I am trying to explain/prove to myself why a root of something is written as $$number^{\frac{1}{root}}$$

I wanted to prove it with the thought in my mind, that "the exponent just says how often the base has to be multiplied with itself". But how does this fit with a fraction in the exponent? Does it (for instance with a square root) mean, that it gets multiplied with itself "halfly" (I don't know how to put this)? I don't think so...

I have tried other things to prove it to me (by going backwards from an exponent of (3/2) to (1/2) and trying to find a pattern), but I just wanted to ask here.


Before I describe my "prove" I want to tell you this:

As with all in mathematics, nothing just happens and everything has a reason. Therefore I thought for example to prove that $$4^0 = 1$$ you can look on equations, find the pattern and realize that it really is consistent.

For instance the above mentioned prove:

4^3 = 64 <-- divide by 4 to get:

4^2 = 16 <-- divide by 4 to get:

4^1 = 4 <-- divide by 4 to get:

4^0 = 1

There is this pattern, and it has no reason to change! So it continues and it comes to the negativ exponent:

4^0 = 1 <-- divide by 4 to get:

4^-1 = 0.25 <-- divide by 4 to get:

4^-2 = 0.125

So lets look on the "prove" I was trying to make to find out whats the pattern with fraction in exponents:

2^(3/2) = 2.8284 = 7071/2500

2^(2/2) = 2 = 7071/3535.50

2^(1/2) = 1.4142 = 7071/5000

But whats the pattern here?

  • $\begingroup$ Are you aware that $a^n\cdot a^m=a^{n+m}$? Then if $a^{1/2}$ makes any sense at all, what do you expect $a^{1/2}\cdot a^{1/2}$ to be? -- But most importantly: Before you can prove anything about $a^{1/n}$, you need to define it. $\endgroup$ – Hagen von Eitzen May 10 '18 at 18:06
  • $\begingroup$ @Hagen It makes sense and I am aware that this equation is true, yes, but your equation doesn't solve my problem. I don't know how to put this... I really know that using a fraction as an exponent works, but it doesn't fit with the idea, that the exponent tells you how often a number as to be multiplied with itself. $\endgroup$ – watchme May 10 '18 at 18:10
  • $\begingroup$ In your last example, the pattern is that you're dividing by $\sqrt 2$. $\endgroup$ – Dando18 May 10 '18 at 18:51
  • $\begingroup$ $2500\times1.4142=3535.50$ and $3535.50\times1.4142=4999.90410\approx5000$. $\endgroup$ – Barry Cipra May 10 '18 at 20:09

You are quite right that exponential notation, when it's first introduced for positive integers, is defined as repeated multiplication. Something similar is done, in elementary school, with multiplication understood as repeated addition, e.g., $2\times3=3+3$ (or $2+2+2$). So your question is analogous to asking, How do you add $a$ to itself half a time to get ${1\over2}a$?

All that's happening is that the definition of $a^n$ as $a\times a\times\cdots\times a$ (with $n$ $a$'s and $n-1$ $\times$'s), which is pertinent for positive integers $n$, no longer applies to $a^r$ for $r\not\in\{1,2,3,\ldots\}$, in the same way that $ra=a+a+\cdots+a$ applies only if $r\in\{1,2,3,\ldots\}$.

  • $\begingroup$ Thank you @Barry! Well written answer :-) I thought I can find a pattern here (You can see my try to find a pattern in my edit). $\endgroup$ – watchme May 10 '18 at 18:54

We use this notation, because it is convenient and consistent with the rules of exponents, i.e.

$$ \left(a^b\right)^c = a^{b\cdot c} $$ so we can apply the same to roots

$$ a = \left(\sqrt{a}\right)^2 = \left(a^{1/2}\right)^2 = \left(a^{1/2\cdot 2}\right) = a^1 = a $$

and you can see for yourself how the same applies to $a^ba^c = a^{b+c}$.

Edit: In response to the question in the comment, the intuition of exponentiation being "multiplying a number with itself $n$ times" only works for exponents in $\mathbb N$.

So to extend to $\mathbb Q$, we have to redefine what exponentiation is. We often define it as follows: If $b,x \in \mathbb R^+$ and $n \in \mathbb N$, then $b^{\frac 1 n} $ is defined such that $\left(b^{\frac 1 n}\right)^n = b$.

We can extend further to $\mathbb R$: for $x\in \mathbb R$, $b\in \mathbb R^+$ we say $b^x = \lim_{r (\in\mathbb Q)\to x} b^r$.

These definitions are made to preserve the original properties of exponents, which you can check for yourself that they do indeed.

  • $\begingroup$ Thank you for your answer Dando! I am aware that all this works, but how does a fraction in the exponent fit with the idea, that the exponent tells you how often a number as to be multiplied with itself (as its often described in books). I know that it works, but I want to get deep down in maths, to master the basics in order to be able to explain all that to students (I have just finished my secondary technical college and now I help some students). Because some students ask all the time "why this why that", and I want to give an answer to all those questions. $\endgroup$ – watchme May 10 '18 at 18:15
  • $\begingroup$ @watchme It doesn't. Your definition of exponentiation is only defined for exponents in $\mathbb N$. When we extend to $\mathbb Q$ we need to redefine what an exponent is. $\endgroup$ – Dando18 May 10 '18 at 18:18
  • $\begingroup$ Yeah... I thought so... But what is that "new definition"? $\endgroup$ – watchme May 10 '18 at 18:21
  • $\begingroup$ @watchme see my edit. $\endgroup$ – Dando18 May 10 '18 at 18:29
  • $\begingroup$ thanks Dando! :-) Good Answer - If you want to you can also look on my edit I've made! $\endgroup$ – watchme May 10 '18 at 18:51

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