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As stated in the title, how is the following simplification performed? $$\frac{\left(\frac{n}{e}\right)^n}{\left(\frac{n}{2e}\right)^n}=2^n$$

This was shown by a student in this Recitation video (29:05) from MIT OpenCourseware.

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@anorton, thank you for clearing up the post, I will restate the question in the body of the post in the future. – user52207 Feb 27 '13 at 16:14
up vote 9 down vote accepted

$$\frac{\left(\frac{n}{e}\right)^n}{\left(\frac{n}{2e}\right)^n} = \frac{n^n (2e)^n}{e^nn^n}$$ by $\frac{a/b}{c/d} = \frac{ad}{bc}$.

Clearly, the $n^n$ terms cancel, and $(2e)^n = 2^ne^n$. Then, the $e^n$ terms cancel.

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Use the rules:

$$\dfrac{\frac{a}{b}}{\frac{c}{d}} = \frac{a d}{b c}$$

$$ \left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$$

$$ (ab)^n = a^n b^n$$

See if you can figure out how it simplifies.

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$$\frac{\left(\frac{n}{e}\right)^n}{\left(\frac{n}{2e}\right)^n} =\frac{\left(\frac{n}{e}\right)^n}{\left(\frac{1}{2}\right)^n\left(\frac{n}{e}\right)^n} =\frac{1}{1/2^n}=2^n$$

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