Roots Formulas

Notation:

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

n,m: powers

Formulas

\( \left( \sqrt[\scriptstyle n]{a} \right)^n = a \)

\( \left( \sqrt[\scriptstyle n]{a} \right)^m = \sqrt[\scriptstyle n]{a^m} \)

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

\( \left( \sqrt[\scriptstyle n]{a^m} \right)^p = \sqrt[\scriptstyle n]{a^{n p}} \)

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

\( \frac{1}{\sqrt[\scriptstyle n]{a}} = \frac{ \sqrt[\scriptstyle n]{a^{n-1}}}{a} \)

\( \sqrt[\scriptstyle n]{\frac{a}{b}} = \frac{\sqrt[\scriptstyle n]{a}}{\sqrt[\scriptstyle n]{b}} \)

\( \sqrt[\scriptstyle n]{\frac{a}{b}} = \frac{\sqrt[\scriptstyle n]{a}}{\sqrt[\scriptstyle n]{b}} \)

\( \frac{\sqrt[\scriptstyle n]{a}}{\sqrt[\scriptstyle m]{b}} = \sqrt[\scriptstyle {nm}]{\frac{a^m}{b^n}} \)

\( \sqrt[\scriptstyle n]{a} \cdot \sqrt[\scriptstyle m]{b} = \sqrt[\scriptstyle{nm}]{a^m b^n} \)

\( \sqrt{ a \pm \sqrt{b}} = \sqrt{ \frac{a + \sqrt{a^2 – b}}{2}} \pm  \sqrt{ \frac{a – \sqrt{a^2 – b}}{2}} \)

\( \frac{1}{\sqrt{a} \pm \sqrt{b}} = \frac{\sqrt{a} \mp \sqrt{b}}{a-b} \)

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