Exponents Formulas

1. If $$p$$ is a positive integer and $$a \in \mathbb{R}$$, then $${a^p} = a \cdot a \cdot a \cdots $$ to $$p$$ factors.

2. $$\forall a \in \mathbb{R}$$,        $$a \ne 0$$, $${a^0} = 1$$

3. $${a^r} \cdot {a^s} = {a^{r + s}}$$,         $$a \in \mathbb{R}$$; $$r,s \in \mathbb{N};a \ne 0$$

4. $${({a^r})^s} = {a^{rs}}$$

5. $${(ab)^r} = {a^r} \cdot {b^r}$$         $$a \in \mathbb{R}$$; $$r \in \mathbb{N};a,b \ne 0$$

6. $${1^n} = 1$$         $$\forall n \in \mathbb{N}$$

7. $$\frac{{{a^r}}}{{{a^s}}} = {a^{r – s}}$$,        $$a \in \mathbb{R}$$; $$r,s \in \mathbb{N};a \ne 0$$

8. $${(\frac{a}{b})^r} = \frac{{{a^r}}}{{{b^r}}}$$         $$a \in \mathbb{R}$$; $$r \in \mathbb{N};a,b \ne 0$$

9. $${a^{ – r}} = \frac{1}{{{a^r}}}$$         $$a \in \mathbb{R}$$; $$r \in \mathbb{N};a \ne 0$$

10. $${a^{\frac{r}{s}}} = \sqrt[s]{{{a^r}}}$$         $$a \in \mathbb{R}$$; $$r,s \in \mathbb{N};a \ne 0$$