Field/Rules for powers/Positive case known/Exercise

Let ${}K$ be a field and let ${}a,b\neq 0$ be elements in ${}K$. Prove the following laws for powers for integer exponents ${}m,n\in \mathbb {Z}$. The corresponding laws for exponents from ${}\mathbb {N}$ can be used and also the facts that the inverse of the inverse is the element itself and that the inverse of ${}u^{k}$ equals ${}{\left(u^{-1}\right)}^{k}$.

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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