The exponential operation possesses the following properties.

• Homogeneity. For $x,y\in\reals^+, p\in \reals$ we have $$(xy)^p = x^p y^p$$
• Exponent additivity. For $x\in\reals^+$ we have $$x^0=1,\quad x^1 = x,\quad x^{p+q} = x^p x^q,\quad (x^p)^q=x^{pq}\qquad p,q\in\reals.$$
• Monotonicity. For $x,y\in\reals^+$ with $x<y$ and $p\in \reals^+$ we have $$x^p < y^p,\qquad x^{-p} > y^{-p}.$$
• Continuity. The exponential operation is continuous with respect to its arguments. To be more precise, the following function is continuous: $$P:\reals^+\times\reals\rightarrow \reals,\qquad P(x,y)=x^y.$$