  Let $J$ be the Jacobian of a superelliptic curve defined by the equation
$y^{\ell} = f(x)$, where $f$ is a separable polynomial of degree non-divisible
by $\ell$. In this article we study the "exponential" (i.e. $\ell$-power)
torsion of $J$. In particular, under some mild conditions on the polynomial
$f$, we determine the image of the associated $\ell$-adic representation up to
the determinant. We show also that the image of the determinant is contained in
an explicit $\mathbb Z_{\ell}$-lattice with a finite index. As an application,
we prove the Hodge, Tate and Mumford-Tate conjectures for a generic
superelliptic Jacobian of the above type.
