Are there ways to think of (integral) exponentiation other than repeated multiplication? Preamble:  
There is a related question to what I'm asking, which has been closed as too broad here: What are some ways of thinking about exponentiation, other than repeated multiplication? 
I would like to limit the scope of my question to natural numbers. I think my question (as presented at the end) is a bit more focused in what kind of answer I'm looking for as well.  
Please comment if you think this needs to be refined further (or my tags could be improved...).

At some point in school, addition is taught, probably using something like an addition table:  
    1 2 3 4 ... 
   --------
1 | 2 3 4 5 
2 | 3 4 6 6 
3 | 4 5 6 7 
4 | 5 6 7 8 ... 
... 

And then it's shown that problems like  
  456 
+ 789 

are composed of smaller related problems: $9+6$ and carry into the shifted (i.e., times 10) $5+8$ ...  
Later on a similar process is used to teach multiplication. A multiplication table is memorized and then it is taught how to solve multi-digit multiplication problems by breaking apart the problem into smaller problems of multiplication and addition. For example  
  456 
× 789 

would first be $9 × 456$ which is then itself broken into $9 × 6$ add that to "shifted" (i.e., times 10) $9 × 5$ ...  
(This may not be an accurate representation of elementary school learning, but the details about the schooling process are not so important here)  
But then this process stops. There is no "power table" (or higher!) learned in school (nor is "exponentiation" as a binary operator associative...).  
Typically when encountering an integral exponential problem there isn't any "exponentiation" that occurs. For example, take $57^4$. This would be solved by squaring $57$ (multiplication) and then squaring (multiplication) the result again.   
Now, I think there is a semantic objection that goes something like: "But you are doing exponentiating, because, by definition, it is simply repeated multiplication." But I would respond by pointing out how there's something rote about multiplication and addition that is simply lacking when applying exponents. So technically, yes, this is exponentiation, but the point of posting this question here is to draw this distinction between applying an algorithm (for exponentiation) and using more fundamental/rote components (for multiplication and addition).  
So my question is something like: In the same way that it's possible to break down addition problems into smaller addition problems, and the same way it's possible to break down multiplication problems into smaller multiplication/addition problems, is it possible to break down exponentiation problems into smaller exponentiation/multiplication/addition problems?  
Or alternatively, am I simply stuck on a semantic issue, and addition, multiplication, and exponentiation are all "the same" in some sense?
 A: The answer is yes, of course. The multiplication process you describe depends on using a radix representation to multiply a given number by single digits, shift them by appropriate number of places and then add the results together. You can do something very similar with exponentiation. For example, suppose you want to evaluate $x^{13}$. Express $13=1101_2$ in binary. Then perform the calculations $a_0=1,\; a_1=a_0^2\cdot x,\; a_2=a_1^2\cdot x,\; a_3=a_2^2,\; a_4=a_3^2\cdot x,\;$ reading the binary digits left to right, with final result $a_4=x^{13}.\;$ A similar process works for any other radix, but it gets more complicated. You can do something similar with Fibonacci numbers. For example, to compute $x^{13},$ do this, $\;a_1=a_2=x,\;a_3=a_2a_1,\;a_4=a_3a_2,\;a_5=a_4a_3,\;a_6=a_5a_4,\;a_7=a_6a_5=x^{13}.$
A: Exponentiation is a function $f$ such that
\begin{align}
f(a+b) &= f(a) \cdot f(b) \\
f(0) &= 1
\end{align}
In other words, $f$ is a homomorphism mapping addition to multiplication. This is the definition used to define an exponential field. $f(1)$ yields what we call the base of the exponential.
