## Wednesday, August 1, 2012

### 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{$993$'s}}4$.