03 - Powers on the real numbers

Positive Powers

Take $a\in \mathbb{R}\setminus \{0\}$
Define

$$a^0=1$$

Then we define the positive powers of $a$ inductively: for integers $k\ge 0$ we define

$$a^{k+1}=a^k\cdot a$$

For integers $l\le -1$ we define

$$a^l=\frac{1}{a^{-l}}$$

Additive Rule for Exponents lemma

For $a\in \mathbb{R}\setminus \{0\}$

$$a^m{a}^n=a^{m+n}\text{ for }m,n\in \mathbb{Z}$$