Deciding if a simple operation involves the associative or commutative laws, and if not, what other axioms can explain it

Question

Can the following two examples be explained by the the commutative and/or associative laws? If not, is there another axiom that does explain them?

$(4 \times 5) + (4 \times 5) + (4 \times 5) + (4 \times 5) + (4 \times 5) = 5 \times 20$

$100+4 = (50 \times 2) + (2+2) $

My confusion is that in the formulations I've seen of each law, terms are reordered but new numbers aren't introduced. For example, $mn = nm$ or $l \times (m \times n) = (l \times m) \times n$. But in my examples at the top, new numbers and even (in the case of the second example) operations are introduced.

(Self learning from Hung-Hsi Wu's Understanding Numbers in Elementary School Mathematics).

Answer 1

The first one can be explained by\begin{align}(4\times5)+(4\times5)+(4\times5)+(4\times5)+(4\times5)&=(4+4+4+4+4)\times5\text{ (distributivity)}\\&=20\times5\\&=5\times20\text{ (commutativity)}.\end{align}The second one has nothing to do with associativity or commutativity. It just uses the fact that $100=50\times2$ and that $4=2+2$.

Answer 2

$(4 \times 5) + (4 \times 5) + (4 \times 5) + (4 \times 5) + (4 \times 5) = 5 \times 20$

What's inside the parentheses is evaluated first, by the order of operations (though in this particular case the parentheses are redundant, since multiplication is evaluated before addition, anyway). Since $\,4 \times 5 =20\,$, the left-hand side becomes:

$$ 20+20+20+20+20 $$

The next step depends on the context, and the definitions you are working with:

• either $\,20+20+20+20+20 = 5 \times 20\,$ if integer multiplication is defined as a repeated addition;

• or $\,1 \times 20 + 1 \times 20 + 1 \times 20 + 1 \times 20 + 1 \times 20=(1+1+1+1+1)\times 20 = 5 \times 20\,$ using the distributivity of multiplication over addition.