<s>[INST] Proof of basic properties Can anyone provide proof of properties such as:
$$a(b+c) = ab+ac$$
$$(a+b)^2 = a^2+2ab+b^2$$
And exponent rules:
$$a^n \cdot a^m = a^{n+m}$$
$$(a^n)^m = a^{n \cdot m}$$
For $a, b, c \in \mathbb{R}$
I'm quite sure that the proof is the same or very similar for the first two and the last two, which is why I pick so 'many' examples.
Thanks.
 [/INST] The first equation is an axiom of $\mathbb{R}$
The second equation is an application of the first one and uses another axiom of $\mathbb{R}$, the commutative property :
$$(a+b)^2 = (a+b)\cdot (a+b) = a\cdot a + a\cdot b + b\cdot a + b\cdot b = a^2+2ab+b^2$$
The two next proves are made for integers. We can extand these results saying
$$\underbrace{a \cdot a \cdot a  \cdot\ldots  \cdot a}_{\lfloor m\rfloor \textrm{ times}}\cdot a^{m-\lfloor m\rfloor} = \underbrace{a \cdot a \cdot a  \cdot\ldots  \cdot a}_{m \textrm{ times}}$$
The third one uses the commutative and associative properties :
$$\underbrace{a \cdot a \cdot a  \cdot\ldots  \cdot a}_{m \textrm{ times}}\cdot \underbrace{a \cdot a \cdot a  \cdot\ldots  \cdot a}_{n \textrm{ times}}= \underbrace{a \cdot a \cdot a  \cdot\ldots  \cdot a}_{m+n \textrm{ times}}$$
The last one is the same of the third one :
$${\underbrace{\underbrace{a \cdot a \cdot a  \cdot\ldots  \cdot a}_{m \textrm{ times}}\cdot \underbrace{a \cdot a \cdot a  \cdot\ldots  \cdot a}_{m \textrm{ times}}\cdot \cdot\ldots \cdot\underbrace{a \cdot a \cdot a  \cdot\ldots  \cdot a}_{m \textrm{ times}}}_{n \textrm{ times}}} = \underbrace{a \cdot a \cdot a  \cdot\ldots  \cdot a}_{m\cdot n \textrm{ times}}$$
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