Root rules

Root rules regulate how roots behave if you use multiplication, division, exponentiation and root extraction.

1. The multiplication of roots

   $\sqrt[n]{a}\cdot\sqrt[n]{b}=\sqrt[n]{a\cdot b}$

2. The division of roots

   $\frac{\sqrt[n]{a}}{\sqrt[n]{b}}=\sqrt[n]{\frac{a}{b}}$

3. The exponentiation of roots

   $(\sqrt[n]{a})^m=\sqrt[n]{a^m}$

4. Root extraction

   $\sqrt[m]{\sqrt[n]{a}}=\sqrt[m \cdot n]{a}$

Examples

• $\sqrt[3]{8}\cdot\sqrt[3]{27}=\sqrt[3]{8\cdot 27}$ $=\sqrt[3]{216}=6$


• $\frac{\sqrt{8}}{\sqrt{32}}=\sqrt{\frac{8}{32}}$ $=\sqrt{\frac{1}{4}}=\frac{1}{2}$


• $(\sqrt{2})^4=\sqrt{2^4}$ $=\sqrt{16}=4$


• $\sqrt{\sqrt{16}} = \sqrt[2 \cdot 2]{16}$ $=\sqrt[4]{16}=2$