Use a subgroup lattice to compute a normalizer The following screenshot is from this page:

To prove $r^2 \in N_G(\langle s,r^4\rangle)$, Why we don't need to consider $(sr^4)^2, (sr^4)^3,\ldots,(sr^4)^n$ and $s^nr^{4m}$ where $n\neq m$ are integers. That is, the multiple combinations of $s$ and $r^4$.
That is, why don't we need to consider the result $r^2(sr^4)^2r^{-2}$,  $r^2(sr^4)^3r^{-2}$,  $r^2(sr^4)^4r^{-2},\ldots$ and $r^2(s^nr^{4m})r^{-2}$? 

Note that if $H$ is a subgroup and $gag^{-1}, gbg^{-1} \in H$ then automatically we have $gabg^{-1} \in H$ because $gabg^{-1} = (gag^{-1})(gbg^{-1})$.  Thus once you have tested that conjugation does not take $sr^4$ outside of your subgroup it is then automatic that for any $n$ it does not take $(sr^4)^n$ out of your subgroup.
In general to test that $gHg^{-1} \subseteq H$ is suffices to test that $gag^{-1} \in H$ for each $a$ in a set of generators of $H$.
