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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.

BWP
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  • it would depend on what the exponents were algebra helps but for example there's http://mathworld.wolfram.com/ComplexExponentiation.html or certain rules that only work with certain sets of numbers. also fractional exponents are roots. –  Aug 05 '17 at 23:25
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    There is an error in line 7. – hamam_Abdallah Aug 05 '17 at 23:27
  • I am aware that fractional exponents are roots, consequently I see no reason to regard exponents and roots as one and the same. Thus roots must have ties to basic exponents – BWP Aug 05 '17 at 23:27
  • @BWP yeah what number do you square ( raise to exponent 2) to get a number ? ... could it be perhaps the square root of that number ? if you allow exponents to multiply then the fractional powers are roots of other powers etc. –  Aug 05 '17 at 23:41
  • Caution! According to your "definition" $a^1$ is meaningless, to say nothing about $a^0$. The most effective definition for natural exponents is the recursive one: $a^0=1;;a^n=a\cdot a^{n-1};;n>0;;n\in\mathbb{N}$ – Raffaele Aug 06 '17 at 13:38

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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.

Ross Millikan
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