Multiplying and dividing roots
With the same index:
$$\sqrt[n]{a}\cdot\sqrt[n]{b}=\sqrt[n]{a\cdot b}\qquad \dfrac{\sqrt[n]{a}}{\sqrt[n]{b}}=\sqrt[n]{\dfrac{a}{b}}$$
If the indices differ, first reduce them to a common index (amplifying). Examples:
$$\sqrt[3]{2^{3}}\cdot\sqrt[3]{2^{7}}=\sqrt[3]{2^{10}}=2^{3}\cdot\sqrt[3]{2}$$
$$\sqrt[3]{2^{2}}\cdot\sqrt{2}=\sqrt[6]{2^{4}}\cdot\sqrt[6]{2^{3}}=\sqrt[6]{2^{7}}=2\cdot\sqrt[6]{2}=2^{7/6}$$
$$\dfrac{\sqrt[3]{20}}{\sqrt[4]{100}}=\dfrac{\sqrt[3]{2^{2}\cdot 5}}{\sqrt[4]{2^{2}\cdot 5^{2}}}=\sqrt[12]{\dfrac{2^{8}\cdot 5^{4}}{2^{6}\cdot 5^{6}}}=\sqrt[12]{\dfrac{2^{2}}{5^{2}}}=\sqrt[6]{\dfrac{2}{5}}$$
Root of a root
$$\sqrt[n]{\sqrt[m]{a}}=\sqrt[n\cdot m]{a}$$
Examples:
$$\sqrt{\sqrt[3]{5}}=\sqrt[6]{5}$$
$$\sqrt{27\cdot\sqrt[3]{3}\cdot\sqrt[4]{27}}=\sqrt{3^{3}\cdot\sqrt[3]{3}\cdot\sqrt[4]{3^{3}}}=\sqrt{3^{3}\cdot\sqrt[12]{3^{4}\cdot 3^{9}}}=\sqrt{3^{3}\cdot\sqrt[12]{3^{13}}}=\sqrt{\sqrt[12]{3^{36}\cdot 3^{13}}}=\sqrt[24]{3^{49}}=3^{2}\cdot\sqrt[24]{3}$$
Addition and subtraction
There is no property for adding or subtracting non-equivalent radicals. Only equivalent radicals (same index and same radicand) can be added:
$$n\cdot\sqrt{a}+m\cdot\sqrt{a}=(n+m)\sqrt{a}$$
Examples:
$$\sqrt{2}+4\sqrt{2}-2\sqrt{2}=3\sqrt{2}$$
$$\sqrt{2}+\sqrt{8}-\sqrt[4]{2^{6}}=\sqrt{2}+\sqrt{2^{3}}-\sqrt{2^{3}}=\sqrt{2}+2\sqrt{2}-2\sqrt{2}=\sqrt{2}$$