In my introductory algebra classes these days, I've switched to providing actual proofs for major principles after about the halfway point in the class. As usual, the point of this is (a) to prepare students for what real math classes are like, and (b) provide insight into why things work as they do.
What I was distressed to find yesterday is that you actually can't find any proofs for the pretty rudimentary notions of the fundamental rule (as I call it) or distribution of exponents or radicals (over multiplication or division), even for just integer powers. Not anywhere online on a Google search. Not in any math text in any of my bookcases at home.
So this in turn caused me to have weird dreams and wake up at 5am all pissed off over that fact, scribbling stuff on note paper to make up the gap, in classic AngryMath style. Just in case anyone else is ever searching for the same thing, I'll present the distribution proofs below. We assume the usual properties of commuting, associating, and distributing multiplication and addition (which for integers can be proven from the Peano axioms).
Definition of Exponents: For positive integers n, a^n = a*a*...a [n times]. For negative integers, a^(-n) = 1/a^n. For the zero power, a^0 = 1.
Theorem: Exponents distribute over multiplication. That is: (ab)^n = a^n * b^n for any integer n.
Proof: For positive powers, (ab)^n = (ab)(ab)...(ab) [n times] = (a*a*...a)(b*b*...b) [by commuting & associating] = a^n * b^n [definition of exponents]. For negative powers, (ab)^(-n) = 1/(ab)^n [definition of negative exponent] = 1/(a^n * b^n) [positive exponents distribute] = 1/a^n * 1/b^n [definition of multiplying fractions] = a^(-n) * b(-n). For the zero power, (ab)^0 = 1 = 1*1 = a^0 * b^0 [definition of zero power].
Definition of Radicals: Square root √a means a positive number x such that x^2 = a.
Theorem: Radicals distribute over multiplication. That is: √(ab) = √a√b for any positive a, b.
Proof: Let x = √a, which means x^2 = a [definition of square root]. Let y = √b, which means y^2 = b [definition of square root]. Note that √(x^2 * y^2) = xy because (xy)^2 = x^2 * y^2 [distribution of exponents], satisfying the definition of square root. So √(ab) = √(x^2 * y^2) = xy = √a√b [substituting in each step].
[Note: Obviously I've skipped cube roots and Nth roots in the foregoing, but they're easily expanded from the above.]
Questions: Did I get those right (e.g., would it be more complete to use induction for the exponents proof)? Do you think they're illuminating for basic algebra students? And more importantly -- Can you find any place online that I missed or in any textbook that presents proofs for the preceding?