Summary: In a number of cases the minimal polynomials of the images of unipotent elements of non-prime order in irreducible representations of the exceptional algebraic groups in good characteristics are found. It is proved that if \(p > 5\) for a group of type \(E_8\) and \(p > 3\) for other exceptional algebraic groups, then for irreducible representations of these groups in characteristic \(p\) with large highest weights with respect to \(p\), the degree of the minimal polynomial of the image of a unipotent element is equal to the order of this element.