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. 



Notice that $0.\overline{14}+0.\overline{41}+0.\overline{15}+0.\overline{51}=1.\overline{2} = 1+\frac{2}{9}$. 
Now $\sqrt{11(1+\frac{2}{9})}=\sqrt{\frac{99}{9}+\frac{22}{9}}=\sqrt{\frac{121}{9}}=\frac{11}{3}=3\frac{2}{3}$
