Why can the root of something be written as a fraction in the exponent?

Question

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.

EDIT:

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?

Answer 1

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 $r\in\{1,2,3,\ldots\}$.

Answer 2

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.