changing the power of 2 to the power of 3

Question

this is a really simple question, I'm solving a time complexity program, to find the order of the program, however when it gets down to simplifying the mathematical part, I get stuck.

I want to get $k^3$ in terms of $n$ where $\dfrac{k^2}{2} = \dfrac{n}{2}$

$\implies k^2 = n$

$\implies k^3 = ??$

originally i had $k^3 = n^{1.5}$ but when I plug in values, it doesn't work out to be the same

Answer 1

If we know (with suitable restrictions on $k$ and $n$) that $k^2 = n$, then we can say $$ k = \sqrt{n} = n^{1/2}$$

So simply cubing both sides yields the desired $$\bbox[10px, border: solid blue 1px]{k^3 = (\sqrt{n})^3 = \sqrt{n^3}= n^{3/2}}$$

So your original answer was correct. $k$ is indeed equal to $n^{1.5}$, your error is probably found elsewhere.

Answer 2

Observe that, using $k\geq0,\,n\geq0$, you have $$ k^2=n \implies k=n^{1/2} $$ giving $$ k^3=\left(n^{1/2}\right)^3=n^{3/2}. $$