I distinctly remember when, in my second year as a college professor, a really good college student — with an SAT Math score over 650 — asked me why and . Of course, he knew that these rules were true and he could apply them in complex problems, but he didn’t know why they were true. And he wanted to have this deeper knowledge of mathematics beyond the ability to solve routine algebra problems.
He also related that he had asked his math teachers in high school why these rules worked, but he never got a satisfactory response. So he asked his college professor.
Looking back on it, I see that this was one of the incidents that sparked my interest in teacher education. As always, I never hold a grudge against a student for asking a question. Indeed, I respected my student for posing a really good question, and I was upset for him that he had not received a satisfactory answer to his question.
This is the second of two posts where I give two answers to this question from two different points of view.
Answer #2. This explanation relies on one of the laws of exponents:
For positive integers and , this can be proven by repeated multiplication:
repeated times and times
Ideally, and should be defined so that this rule still holds even if one (or both) of and is either zero or a negative integer. In particular, we should define so that the following rule holds:
In other words, the product of something with should be the original something. Clearly, the only way to make this work is if we define .
In the same way, we should define so that the following rule holds:
Dividing, we see that