Is $(n^5 + n^7)\in \Omega(n^7)$? Shouldn't it be in $\Omega(n^5)$?
The function $n^5 + n^7$ is big-$Ω$ of both. However, it is little-$\omega$ of $n^5$.

It happens that $n^5 + n^7 \in \Theta(n^7)$. These asymptotic-notations only describe the behavior of the function $f(n) = n^5 + n^7$ for large $n$, aka "for all $n &gt; \textrm{some value}$". The function is a polynomial function, and the highest-degree term determines the order of growth.

From the definition of $Θ(g)$, it means that function $f$ is both big-$O$ and big-$Ω$ of $g$. The function is bounded by some constant multiple of $n^7$ for large $n$. Namely, the function is bounded above by $2n^7$ and below by $n^7$.

Nevertheless, $n^5 + n^7 \in \Omega{(n^5)}$; the function is of [strictly] greater order than a quintic. Equivalently, $n^5 \in O(n^5 + n^7)$.