the idea being, hmmm... you'd multiply the value by some value of 10 raised at some power, that way the repeating part, moves over to the left-side of the decimal..... so let's do so, in this case the repeating part is 52, so we need to move two fellows over, so, we'll use 100, 10 at the second power then.
[tex]\bf 0.5252\overline{52}\qquad \boxed{x=0.5252\overline{52}}
\\\\\\
\begin{array}{llll}
100\cdot x&=&52.5252\overline{52}\\
&&52+0.5252\overline{52}\\
&&52+\boxed{x}\\
100x&=&52+x
\end{array}
\\\\\\
100x-x=52\implies 99x=52\implies x=\cfrac{52}{99}[/tex]
