Radicals

For positive integers

and real numbers

such that $\sqrt[n]{a}$ and $\sqrt[n]{b}$ are real:

	$\displaystyle \sqrt{a}=a^{1/2}$		$\displaystyle \sqrt[n]{a}=a^{1/n}$		$\displaystyle a^{m/n}=(a^{1/n})^m=(a^m)^{1/n}=\sqrt[n]{a^m}$
	$\displaystyle \sqrt[n]{ab}=\Big(\sqrt[n]{a}\Big)\Big(\sqrt[n]{b}\Big)$		$\displaystyle \sqrt[n]{\frac{a}{b}}=\frac{\sqrt[n]{a}}{\sqrt[n]{b}},b\ne 0$		$\displaystyle \sqrt[m]{\sqrt[n]{a}}=\sqrt[mn]{a}$

$\quad\quad\quad\quad (\sqrt[n]{a})^n=a \quad\quad\quad\quad\quad\quad\quad \s... ...ght\rvert , & \text{for $n$ even} \\ a, & \text{for $n$ odd} \\ \end{cases}$