Finding right quotient of $a^*b^*/b^*.$
$a^*b^* / b^* = \{ w \ | \ \exists \ x, \ x \in b^* \land wx \in a^*b^*\} = a^*b^*$. This is because $wx \in a^*b^* \text{ AND } x \in b^*$ means $w$ is of the form $a^ib^j$ for some $i,j \in \mathbb{N} \cup \{0\}$. Finish it off now.