What is $x^{1/2}$? I was wondering what $x^{1/2}$ is. You know, when I say $x^2$ it's $x \cdot x$ or $x^3 = x \cdot x \cdot x$ etc.
But what is $x ^{1/2}$? 
I know it's $$\sqrt x$$ but I mean when you want to explain it like with $$x^2 = x \cdot x$$
How do you explain it?
 A: You probably know the rule $x^n\cdot x^m=x^{n+m}$ which can simply be motivated for natural numbers $n$ and $m$ from the intuitive undertstanding via
$$x^n\cdot x^m=\underbrace{x\cdots x}_{\text{$n$ times}}\cdot \underbrace{x\cdots x}_{\text{$m$ times}}=\underbrace{x\cdots x}_{\text{$n+m$ times}}=x^{n+m}.$$
Now you introduce $x^{1/2}$ for what reason whatsoever and ask yourself what value this should give. One sensible way could be to do it in such a way so that $x^{1/2}\cdot x^{1/2}=x^{1/2+1/2}=x^1=x$. This means we try to keep the rule from above. In other words: we want
$$(x^{1/2})^2=x,\qquad\text{and $x^{1/2}=\sqrt{x}$ is one reasonable way to do it}.$$
I cannot think of another sensible way to define the meaning of $x^{1/2}$.
A: We define $x^{\frac12}$ to be equal to $\sqrt x$ because we like the property $$\forall x\in\mathbb R:\forall a,b\in\mathbb N: (x^a)^b = x^{a\cdot b}$$ and we want this equality to hold even if $a$ and $b$ are not integers.
So, whaterver $y=x^\frac12$ is equal to, we want $y^2$ to be equal to $x$, and that means that $y$ must be $\pm \sqrt x$. The fact that there are two possibilities already presents a problem, and in fact the rule cannot be extended as-is, but rather in the slightly changed form
$$\forall x>0:\forall a,b\in\mathbb Q: (x^a)^b=x^{a\cdot b}$$
(notice that it's only true for $x>0$ from now on).
We choose the positive one because if we choose the negative, we wouldn't be able to apply the "$^\frac12$" twice in a row.
A: I was once asked this question by a Chinese high school student who was quite bright, but couldn't speak English worth beans.
He wrote the following on a piece of paper and looked at me questioningly:
$$2^{1/2}=?$$
Here is how I answered him.
(I wrote each of the following on a sheet of paper one after another and pushed the paper at him, and he wrote the answer to the question mark.)  (And actually I used dots for multiplication rather than $\times$, but whatever.)
$$5\times5\times5=5^?$$
$$7\times7\times7\times7\times7=7^?$$
$$(5\times5)\times(5\times5\times5)=5^?$$
$$5^2\times5^3=5^?$$
$$7^4\times7^5=7^?$$
$$37^4\times37^{13}=37^?$$
$$17^5\times17^5\times17^5=17^?$$
$$(17^5)^3=17^?$$
$$(17^3)^5=17^?$$
$$(1357^{17})^{10}=1357^?$$
$$(x^2)^{1/2}=x^?$$
At this point he grabbed the paper from me and wrote:
$$2^{1/2}=\sqrt 2$$
(Note: it was implied by our written dialogue that $x$ represented some counting number.  Not a negative number!  When you allow negative numbers to get involved with fractional exponents, things get more interesting—and also very beautiful—but that's a different lesson.  :)  )
A: In the same way you explain $\frac{1}{2}$ formally: it's a solution to $y + y = 1$. (This formal stipulation works even when you don't have a sensible way to cut your representations of whole numbers in half.)
In your case, $x^{\frac{1}{2}}$ describes the solutions to $y \times y = x$.
A: It's a positive number (remember that square roots are "always" positive, and are defined for positive numbers if you want real solution) y such that
y*y = x ;
