This may be a trivial question, but somehow I just can't seem to find an easy enough argument, and I lose myself in complicated constructions.
Let $k$ be a field and let $k\langle x,y\rangle$ be the free non-commutative, associative algebra on two generators. I would like to show that $x^n\notin (xy-yx-1)$, where $(xy-yx-1)$ is the two sided ideal generated by $xy-yx-1$. How to do so?