A Family of surjective word map on SU(2)

Let $\mathbf{F}$ be the free group on two generators $a,b$ and let a family of words $w=[[a,b],[a^3,b^n]]$ in $\mathbf{F}$. In this paper we examine surjectivity of word map $w$ on special unitary group SU(2) over complex field $\mathbf{C}$.