Is star closure of reverse of language equivalent to reverse of closure of that language
Take any $w\in {(L^*)}^R$. Then $w$ can be writen as $w=(u_1\cdot u_2\cdots u_n)^R$, with $u_i\in L$. We have

$$
w =(u_1\cdot u_2\cdots u_n)^R =u_n^R \cdot u_{n-1}^R \cdots u_1^R,
$$

and therefore $w\in (L^R)^*$.

Assume now that  $w\in {(L^R)}^*$, then by the same argument 
$$
w =u_1^R \cdot u_{2}^R \cdots u_n^R= (u_n\cdot u_{n-1}\cdots u_1)^R ,
$$
and hence $w\in {(L^*)}^R$.

As a consequence ${(L^*)}^R={(L^R)}^*$