That's a lot of logging

\log _{ 2 }{ 3 } =\frac { \log { 3 } }{ \log { 2 } }

Similarly, \log _{ 3 }{ 4 } =\frac { \log { 4 } }{ \log { 3 } }

\log _{ 511 }{ 512 } =\frac { \log { 512 } }{ \log { 511 } }

After multiplying all of them the result will be \log _{ 2 }{ 512 }

which is equal to 9

Therefore, a=9 \sqrt { 9 } =3

$\begin{aligned} a &= (\log_2 3)(\log_3 4)(\log_4 5)\cdots(\log_{511} 512) \\ & = \dfrac{\cancel{\log 3}}{\log 2} \cdot \dfrac{\cancel{\log 4}}{\cancel{\log 3}} \cdot \dfrac{\cancel{\log 5}}{\cancel{\log 4}} \cdots \dfrac{\log {512}}{\cancel{\log {511}}} \\ &= \dfrac{\log {512}}{\log 2} \\ &= \log_2 512 \\ &=9 \end{aligned} \\ \therefore \sqrt{a} = \sqrt{9} = \boxed{3}$