Find square root of 110.6 and 10080 without using a calculator.

Let N be the number you want to get the square root. Assume E1 as the initial estimate for `sqrtN` .

So the next estimate E2 will be;

`E2 = (N/(E1)+E1)/2`

Similarly E3 will be;

`E3 = (N/(E2)+E2)/2`

For 110.6;

We know that `sqrt100 = 10` and `sqrt121 = 11`

So let us estimate E1 = 10 for ease of calculation.

`E2 = (110.6/10+10)/2 = 10.53`

If you need further accuracy you can estimate E3 as well.

`E3 = (110.6/10.53+10.53)/2 = 10.517`

For 10080;

We know that `sqrt10000 = 100`

So let us say E1 = 100.

`E2 = (10080/100+100)/2 = 100.4`

Similarly;

`E3 = (10080/100.4+100.4)/2 = 100.399`

So the answers are;

`sqrt110.6 = 10.517`

`sqrt10080 = 100.399`