Without relying on a background in calculus, analysis, abstract algebra etc., is there an elementary way to extend the definitions of basic operations from $\mathbb{Z}$ to $\mathbb{R}$. For example, the "repeated addition is multiplication" approach coincides with the typical definition in $\mathbb{N}$, but this clearly isn't sufficient (it's straight up incorrect.)
How can one teach multiplication without depending on an axiomatic treatment of the operation? Same with exponentiation, I clearly do not want to rely on $\ln$ or limits. This has more or less immediate import for students, who are say, learning about $\pi$. Take the commutative property of multiplication:
Clearly, if you have 5 baskets of 3 apples, it is abundantly clear that $3\cdot 5=5 \cdot 3$. However, while one can have $3 \pi$'s or $3/4(\pi)$'s but the reverse order doesn't have an easy justification (intuitively or otherwise.)
alternative answer: Is there a way to convince students that the basic laws of commutativity, associativity, etc. still work? I think we would have to certainly undo the "repeated operation $-$" mentality first. But then how can I "fill the void," and explain what each is doing?
One geometric possibility is to describe $\times: \mathbb{R}^2 \to \mathbb{R}$, as taking two points on the $y$-axis and $x$-axis and defining $x\times y$ to be the $x$-intercept of line parallel to the segment joining $x$ and $(0,1)$, but also going through the point $(0,y)$. This geometric approach might help, although we would want to show that it coincides with our current definition, it has the properties we want, etc.
Here is a resource that continues in this way: http://www.math.csusb.edu/faculty/pmclough/MP.pdf
Let me offer some thoughts, in the hope that others will do better.
It all depends, doesn’t it, on what your students’ understanding is of the behavior and nature of real numbers? Let’s suppose you make a good correspondence between real numbers and lengths of segments. Then you can define multiplication by means of ratios: $ab$ is defined to be the thing that fits into the proportionality $a:1 = ab:b$, and this can be gotten computationally and defined by using similar triangles. Then if they believe the basic facts about similarity, they should be able to see (or prove) that you get the same thing with the proportionality $a:1=ab:b$.
For exponentiation, I think you’ll have to be satisfied with integer (or, conceivably, rational) exponents. The definition of $3^{\sqrt2}$ must certainly wait for analytic sophistication on the part of the student.