$(M^n,g)$ is a compact $n$ dimensional manifold of negative curvature with n>2 . let $\alpha$ be a simple closed geodesic loop in $M$ based at a point $p$

1. will the geodesic in the free homotopy class of alpha be simple ?

2. can $\alpha$ be homotopic ( with respect to $p$ ) to a power of another closed curve at $p$