(i) $a^{x}a^{y}=a^{x+y}$;

(ii) $a^{x}b^{x}=(ab)^{x}$;

(iii) $a^{0}=1$;

(iv) $a^{-x}=1/a^{x}$; for all $a,b>0$ and $x\in{\mathbb{R}}$.

These properties follow easily from the properties of exp and log that we have seen in 2.32 and 2.33.

The formula for powers is also useful for calculating some derivatives.

(ii) We have