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I have been trying to figure out how the following has been simplified, but I am getting nowhere with it. Anyone have any ideas?

$9(n/3)^{5/2}$ to $(1/3)^{1/2} f(n)$

It is given that $f(n) = n^{2.5}$.

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Neither algorithms nor computer-science seem like appropriate tags. – mrf Jan 9 '13 at 23:41
up vote 4 down vote accepted

$$\begin{align*} 9\left(\frac{n}3\right)^{5/2}&=9\left(\frac13\right)^{5/2}n^{5/2}\\\\ &=3^2\cdot 3^{-5/2}f(n)\\\\ &=3^{2-5/2}f(n)\\\\ &=3^{-1/2}f(n)\\\\ &=\left(\frac13\right)^{1/2}f(n) \end{align*}$$

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$$9\left(\frac{n}{3}\right)^{5/2} = 9\cdot \left(\frac13\right)^{5/2} n^{5/2}$$ $$=9\cdot\dfrac{1}{3^{5/2}} f(n)$$ $$= \dfrac{3^2}{3^2\cdot3^{1/2}}f(n)$$ $$= \frac{1}{3^{1/2}}f(n)$$ $$=\left(\dfrac{1}{3}\right)^{1/2} f(n)$$

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