From Förberedande kurs i matematik 1

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

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

we see that

\displaystyle \sqrt[3]{8} = \sqrt[3]{2^{3}} = \bigl(2^{3}\bigr)^{1/3} = 2^{3\cdot\frac{1}{3}} = 2^{1} = 2\,\textrm{.}

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