From Förberedande kurs i matematik 1

The cube root of a number is the same thing as the number raised to the power \displaystyle {1}/{3}\;, i.e. \displaystyle \sqrt[3]{a}=a^{{1}/{3}\;} If we therefore write the number \displaystyle \text{8} as a product of its smallest possible integer factors

\displaystyle 8=2\centerdot 4=2\centerdot 2\centerdot 2=2^{3}

we see that

\displaystyle \sqrt[3]{8}=\sqrt[3]{2^{3}}=\left( 2^{3} \right)^{{1}/{3}\;}=2^{3\centerdot \frac{1}{3}}=2^{1}=2.

NOTE: Taking the cube root can thus be seen as cancelling the operation of raising a number to the power \displaystyle \text{3}, i.e. \displaystyle \sqrt[3]{5^{3}}=5,\quad \sqrt[3]{6^{3}}=6 etc.