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Whenever I ask what is the meaning of fractional and negative exponents, the answer is always that they are defined with these two properties. $$\ a^{x+y} = a^x a^y $$ $$\ a^{xy} = (a^x)^y $$

But why are these Properties (That positive exponents have) used to define negative and fractional exponents ???

I am well aware that these 2 equations are used to extend the definition of exponents to real numbers, but my question is why ?? I mean why did they choose these equations ?, just because they apply to positive numbers, or there is another reason . I mean they could have just gave new definition to fractional and negative exponents ?

Djebbar Abderrahmene
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  • @lulu yes,I will edit the question thank you, – Djebbar Abderrahmene Feb 11 '23 at 20:20
  • I don't get your question, $a^{x+y}=a^x a^y$ is not a definition. The usual definition is $a^x=e^{x\log a}$ for any $x\in\mathbb{R}$ and $a>0$ and, using this definition, we have the equality $a^{x+y}=a^x a^y$. – Tuvasbien Feb 11 '23 at 20:22
  • @Tuvasbien I didn't say it is a definition, I said that they use this property to define or calculate negative and fractional exponenets – Djebbar Abderrahmene Feb 11 '23 at 20:34
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    If you want to extend the positive exponents to real exponents, you want something that behaves the samen and the equality $a^{x+y}=a^xa^y$ is one (if not the) property that is important when using exponents. It would be odd to have exponents that don't satisfy this equality (though it exists with complex roots) – Tuvasbien Feb 11 '23 at 20:37
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    I'd like to direct you to my answer here. The first half doesn't answer your question, but the second half seems on point for what you want to know. – MJD Feb 12 '23 at 19:33
  • This boils down to "why we define integers using subtraction on natural numbers and why define rationals via division of integers". – Paramanand Singh Feb 13 '23 at 09:06
  • Extending the definitions so that the familiar useful laws would continue to hold must have seemed like a good idea at the time. And they got lucky and it worked out. Good thing you weren't there to throw a wet blanket on the idea. By the way, what alternative definition did you have in mind? – user14111 Feb 13 '23 at 09:13
  • Not sure what is your confusion about. The above rules apply to negative and positive: $2^{2-3}=2^{-1}:and:2^2:2^{-3}=2^{-1}:and:\left(2^2\right)^{-3}=4^{-3}=2^{-6}$ – NoChance Feb 13 '23 at 11:42

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When you start defining powers, they are just shorthands for repeated multiplication, i.e. $$a^n=\underbrace{a\cdot a\dotsm a}_{n \text{ times}}\,.$$ Clearly, this only works for $n$ being a positive integer. (On the other hand, $a$ can be any number.)

From this definition, you can derive two rules, $$ a^{n+m}=\underbrace{a\cdot a\dotsm a}_{n+m \text{ times}}=\underbrace{a\cdot a\dotsm a}_{n \text{ times}}\cdot \underbrace{a\cdot a\dotsm a}_{m \text{ times}}=a^n\cdot a^m \tag{*}$$ and $$\left(a^n\right)^m=\underbrace{\underbrace{\left(a\cdot a\dotsm a\right)}_{n \text{ times}}\dotsm \underbrace{\left(a\cdot a\dotsm a\right)}_{n \text{ times}}}_{m \text{ times}}=\underbrace{a\cdot a\dotsm a}_{n\times m \text{ times}}=a^{nm}\,.\tag{#}$$ At first, these are still only defined for $n$ and $m$ positive integers, but they seem pretty basic rules for taking powers.

Now, you can try to extend this definition of powers to other exponents. In some sense, that means that you do not think of powers just as shorthand notation for repeated multiplication, but as a mathematical operation in its own right. You can denote this as a function$$f_a(n): n\mapsto a^n\,,$$which you have already defined for positive integer $n$. Of course, you want to do this in such a way that the integer powers still work as before and that the rules carry over as well as possible. Then, Equations (*) and (#) form the natural "functional equations" to define $f_a$ for other arguments. (In other words: It's a choice, but it's the obvious one.)

For example, for zero and negative integers, Equation (*) you can deduce that $a^0=1$ and $a^{-n}=1/a^n$ (do you see how?). For fractions, you can the use (#).

Note that

  • you pay a price in that now, $a$ is more restricted -- you cannot take a real root of a negative number, for example, and neither can you define a continuous square root function $\mathbb{C}\to\mathbb{C}$,
  • there is extra work to do to go to real (as opposed to rational) $n$.

Anyway, upshot: Equations (*) and (#) are rather natural rules, and help generalise powers to rational exponents.

Toffomat
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  • I am well aware that equation (*) and (#) are used to extend the definition of exponents to real numbers, but my question was why ?? I mean why did they choose these two equations ?, just because they apply to positive numbers, or there is another reason . I mean they could have just gave new definition to fractional and negative exponents ? – Djebbar Abderrahmene Feb 12 '23 at 18:49
  • thank you btw, I forgot about equation (#) – Djebbar Abderrahmene Feb 12 '23 at 19:04
  • @DjebbarAbderrahmene I have added a sentence about that – Toffomat Feb 13 '23 at 09:03