  Let $S_g$ be a closed orientable surface of genus $g \geq 2$ and $C$ a simple
closed nonseparating curve in $F$. Let $t_C$ denote a left handed Dehn twist
about $C$. A \textit{fractional power} of $t_C$ of \textit{exponent}
$\fraction{\ell}{n}$ is an $h \in \Mod(S_g)$ such that $h^n = t_C^{\ell}$.
Unlike a root of a $t_C$, a fractional power $h$ can exchange the sides of $C$.
We derive necessary and sufficient conditions for the existence of both
side-exchanging and side-preserving fractional powers. We show in the
side-preserving case that if $\gcd(\ell,n) = 1$, then $h$ will be isotopic to
the $\ell^{th}$ power of an $n^{th}$ root of $t_C$ and that $n \leq 2g+1$. In
general, we show that $n \leq 4g$, and that side-preserving fractional powers
of exponents $\fraction{2g}{2g+2}$ and $\fraction{2g}{4g}$ always exist. For a
side-exchanging fractional power of exponent $\fraction{\ell}{2n}$, we show
that $2n \geq 2g+2$, and that side-exchanging fractional powers of exponent
$\fraction{2g+2}{4g+2}$ and $\fraction{4g+1}{4g+2}$ always exist. We give a
complete listing of certain side-preserving and side-exchanging fractional
powers on $S_5$.
