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During calculations I got this step $$(e^m/((m+1)^{m+1}) )^{3n/4} = 1/2^n$$

I want the value of m here??

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  • $\begingroup$ Assuming $n,m\in\Bbb R$ (or some subset), take the $\frac {3n}4$ root of both sides, then see what else makes sense from there... $\endgroup$ – abiessu Oct 28 '13 at 1:59
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Well, you can eliminate $n$ by raising both sides to $4/(3n)$:

$$\frac{e^m}{(m+1)^{m+1}} = \frac{1}{2^{4/3}}.$$

I don't think there's much hope of a closed-form solution, but Wolfram Alpha can easily find $m$ numerically.

Note that more, (likely complex, depending on $n$) solutions also exist -- multiply the right-hand side by $4/(3n)$th roots of unity.

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