  We study the growth rate of some power-free languages. For any integer $k$
and real $\beta>1$, we let $\alpha(k,\beta)$ be the growth rate of the number
of $\beta$-free words of a given length over the alphabet $\{1,2,\ldots, k\}$.
Shur studied the asymptotic behavior of $\alpha(k,\beta)$ for $\beta\ge2$ as
$k$ goes to infinity. He suggested a conjecture regarding the asymptotic
behavior of $\alpha(k,\beta)$ as $k$ goes to infinity when $1<\beta<2$. He
showed that for $\frac{9}{8}\le\beta<2$ the asymptotic upper-bound holds of his
conjecture holds. We show that the asymptotic lower-bound of his conjecture
holds. This implies that the conjecture is true for $\frac{9}{8}\le\beta<2$.
