<s>[INST] express the value of an expression as a common factor In the following problem, by adding $0.141414$..., $0.414141..., 0.151515...$, and $0.515151...$, I get $1.111....$ Then the expression becomes square root ($11 \times 1.1111$....). My answer is $11 \times \sqrt{0.1010101...}$, which is different from the answer sheet. Any help is highly appreciated.

What is the value of the following expression? Express your answer as a common fraction. 


 [/INST] I like @ncmathsadist's suggestion.  If you can reason that $0.\overline{14} + 0.\overline{41} = 0.\overline{5}$ and $0.\overline{15} + 0.\overline{51} = 0.\overline{6}$  Then think about it:
\begin{align}
&0.555 \\
+&0.666 \\\hline
&1.221
\end{align}
Clearly if it repeats forever, $0.\overline{5} + 0.\overline{6} = 1.\overline{2}$.  Then we can solve by doing the following:
$$
x = 1.\overline{2} \\
10x = 12.\overline{2}\\
10x - x = 9x \rightarrow  9x = 12 - 1 = 11
$$
Therefore $x = 1.\overline{2} = \frac{11}{9}$.  Then we have:
$$
\sqrt{11\cdot\frac{11}{9}} = \sqrt{\frac{11^2}{3^2}} = \frac{11}{3} = 3\frac{2}{3}
$$
</s>