  We provide an upper bound on the efficient irrationality exponents of cubic
algebraics $x$ with the minimal polynomial $x^3 - tx^2 - a$. In particular, we
show that it becomes non-trivial, i.e. better than the classical bound of
Liouville in the case $|t| > 19.71 a^{4/3}$. Moreover, under the condition $|t|
> 86.58 a^{4/3}$, we provide an explicit lower bound on the expression $||qx||$
for all large $q\in\mathbb{Z}$. These results are based on the recently
discovered continued fractions of cubic irrationals and improve the currently
best-known bounds of Wakabayashi.
