if $a=O(N^2) $ then according to the big oh definition I didn't see why we can't say $a= O(N^4)$ or $= O(N^8)$
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Yes, you can say either of those. They don't tell you as much about $a(N)$ as $a(N)=\mathcal O(N^2)$ does, but what little they do say is true.