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We define natural volume forms on $n$-dimensional oriented pseudo-Finslerianmanifolds with non-degenerate $m$-th root metrics. Our definitions of thenatural volume forms depend on the parity of the positive integer $m>1$. Theadvantage of the stated definitions is their algebraic structure. The naturalvolume forms are expressed in terms of Cayley hyperdeterminants. In particular,the computation of the natural volume form does not require the difficultintegration over the domain within the indicatrix in the tangent space $T_xM^n$ of the pseudo-Finslerian manifold at a point $x$.

Natural volume forms on pseudo-Finslerian manifolds with $m$-th root metrics