Math Calculators, Lessons and Formulas

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# Math Formulas: Roots Formulas

### Notation:

$a , b$ : bases $( a \geq 0 , b \geq 0 ~~\text{if} ~~ n = 2k )$

$n , m$: powers

### Formulas

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