Q:

general rule for x raised to any exponent y

This is a pretty basic question and I bet that when it is answered it will strike me as intuitive and obvious. Nevertheless, this is still a question that has bugged me for quite a while. Basic exponents (where the exponent is a whole number) work as such: $x^a = \underbrace{x \cdot x \cdot x \ldots x}_a$. Prealgebra tells us that $x^{1/a} = b \text{ where } b^a = x$ and that $x^{a/b} = (x^{1/b})^a$ and that $x^{-a} = \frac{1}{x^a}$. However, I fail to find much common ground between the way basic exponents work and the ways those of prealgebra work. I realize the logic behind negative exponents: $x^{a-1}=\frac{x^a}{x}$ yet I fail to see a corollary between the basic exponents and the fractional ones.  

A:

Take the expression you are happy with, $x^a = \underbrace{x \cdot x \cdot x \ldots x}_a$ and think about what $x^{1/a}$ might mean.  Let's call it $y$ and note that $y^a = \underbrace{y \cdot y \cdot y \ldots y}_a=\underbrace{x^{1/a} \cdot x^{1/a} \cdot x^{1/a} \ldots x^{1/a}}_a=x^1$ where the last comes from the exponent law that a product of terms is the same as adding the exponents.  This gives us a good way to define fractional powers and shows the connection to roots.  

