From Förberedande kurs i matematik 1

\displaystyle \sqrt[3]{3} means \displaystyle \text{3}^{\text{1}/\text{3}}~ and then

\displaystyle \left( \sqrt[3]{3} \right)^{4}=\left( 3^{{1}/{3}\;} \right)^{4}=3^{\frac{1}{3}\centerdot 4}=3^{\frac{4}{3}}