Root formulas

Root, in mathematics, a solution to an equation, usually expressed as a number or an algebraic formula.

Bases : a, b

Powers (rational numbers) : n, m

a,b ≥ for even roots(n = 2k, k ∈ N)

$$ 1.\ \sqrt[n]{ab} = \sqrt[n]{a} \sqrt[n]{b}$$

$$ 2.\ \sqrt[n]{a} \sqrt[m]{b} = \sqrt[nm]{a^mb^n}$$

$$ 3.\ \sqrt[n]{\frac ab} = {\sqrt[n]{a} \over \sqrt[n]{b}}, b \neq 0 $$

$$ 4.\ {\sqrt[n]{a} \over \sqrt[n]{b}} = {\sqrt[nm]{a^m} \over \sqrt[nm]{b^n}} = \sqrt[nm]{\frac {a^m}{b^n}} , b \neq 0 $$

$$ 5.\ ( \sqrt[n]{a^m} )^p = \sqrt[n]{a^{mp}} $$

$$ 6.\ ( \sqrt[n]{a} )^n = a $$

$$ 7.\ \sqrt[n]{a^m} = \sqrt[np]{a^{mp}} $$

$$ 8.\ \sqrt[n]{a^m} = a^{\frac mn} $$

$$ 9.\ \sqrt[m]{\sqrt[n]{a}} = \sqrt[mn]{a} $$

$$ 10.\ ( \sqrt[n]{a} )^m = \sqrt[n]{a^m} $$

$$ 11.\ {1 \over \sqrt[n]{a}} = {\sqrt[n]{a^{n-1}} \over a}, a \neq 0 $$

$$ 12.\ \sqrt{a \pm \sqrt{b}} = \sqrt{\frac {a + \sqrt{a^2 - b}}{2}} \pm \sqrt{\frac {a - \sqrt{a^2 - b}}{2}} $$

$$ 13.\ {1 \over \sqrt{a \pm \sqrt{b}}} = {\sqrt{a} \pm \sqrt{b} \over a-b} $$

$$ 14.\ \sqrt[n]{a} = a^{\frac 1n} $$