Root closed function algebras on compacta of large dimension

Let $X$ be a Hausdorff compact space and let $C(X)$ be the algebra of all continuous complex-valued functions on $X$, endowed with the supremum norm. We say that $C(X)$ is (approximately) $n$-th root closed if any function from $C(X)$ is (approximately) equal to the $n$-th power of another function. We characterize the approximate $n$-th root closedness of $C(X)$ in terms of $n$-divisibility of the first Cech cohomology groups of closed subsets of $X$. Next, for each positive integer $m$ we construct an $m$-dimensional metrizable compactum $X$ such that $C(X)$ is approximately $n$-th root closed for any $n$. Also, for each positive integer $m$ we construct an $m$-dimensional compact Hausdorff space $X$ such that $C(X)$ is $n$-th root closed for any $n$.