**Definition (Exponentiation to a natural number)** Let $x$ be a [[Real numbers]]. To raise $x$ to power 0, we define $x^{0}:=1$. Now suppose inductively that $x^{n}$ has been defined for some natural number $n$, then we define $x^{n+1}:=x^{n}\times x$. **Definition (Exponentiation to a negative integer)** Let $x$ be a non-zero real number. Then for any negative integer $-n$, we define $x^{-n} = 1/x^{n}.$ See [[Existence & uniqueness nth root of positive reals]]. See [[Inequalities for powers of positive reals with natural number exponents]].