a)
\sqrt{x}=x^{\frac{1}{2}}
\sqrt[3]{y}=y^{\frac{1}{3}}
2\sqrt[4]{x}=2x^{\frac{1}{4}}
\sqrt[3]{\frac{1}{3}}=(\frac{1}{3})^{\frac{1}{3}}
\sqrt{3^{-1}}=\sqrt{\frac{1}{3}}=(\frac{1}{3})^{\frac{1}{2}}
\sqrt[3]{(\frac{1}{2})^{-2}}=\sqrt[3]{2^2}=2^{\frac{2}{3}}
b)
\sqrt[3]{a^2}=a^{\frac{2}{3}}
\sqrt{a^3}=a^{\frac{3}{2}}
4\sqrt{a^5}=4a^{\frac{5}{2}}
\frac{1}{2}\sqrt[3]{a^6}=\frac{1}{2}*a^{\frac{6}{3}}=\frac{1}{2}a^2
c)
\sqrt{3+\sqrt{2}}=(3+2^{\frac{1}{2}})^{\frac{1}{2}}
\sqrt[3]{2+\sqrt[3]{3}}=(2+3^{\frac{1}{3}})^{\frac{1}{3}}
\sqrt{3+\sqrt{3+\sqrt{3}}}=[3+(3+3^{\frac{1}{2}})^{\frac{1}{2}}]^{\frac{1}{2}}
\sqrt{2+\sqrt[3]{2+\sqrt[3]{2}}}=[2+(2+2^{\frac{1}{3}})^{\frac{1}{3}}]^{\frac{1}{2}}