<s>[INST] What exactly IS a square root? It's come to my attention that I don't actually understand what a square root really is (the operation). The only way I know of to take square roots (or nth root, for that matter) it to know the answer! Obviously square root can be rewritten as $x^{1/2}$ , but how does one actually multiply something by itself half a time? 
How do calculators perform the operation?
 [/INST] I don't believe any computer actually does it this way, but you can compute a continued fraction to get an exact representation of a square root.  For example, since $10^2$ $<$ $111$ $<$ $11^2$:
\begin{eqnarray}
\sqrt{111}
&=& 10 + (\sqrt{111} - 10) \\
&=& 10 + \frac{11}{\sqrt{111} + 10} = 10 + \frac{1}{\frac{\sqrt{111} + 10}{11}} \\
\frac{\sqrt{111} + 10}{11} &=& 1 + \frac{\sqrt{111} - 1}{11} \\
&=& 1 + \frac{10}{\sqrt{111} + 1} = 1 + \frac{1}{\frac{\sqrt{111} + 1}{10}} \\
\frac{\sqrt{111} + 1}{10} &=& 1 + \frac{\sqrt{111} - 9}{10} \\
&=& 1 + \frac{3}{\sqrt{111} + 9} = 1 + \frac{1}{\frac{\sqrt{111} + 9}{3}} \\
\frac{\sqrt{111} + 9}{3} &=& 6 + \frac{1}{\frac{\sqrt{111} + 9}{10}} \\
\frac{\sqrt{111} + 9}{10} &=& 1 + \frac{1}{\frac{\sqrt{111} + 1}{11}} \\
\frac{\sqrt{111} + 1}{11} &=& 1 + \frac{1}{\sqrt{111} + 10} \\
\sqrt{111} + 10 &=& 20 + \frac{1}{\frac{\sqrt{111} + 10}{11}}
\end{eqnarray}
This yields the continued fraction $[10, \dot1, 1, 6, 1, 1, \dot{20}]$, and it looks like:
$$
\sqrt{111} = 10 + \cfrac{1}{1 + \cfrac{1}{1 + \cfrac{1}{6 + \cfrac{1}{1 + \cfrac{1}{1 + \cfrac{1}{20 + \cfrac{1}{...}}}}}}}
$$
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