Today, I’m going to estimate the square root of 105. We first need to set up out barriers, the square root of 100 on the left, and the square root of 121 on the right. That leaves us with 10 on the left, and 11 on the right. We easily know that half of 10 and 11 is 10.5, and that 10.5 is a lot closer to 100 than it is to 121. So we can’t use 10.6-10.9, those numbers are too high. Our options are 10.1-10.4. Let’s find the middle number between 100 and 121.

121-100 is 21, take half of that and we got 10.5.

Add 100+10.5 and our total is 110.5.

Even though 100 is still closer, it is by a super tad bit, by 0.5. So lets eliminate 10.1 and 10.4, there not good enough, so it’s either 10.3 or 10.4. Even though 100 is closer to 105 than 121, but almost a tie with the half number, 110.5, lets go with 10.24. That sounds pretty close to the answer. The answer is 10.25, we were .1 off which is really good.