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?

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).
