  This paper contains two topics, the index of a power subgroup in the mapping
class group $\mathcal{M}(0,2n)$ of a $2n$-punctured sphere and in the
hyperelliptic mapping class group $\Delta(g,0)$ of an oriented closed surface
of genus $g$. The main tool is a projective representation of
$\mathcal{M}(0,2n)$ obtained through the Kauffman bracket skein module. For
$\mathcal{M}(0,2n)$, we prove that the normal closure of the fifth power of a
half-twist has infinite index. This is the remaining case of a Masbaum's work.
For $\Delta(g,0)$, we consider the normal closure of $m$-th powers of Dehn
twists along all symmetric simple closed curves. We show the subgroup has
infinite index if $m\geq 5$ and $m\neq 6$ for any $g\geq 2$.
