Infinite product

$\huge \mathfrak K=\prod _{ i=0 }^{ \infty }{ { \left( 512 \right) }^{^{{ \left( \frac { 1 }{ 4 } \right) }^{ i } } }}$ $\huge =512^{^{\small{\color{#D61F06}{\displaystyle\sum_{i=0}^{\infty}(\dfrac{1}{4})^i}}}}$ $\color{#D61F06}{\displaystyle\sum_{i=1}^{\infty}(\dfrac{1}{4})^i}$ denote a sum of infinite GP given by:- $\color{#D61F06}{\dfrac{1}{1-\frac{1}{4}}=\dfrac 43}$ . Hence, $\huge \mathfrak K=512^{^{\color{#D61F06}{\frac 43}}}=(2^{~9})^{^{\color{#D61F06}{\frac 43}}}$ $~~~~~~~~~~~~~\huge =\boxed{4096}$