Basic Laws of Exponents and Radicals
$\tcBal{Laws of Exponents}$ $$ \begin{align} a^m \cdot a^n &= a^{m+n} \\ \frac{a^m}{a^n} &= a^{m-n} \\ (a^m)^n &= a^{mn} \\ \br{ \frac{a}{b} }^n &= \frac{a^n}{b^n} \\ a^{\frac{m}{n}} &= \sqrt[n]{a^m} \\ a^{-n} &= \frac{1}{a^n} \\ a^0 &= 1 \end{align} $$
$\tcBal{Laws of Radicals}$ $$ \begin{align} \sqrt[n]{a} &= a^{\frac{1}{n}} \\ \sqrt[n]{a^m} &= \br{ \sqrt[n]{a} }^m \\ \sqrt[n]{a^n} &= a \\ \sqrt[n]{a} \cdot \sqrt[n]{b} &= \sqrt[n]{ab} \\ \frac{\sqrt[n]{a}}{\sqrt[n]{b}} &= \sqrt[n]{\frac{a}{b}} \; : b\neq 0 \\ \sqrt[-n]{a} &= \frac{1}{\sqrt[n]{a}} \\ \sqrt[1]{a} &= a \end{align} $$