From Förberedande kurs i matematik 1

If we write the numbers in power form

\displaystyle \sqrt[3]{5}=5^{1/3}\quad and \displaystyle \quad\sqrt[3]{6}=6^{1/3}

we see that \displaystyle \sqrt[3]{6} is larger since 6 is larger than 5 and both numbers have the same positive exponent, 1/3.