Doubt regarding "Elementary approach to proving that a group of order 9 is Abelian"

Question

I am trying to understand the solution of this problem . I am unable to understand why :

If $yx=x^2y$, then $yxy^{-1}=x^2$. This means that $y^3xy^{-3}=x^8 $

It seems like I am missing something silly , but why is this true ?

Answer 1

$$ \begin{align} y^3xy^{-3}&=y^2(yxy^{-1})y^{-2}\\ &=y^2x^2y^{-2}\\ &=y(yx^2y^{-1})y^{-1}\\ &=y(yxy^{-1})^2y^{-1}\\ &=y(x^2)^2y^{-1}\\ &=yx^4y^{-1}\\ &=(yxy^{-1})^4\\ &=(x^2)^4\\ &=x^8 \end{align} $$