Radicals

For positive integers $m,n$ and real numbers $a,b$ such that $\sqrt[n]{a}$ and $\sqrt[n]{b}$ are real:

$$\sqrt{a}=a^{1/2} \sqrt[n]{a}=a^{1/n} a^{m/n}=(a^{1/n})^m=(a^m)^{1/n}=\sqrt[n]{a^m}$$

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

$\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}$