Field/Rules for powers/Natural exponents/Fact/Proof/Exercise

Let ${}K$ be a field and let ${}a,b\neq 0$ be elements in ${}K$. Prove the following laws for powers for natural exponents ${}m,n\in \mathbb {N}$.

1. ${}a^{m+n}=a^{m}\cdot a^{n}\,.$

2. ${}{\left(a^{m}\right)}^{n}=a^{mn}\,.$

3. ${}(a\cdot b)^{n}=a^{n}\cdot b^{n}\,.$

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