a)
2^{\sqrt7-6}*4^3=2^{\sqrt7-6}*(2^2)^3=2^{\sqrt7-6}*2^6=2^{\sqrt7-6+6}=2^{\sqrt7}
b)
2^{\sqrt3+3}:8=2^{\sqrt3+3}:2^3=2^{\sqrt3+3-3}=2^{\sqrt3}
c)
9^{\frac{\sqrt5}{2}}*27^{\frac{\sqrt5}{3}}=(3^2)^{\frac{\sqrt5}{2}}*(3^3)^{\frac{\sqrt5}{3}}=3^{\sqrt5}*3^{\sqrt5}=3^{\sqrt5+\sqrt5}=3^{2\sqrt5}
d)
27*(3^{\sqrt3})^2=3^3*3^{2\sqrt3}=3^{3+2\sqrt3}
e)
\frac{1}{3}*9^{\pi+\frac{1}{2}}:81^{2\pi}=3^{-1}*3^{2(\pi+\frac{1}{2})}:(3^4)^{2\pi}=3^{-1}*3^{2\pi+1}:3^{{8\pi}}=3^{-1+2\pi+1-8\pi}=3^{-6\pi}
f)
7^{-2\sqrt2}:49^{\pi+\sqrt2}=7^{-2\sqrt2}:7^{2(\pi+\sqrt2)}=7^{-2\sqrt2}:7^{2\pi+2\sqrt2}=7^{-2\sqrt2-(2\pi+2\sqrt2)}=
=7^{-2\sqrt2-2\pi-2\sqrt2}=7^{-4\sqrt2-2\pi}