Let me first state the definitions :
A not-nullhomotopic closed curve / loop $c$ on a an orientable surface $X,c:[0,1]\to X$ is called simple closed curve is $c|[0,1)$ is injective and [ $c(0)=c(1) ] ; $ A closed curve / loop $c$ is called primitive if in the fundamental group $\pi_1(X,c(1)),$ the homotopy class $[c]$ can NOT be written as $[c]= [\gamma]^n$ for some closed curve $\gamma$ with $\gamma(0)=\gamma(1)=c(0)=c(1) $ and for some $n\ge 2$.
My question is : rigorously prove that simple closed curves are primitive .
It is visually pretty clear , but I have difficulty proving it. Thanks !