I've been working on a formula, which I have managed to simplify to the following expression, but I wonder if anyone can spot a way to simplify it further?
$$2^{1 -\frac{1}{2}\sum_i \log_2 \frac{(a_i + c_i)^{(a_i + c_i)}}{a_i^{a_i}c_i^{c_i}}}$$
I've been working on a formula, which I have managed to simplify to the following expression, but I wonder if anyone can spot a way to simplify it further?
$$2^{1 -\frac{1}{2}\sum_i \log_2 \frac{(a_i + c_i)^{(a_i + c_i)}}{a_i^{a_i}c_i^{c_i}}}$$
This looks a little simpler to me: $$ 2\prod_i\left(\frac{a_i}{a_i+c_i}\right)^{a_i/2}\left(\frac{c_i}{a_i+c_i}\right)^{c_i/2}\tag{1} $$ If you don't mind introducing new variables, let $u_i=\frac{a_i}{a_i+c_i}$ and $v_i=\frac{c_i}{a_i+c_i}$, then $(1)$ becomes $$ 2\prod_i\left(u_i^{u_i}v_i^{v_i}\right)^{(a_i+c_i)/2}\tag{2} $$
For $\ b_i\ :=\ c_i/a_i\ $ it is
$$ 2\ \prod_i\left(\frac{b_i^{b_{\:i}}}{(b_i+1)^{b_{\:i}+1}}\right)^{{a_{\:i}}/2}$$
Does this look simpler to you? $$ 2 \left( \prod_i \frac{a_i^{a_i}c_i^{c_i}}{(a_i + c_i)^{(a_i + c_i)}} \right)^\frac{1}{2} $$