  In this paper we study minimal free resolutions of some classes of monomial
ideals. we first give a sufficient condition to check the minimality of the
resolution obtained by the mapping cone. Using it, we obtain the Betti numbers
of max-path ideals of rooted trees and ideals containing powers of variables.
In particular, we discuss about resolutions of ideals of the form
$J_{\mathcal{H}}+(x_{i_1}^2,\ldots, x_{i_m}^2)$ where $J_{\mathcal{H}}$ is the
edge ideal of a hypergraph $\mathcal{H}$.
