# What does $2^{2.5}$ mean [duplicate]

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We know that $2^2$ means $2\cdot 2$, $2^3$ means $2\cdot 2\cdot 2$, and $2^5$ means $2\cdot 2\cdot 2\cdot 2\cdot 2$ and so on and so forth. But what does $2^{2.5}$ mean? I don't mean to say "How to find it out". I mean, what is the idea behind it? Is it something like "If $S=V\cdot T$, then $T=\frac SV$ and $T$ gets smaller as $V$ gets bigger, and theoretically $V$ can get so big that $T$ might become $0$. (that is, $\lim_{V\to \infty} \frac SV = 0, (T = \frac SV))$" but can't be shown physically?

## marked as duplicate by Zev Chonoles, Community♦Jan 19 '17 at 6:50

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• It means you need braces to get more than one character in a superscript, thus to get $2^{2.5}$ you should put in 2^{2.5}. – Oscar Lanzi Jan 21 '17 at 1:49

## 1 Answer

You multiply 2*2*2^(1/2) in other words when you do 2^(2.5).

2^(1/2) is sqrt(2) by definition. There is no explanation other than the definition of a square root. 2^(2.5)=4sqrt2 but you can't show it physically; 4sqrt2 and sqrt2 are indeed constructible, but you can't say "I take one object, double the amount I have, double it again, then take the amount of objects I have and multiply that by sqrt2; now I have 4sqrt2 objects."