3\sqrt3=3^1*3^{\frac{1}{2}}=3^{1+\frac{1}{2}}=3^{\frac{2+1}{2}}=3^{\frac{3}{2}}
b)
4^3(\sqrt[3]{4})^2 = 4^3 * 4^{\frac{2}{3}}=4^{3+\frac{2}{3}}=4^{\frac{9+2}{3}}=4^{\frac{11}{3}}=(2^2)^{\frac{11}{3}}=2^{\frac{22}{3}}
c)
\sqrt{2\sqrt2} = 2^{\frac{1}{2}} * (2^{\frac{1}{2}})^{\frac{1}{2}}=2^{\frac{2}{4}+\frac{1}{4}} = 2^{\frac{3}{4}}
d)
\sqrt[3]{7\sqrt[4]7} = 7^{\frac{1}{3}} * (7^{\frac{1}{4}})^{\frac{1}{3}} = 7^{\frac{4}{12}+\frac{1}{12}} = 7^{\frac{5}{12}}