[m]5^{6log_{5}sqrt{2}+\frac{1}{3}log_{5}27}=5^{log_{5}(2^{\frac{1}{2}})^6+log_{5}(3^3)^{\frac{1}{3}}}=5^{log_{5}2^{\frac{1}{2}\cdot 6}+log_{5}3^{3\cdot \frac{1}{3}}}=5^{log_{5}(2^3\cdot 3)}=2^3\cdot 3=24[/m]
