The textbook I'm reading through at the moment is describing the axioms of exponentiation. Quoting directly from the book:
By the definition of the nth power of a number and the use of the commutative axiom, we can prove that
(ab)n = anbn
Proof →
- (ab)n = ab ⋅ ab ⋅ ab ⋅ ⋅ ⋅ to n factors
- ____ = (a ⋅ a ⋅ a ⋅ ⋅ ⋅ to n factors) (b ⋅ b ⋅ b ⋅ ⋅ ⋅ to n factors)
- ____ = anbn
I'm a bit confused by how this is worded. Is exponentiation distributive because (like multiplication) it's an iteration of commutative operations?
The wiki page on this topic is absolutely horrendous. I've tried googling for the answer to this, but all websites discussing it seem to be written for the common idiot, and only talk about the distributive property of multiplication, not the distributive property in general.
It doesn’t actually make sense to say that an operation is distributive: distributivity is a joint property of two operations. For example, in the ring of $n\times n$ real matrices matrix multiplication is distributive over matrix addition, because $A(B+C)=AB+AC$ and $(B+C)A=BA+CA$ for all $n\times n$ real matrices $A,B$, and $C$.
Exponentiation of real numbers is distributive over multiplication of real numbers whenever it makes sense: $(ab)^c=a^cb^c$ whenever all of the exponentiations are defined. However, the argument quoted in your question proves this only for the special case in which $c$ is a positive integer. That special case is indeed a consequence of the commutativity of multiplication and the definition of $a^n$ for positive integers $n$ as repeated multiplication. We then go on to define $a^r$ for arbitrary rational numbers $r$ in such a way that all of the usual laws of exponents true for positive integer exponents remain true; in particular, this distributive law of exponentiation over multiplication remains true whenever the exponentiations involved are defined. This isn’t directly a consequence of the commutativity of multiplication: that’s true only for positive integer exponents. It’s true because we define exponentiation when the exponent is an arbitrary rational number in such a way as to make it true.
The last step is extending the definition to allow irrational exponents as well. We do this in such a way as to make the exponential function continuous, and then we prove that this preserves the usual laws of exponents, including the distributivity of exponentiation over multiplication.