What is the square root of 154.

 is not rational.  That mean that there is no exact number x itself that will give you 154 because the answer cannot be written in rational form (a/b - a fraction)   In order to approximate it we can say that it is somewhere between 12 ( and 13 .
Since it is closer to 144 (only 10 more) than 169 (15 away), a good guess would be a little less than 12.5  If we guess 12.4 and do 154/12.4  - we get about 12.41
Now we find the average of 12.40 and 12.41 and try 12.405 as our next divisor.  This process goes on and on because you can never get the exact answer which is a decimal that goes on forever without repeating.
Using a calculator we can approximate the value.
= 12.409673645990856596133241955449  - but the answer goes on an on.

√154 = 12.409674

To see if it is right, multiply the answer by itself.

12.409674 × 12.409674 = 154.