= Solution
{c}
Fix $x\in G$. The <integer powers of a group element> satisfy
$$
x^{z+w}=x^zx^w,\qquad (x^w)^z=x^{wz}=x^{zw}\qquad(z,w\in\mathbb Z).
$$
For <nonnegative integers> these rules follow by <associativity> and counting factors; negative <integers> are handled by $x^{-r}=(x^{-1})^r$, and opposite signs by cancellation of adjacent <inverse elements>. All factors are powers of the same element, so no <abelian group> assumption is involved.
It follows that $z\mapsto\rho(z,x)=x^z$ is a <group homomorphism> from the <additive group of integers> to $G$, and
$$
\boxed{\rho(z,\rho(w,x))=\rho(zw,x).}
$$
The homomorphism uses ordinary <addition> on $\mathbb Z$; composition of the power <functions> corresponds to ordinary <multiplication> of exponents.
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