Powers, Roots, Indices & Standard Form

Laws of indices

For any base $a$ (with the usual restrictions):

$$a^m \times a^n = a^{m+n}, \qquad \frac{a^m}{a^n} = a^{m-n}, \qquad (a^m)^n = a^{mn}$$

$$a^0 = 1, \qquad a^{-n} = \frac{1}{a^n}, \qquad a^{\frac{1}{n}} = \sqrt[n]{a}, \qquad a^{\frac{m}{n}} = \left(\sqrt[n]{a}\right)^m$$

So $16^{\frac{3}{4}} = \left(\sqrt[4]{16}\right)^3 = 2^3 = 8$, and $5^{-2} = \frac{1}{25}$.