Is $O(2^{n/2})$ the same as $O(2^n)$?

Question

Why or why not? It seems like the answer should be no, but on the other hand, it's weird that you'd reach the same value in a constant multiple of n.

Answer 1

$2^{n}$ is $O(2^{n})$ but it is not $O(2^{n/2})$.

Answer 2

I made the same mistake once but if you write $x=2^n$, then you would have $O(x) = O(x^2)$ which of course doesn't make sense.

Answer 3

By exponent rules, $O(2^{n/2}) = O((2^{1/2})^n) \approx O(1.414^n)$ which clearly differs from $O(2^n)$ by more than a constant factor (consider the ratio).