  In this article we apply a formula for the $n$-th power of a $3\times 3$
matrix (found previously by the authors) to investigate a procedure of
Khovanskii's for finding the cube root of a positive integer. We show, for each
positive integer $\alpha$, how to construct certain families of integer
sequences such that a certain rational expression, involving the ratio of
successive terms in each family, tends to $\alpha^{1/3}$. We also show how to
choose the optimal value of a free parameter to get maximum speed of
convergence. We apply a similar method, also due to Khovanskii, to a more
general class of cubic equations, and, for each such cubic, obtain a sequence
of rationals that converge to the real root of the cubic. We prove that
Khovanskii's method for finding the $m$-th ($m \geq 4$) root of a positive
integer works, provided a free parameter is chosen to satisfy a very simple
condition. Finally, we briefly consider another procedure of Khovanskii's,
which also involves $m \times m$ matrices, for approximating the root of an
arbitrary polynomial of degree $m$.
