A Square Root Calculation

Calculate \[ \sqrt{\phantom{XXXXXXXXXXXXX}} \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! {(\underbrace{111\ldots1}_{\mbox{$100$ $1$'s}})(1\underbrace{000\ldots0}_{\mbox{$99$ $0$'s}}5)+1} \,\,. \]

Solution: We can write the expression inside the square root as \[ \frac{1}{9}(10^{100}-1)(10^{100}+5)+1=\frac{(10^{100}+2)^2}{9}. \] Hence the answer is $(10^{100}+2)/3=\underbrace{3\ldots3}_{\mbox{$99$ $3$'s}}4$.

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