$a){\left(3^{\sqrt{3}}\right)}^{\sqrt{12}}=3^{\sqrt{3}\cdot \sqrt{12}}=3^{\sqrt{3}\cdot \sqrt{3}\cdot \sqrt{4}}=3^{3\cdot 2}=3^6=729$  

 

$b){\left(2^{\sqrt{2}}\right)}^{-\frac{1}{\sqrt{8}}}=2^{-\frac{\sqrt{2}}{\sqrt{8}}}=2^{-\frac{\sqrt{2}}{2\sqrt{2}}}=2^{-\frac{1}{2}}=\frac{1}{\sqrt{2}}=\frac{\sqrt{2}}{2}$  

 

$c){\left(4^{\frac{\sqrt{3}}{2}}\right)}^{\frac{\sqrt{3}}{2}}=4^{\frac{\sqrt{3}}{2}\cdot \frac{\sqrt{3}}{2}}={\left(2^2\right)}^{\frac{3}{4}}=2^{2\cdot \frac{3}{4}}=2^{\frac{3}{2}}=\sqrt{8}=2\sqrt{2}$  

 

$d){\left(5^{\sqrt[3]{4}}\right)}^{\sqrt[3]{16}}=5^{\sqrt[3]{4}\cdot \sqrt[3]{16}}=5^{\sqrt[3]{64}}=5^4=625$