# Should brackets be placed around an exponentiated factorial?

For example, one can derive an approximation of $\pi$ from Stirling's approximation with one additional term as $$\lim_{n \to \infty} \frac{72n(n!)^2}{n^{2n} e^{-2n} (12n+1)^2}$$

but is it correct to write

$$\lim_{n \to \infty} \frac{72n(n!^2)}{n^{2n} e^{-2n} (12n+1)^2}?$$

Is the ! sufficient to separate the factoriand (if that's the word) from the exponent? I've seen both used in various places.

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I changed \frac{72n(n!)^2}{{n^{2n}} {e^{-2n}}{(12n+1)^2}} to \frac{72n(n!)^2}{n^{2n} e^{-2n} (12n+1)^2}. The former is hard to read and edit. – Michael Hardy May 1 '13 at 21:57

This is really a matter of preference. I think either one is fine since $!^2$ has no meaning on its own, so the only way to interpret $n!^2$ is as $(n!)^2$. That being said, I think $(n!)^2$ looks a bit nicer.
I would say yes, since $n n!$ could be interpreted as $n(n!)$ or $(nn)!$. Whenever there's a chance for ambiguity, you should use parentheses. – Alistair Savage May 1 '13 at 23:37