If the associative law doesn't hold, can we define $a^n$?

Question

I am reading "Higher Algebra" by A. Kurosh.

The following sentence is in this book:

Analogously, the associative law of addition leads to the concept of a multiple, $na$, of the element $a$ by a positive integral coefficient $n$

If the associative law doesn't hold, then we cannot define $a^n$.
Is it really true?

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Original image: https://i.stack.imgur.com/YbaAy.png

Answer 1

Suppose we define a non-associative binary operation $\circ$ on $\Bbb R$.
$x\circ y:=x-y$
trying to figure out the third (or higher) power causes problems.
$1\circ (1\circ 1)=1-(1-1)=1$
$(1\circ 1)\circ 1=(1-1)-1=-1$
$(\Bbb R,\circ)$ is a non-associative magma, so it is not a semigroup. it is not an example of a non-associative ring or algebra though.