Calculate $\sqrt{\phantom{XXXXXXXXXXXXX}} \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! {(\underbrace{111\ldots1}_{\mbox{100 1s}})(1\underbrace{000\ldots0}_{\mbox{99 0s}}5)+1} \,\,.$ 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 $\frac{10^{100}+2}{3}=\underbrace{3\ldots3}_{\mbox{99 3s}}4.$