Let $F$ be a hyperbolic surface and $p\in F$ be a point. Consider $\pi_1(F,p)$, the fundamental group of $F$ with base point $p$. Let $x,y\in \pi_1(F,p)$ and $z$ be a simple closed curve in $F$ such that $i(z,x*y)\neq 0$ where $i(~,~)$ denotes the geometric intersection number (i.e. minimum number of possible intersection points between the free homotopy classes of each curves).
Q) Is $i(z,[(x*y)^n*(y*x)])\neq 0$ for all $n\in \mathbb{N}$?
My intuition is, as $i(z,x*y)\neq 0$, powers of the same curve intersects $z$ non-trivially and as (pictorially) there is no back tracking, the answer should be true. But I am not sure whether this could give a proof.
(P.S. As Ian Agol pointed out in the comments, this is not true for non-orientable case. Therefore assume $F$ to be orientable.)
A more general question is the following.
Let $z$ be a simple closed curve and $x,y\in \pi_1(F,p)$ such that $i(z,x*y)= 0$ and $y$ is not conjugate to $x^{-1}$. Then $i(z,x)=i(z,y)=0$.
But this general statement is false. A counter example is given in the following picture.
