiplications \(x\mapsto bx\) where \(b\) is a positive algebraic number. It is proved that \(gp(P_ 0,T)=P_ 0*T\) and \(gp(P_ 0,M,T)=P_ 0M*_ M MT\) (Theorems \(B\) and \(C\)). In particular, Theorem B implies immediately that the translation \(x\mapsto x+1\) and the power function \(x\mapsto x^ n\) are generators of a free group of degree 2 whenever \(n\geq 3\) is a fixed odd integer (Theorem A); this was known earlier only for prime \(n\) [\textit{S. White}, J. Algebra 118, 408-422 (1988; Zbl 0662.20024)].