Explaining something to the half I'm a private tutor in my free time, teaching some basic high school mathematics and I've often been asked: ''Why is something to the half equal to the root of that something?''.
And I'm having problems explaining it. I have an idea of why in my head but obviously this idea is not strong enough, as I can't explain it properly. Can anyone lay it out?
 A: You could try explaining it this way:
Multiplying is adding of powers:
$x^\frac{1}{2}\times x^\frac{1}{2} = x^1 = x$
$\sqrt x\times \sqrt x= x$
A: We have for $x\in\Bbb{R}_{>0}$  the functional equation $x^ax^b =x^{a+b}$, so $x^{\frac{1}{2}}x^{\frac{1}{2}}=x^{\left(\frac{1}{2}+\frac{1}{2}\right)}=x^{1}$. Since finding a square root of $x$ is equivalent to finding an $y\in\Bbb{R}$ with $y\cdot y=x$, we can conclude $\sqrt{x}=x^{\frac{1}{2}}$ (for the standard branch of the root and the $\exp$-function).
A: Well if algebra does not work for you, then here is a weird approach. Give them a stick of length (say $9 (=3^2)m$) and measurement tape. And assuming your student know the meaning of $a^2$, and ask them if length of the stick is equal to some number $a^2$, by using the measurement tape find $a$. The change the length and rerun this experiment with a different stick length. 
After all this, tell them $a$ is the root of stick length as squaring the root will give me the length. I know it is a funny experiment but this is something they can feel about!!
