We will try to reduce several radicals to common index , and we can do this by converting the fraction exponents of their expression as a power into a common denominator.

$$\displaystyle \sqrt{5}\cdot \sqrt[3]{2}=5^{\frac{1}{2}}\cdot 2^{\frac{1}{3}}=5^{\frac{3}{6}}\cdot 2^{\frac{2}{6}}=\sqrt[6]{5^3}\cdot\sqrt[6]{2^2}=\sqrt[6]{5^3\cdot 2^2}=\sqrt[6]{500} $$