  In this paper, we establish improved effective irrationality measures for
certain numbers of the form $\sqrt[3]{n}$, using approximations obtained from
hypergeometric functions. These results are very close to the best possible
using this method. We are able to obtain these results by determining very
precise arithmetic information about the denominators of the coefficients of
these hypergeometric functions. Improved bounds for $\theta(k,l;x)$ and
$\psi(k,l;x)$ for $k=1,3,4,6$ are also presented.
