Notation:

a,b : bases (a≥0,b≥0 if n=2k)
n,m: powers

Formulas:

$\left( \sqrt[\scriptstyle n]{a} \right)^n = a$ $\left( \sqrt[\scriptstyle n]{a} \right)^m = \sqrt[\scriptstyle n]{a^m}$ $\sqrt[\scriptstyle m]{ \sqrt[\scriptstyle n]{a}} = \sqrt[\scriptstyle {n m}]{a}$ $\left( \sqrt[\scriptstyle n]{a^m} \right)^p = \sqrt[\scriptstyle n]{a^{n p}}$ $\sqrt[\scriptstyle n]{a^m} = \sqrt[\scriptstyle n p]{a^{n p}}$ $\frac{1}{\sqrt[\scriptstyle n]{a}} = \frac{ \sqrt[\scriptstyle n]{a^{n-1}}}{a}$ $\sqrt[\scriptstyle n]{ab} = \sqrt[\scriptstyle n]{a} \cdot \sqrt[\scriptstyle n]{b}$ $\sqrt[\scriptstyle n]{\frac{a}{b}} = \frac{\sqrt[\scriptstyle n]{a}}{\sqrt[\scriptstyle n]{b}}$ $\frac{\sqrt[\scriptstyle n]{a}}{\sqrt[\scriptstyle m]{b}} = \sqrt[\scriptstyle {nm}]{\frac{a^m}{b^n}}$ $\sqrt[\scriptstyle n]{a} \cdot \sqrt[\scriptstyle m]{b} = \sqrt[\scriptstyle{nm}]{a^m b^n}$ $\sqrt{ a \pm \sqrt{b}} = \sqrt{ \frac{a + \sqrt{a^2 - b}}{2}} \pm \sqrt{ \frac{a - \sqrt{a^2 - b}}{2}}$ $\frac{1}{\sqrt{a} \pm \sqrt{b}} = \frac{\sqrt{a} \mp \sqrt{b}}{a-b}$