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For an integer $M\geq 2$ and a finite group $G$, an element $\alpha\in G$ iscalled an $M$-th power if it satisfies $A^M=\alpha$ for some $A\in G$. In thisarticle, we will deal with the case when $G$ is finite symplectic or orthogonalgroup over a field of order $q$. We introduce the notion of $M^*$-power SRIMpolynomials. This, amalgamated with the concept of $M$-power polynomial, weprovide the complete classification of the conjugacy classes of regularsemisimple, semisimple, cyclic and regular elements in $G$, which are $M$-thpowers, when $(M,q)=1$. The approach here is of generating functions, as workedon by Jason Fulman, Peter M. Neumann, and Cheryl Praeger in the memoir "Agenerating function approach to the enumeration of matrices in classical groupsover finite fields". As a byproduct, we obtain the corresponding probabilities,in terms of generating functions.

Powers in finite orthogonal and symplectic groups: A generating function approach