From Förberedande kurs i matematik 1

In order to eliminate \displaystyle \sqrt[3]{7}=7^{{1}/{3}\;} from the denominator, we can multiply the top and bottom of the fraction by \displaystyle 7^{{2}/{3}\;}. The denominator becomes \displaystyle 7^{{1}/{3}\;}\centerdot 7^{{2}/{3}\;}=7^{{1}/{3+{2}/{3}\;}\;}=7^{1}=7 and we get

\displaystyle \frac{1}{\sqrt[3]{7}}=\frac{1}{7^{{1}/{3}\;}}=\frac{1}{7^{{1}/{3}\;}}\centerdot \frac{7^{{2}/{3}\;}}{7^{{2}/{3}\;}}=\frac{7^{{2}/{3}\;}}{7}.