Quoted By:
>What are we doing when we take the square root of something
Solving the equation $a=x^2$
>What does it mean to raise a number to a power of $\frac{1}{2}?
That's simply a useful generalization of power to rational numbers. AFAIK (that may be false) that's not a "law", but rather a useful decision.
> How do we do square roots without calculators and with accuracy?
Use Newton's method:
Let $f(x)$ be $a-x^2$. Then $f'(x)=-2x$.
Take a number $x_0$, say $1$.
On each step, $x_{n+1}=x_n-\frac{f(x)}{f'(x)}$. Repeat to desired precision, there should be precision formula on Wikipedia.
