$\underset{99}{\underbrace{111....11}}0\underset{99}{\underbrace{88...8}}9=\underset{100}{\underbrace{111....11}}\underset{100}{\underbrace{000...000}}-\underset{100}{\underbrace{111....11}}=\underset{100}{\underbrace{111....11}}(1\underset{100}{\underbrace{00...0}}-1)=\underset{100}{\underbrace{111....11}}.\underset{100}{\underbrace{99...9}}=\underset{100}{\underbrace{111....11}}^2.9 \Rightarrow \sqrt{\underset{99}{\underbrace{111....11}}0\underset{99}{\underbrace{88...8}}9}=\sqrt{\underset{100}{\underbrace{111....11}}^2.9}=\underset{100}{\underbrace{33...3}}$
