$\sum_{ i=0 }^{2016} \binom{2016}{i}$

The expression above can be presented in the form $x^y$ , where $x,y$ are positive integers. Among all such solutions, take the one with the smallest $|x-y|,$ and enter your answer as $x+y.$

The answer is 508.

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Clearly given is the expansion of $(1+1)^{2016}$ i.e $\Large \sum_{ i=0 }^{2016} \binom{2016}{i} =(1+1)^{2016}$ and this is $x^y$ when |x-y| is least possible, this is the case when $\\2^{2016}=(2^8)^{252}=256^{252}$ Hence $\huge 256+252=\boxed{ \color{#D61F06}{\boxed{508}}}$