Real exponential function/Base/Rules/Fact/Proof/Exercise

Prove that for the exponential function

$\mathbb {R} \longrightarrow \mathbb {R} ,x\longmapsto a^{x},$

the following calculation rules hold (where ${}a,b\in \mathbb {R} _{+}$ and ${}x,y\in \mathbb {R}$).

1. ${}a^{x+y}=a^{x}\cdot a^{y}\,.$

2. ${}a^{-x}={\frac {1}{a^{x}}}\,.$

3. ${}(a^{x})^{y}=a^{xy}\,.$

4. ${}(ab)^{x}=a^{x}b^{x}\,.$

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