We study the growth rate of some power-free languages. For any integer $k$ and real $β>1$, we let $α(k,β)$ be the growth rate of the number of $β$-free words of a given length over the alphabet $\{1,2,\ldots, k\}$. Shur studied the asymptotic behavior of $α(k,β)$ for $β\ge2$ as $k$ goes to infinity. He suggested a conjecture regarding the asymptotic behavior of $α(k,β)$ as $k$ goes to infinity when $1<β<2$. He showed that for $\frac{9}{8}\leβ<2$ 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 $\frac{9}{8}\leβ<2$.