  Let $w$ be a Muckenhoupt $A_2(\mathbb{R}^n)$ weight and
$L_w:=-w^{-1}\mathop\mathrm{div}(A\nabla)$ the degenerate elliptic operator on
the Euclidean space $\mathbb{R}^n$, $n\geq 2$. In this article, the authors
establish some weighted $L^p$ estimates of Kato square roots associated to the
degenerate elliptic operators $L_w$. More precisely, the authors prove that,
for $w\in A_{p}(\mathbb{R}^n)$, $p\in(\frac{2n}{n+1},\,2]$ and any $f\in
C^\infty_c(\mathbb{R}^n)$, $\|L_w^{1/2}(f)\|_{L^p(w,\,\mathbb{R}^n)}\sim
\|\nabla f\|_{L^p(w,\,\mathbb{R}^n)}$, where $C_c^\infty(\mathbb{R}^n)$ denotes
the set of all infinitely differential functions with compact supports.
