Justify the order of asymptotic growth for the following pairs of functions by giving a brief explanation.

$($a$)1000nlgn$ and $n^2,$

$($b$)\sqrt[\phantom{2}]{10n^2+200n^3}$ and $\sqrt[3]{n^4+20n+30n^5+100}$

$($c$)n!$ and $n^n,$

$($d$)n^2$ and $n*2^n$

$($e$)n^{n^n}$and $2^{2^n}$