On the motive of the Quot scheme of finite quotients of a locally free sheaf

Let $X$ be a smooth variety, $E$ a locally free sheaf on $X$. We express the generating function of the motives $[{\text{Quot}}_X(E,n)]$ in terms of the power structure on the Grothendieck ring of varieties. This extends a recent result of Bagnarol, Fantechi and Perroni for curves, and a result of Gusein-Zade, Luengo and Melle-Hern'{a}ndez for Hilbert schemes. We compute this generating function for curves and we express the relative motive $[$\text{Quot}$_{\mathbb A^d}(\mathscr{O}^{\oplus r}) \to $\text{Sym}$, \mathbb A^d]$ as a plethystic exponential.