Prove that $(L^R)^* = (L^*)^R$
If $w \in (L^R)^*$, then $w = w_1 \ldots w_\ell$ for some $w_1,\ldots,w_\ell \in L^R$. Hence $w_1^R,\ldots,w_\ell^R \in L$, and so $w_\ell^R \ldots w_1^R \in L^*$. This shows that $w = (w_\ell^R \ldots w_1^R)^R \in (L^*)^R$.
Similarly, if $w \in (L^*)^R$ then $w = x^R$ for $x \in L^*$. Since $x \in L^*$, $x = x_1 \ldots x_\ell$ for some $x_1,\ldots,x_\ell \in L$. Then $x_1^R,\ldots,x_\ell^R \in L^R$, and so $w = (x_1 \ldots x_\ell)^R = x_\ell^R \ldots x_1^R \in (L^R)^*$.